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Special Features and Supplements Geometric Formulas TWO DIMENSIONAL Square: Length of one side: s Perimeter: P 5 4s Area: A 5 s2 Rectangle: Length: l; width: w Perimeter: P 5 2l 1 2w Area: A 5 lw Triangles: Lengths of sides: a, b, c; base: b; height: h Perimeter: P 5 a 1 b 1 c Area: A 5 (1/2)bh Pythagorean Theorem (for right triangles only, c is the hypotenuse): a2 1 b2 5 c2 Circle: Radius: r Diameter: d 5 2r Circumference: C 5 2r 5 d Area: A 5 r 2 THREE DIMENSIONAL Rectangular Solids: Length: l; width: w; height: h Volume: V 5 lwh Surface area: S 5 2lw 1 2lh 1 2wh Sphere: Radius: r Volume: V 5 (4/3)r3 Surface area: S 5 4r2 Right Circular Cylinder: Radius of base: r; height: h Volume: V 5 r2h Surface area: S 5 2rh 1 2r2 Right Circular Cone: Radius of base: r; height: h; slant length: l Volume: V 5 (1/3)r2h Surface area: S 5 rl 1 r
Unit Conversion Relationships
An instructor is free to choose among a number of special features. The Instructor’s Teaching and Solutions Manual provides extensive teaching ideas and support for using these features. The manual also contains solutions to the even-numbered problems.1
1 in 5 2.54 cm 5 0.0254 m 1 ft 5 30.5 cm 5 0.305 m 1 yd 5 91.4 cm 5 0.914 m 1 mi 5 1609 m 5 1.609 km
Exploring Mathematical Ideas and Skill Building NEW! Explore & Extend are short explorations that provide students and instructors
1 qt 5 946 ml 5 0.9461 1 oz 5 28,350 mg 5 28.35 g
with ideas for going deeper into topics or previewing new concepts. They can be found in almost every section.
1 lb 5 453.6 g 5 0.4536 kg
Explorations are extensive problem-solving situations at the end of each chapter that
1 tn 5 907 kg 5 0.907 metric tn
can be used for small group or individual projects. Algebra Aerobics are collections of skill-building practice problems found in each
12
1 light year 5 9.46 ? 10 km or 5.88 trillion miles
section. All of the answers are in the back of the text. Check Your Understanding is a set of mostly true/false questions at the end of each
Metric Prefixes for Powers of 10 attofemtopiconanomicromillicentidecidekahectokilomegagigaterapetaexa-
a f p n m m c d (unit) da h k M G T P E
10218 10215 10212 1029 1026 1023 1022 1021 100 101 102 103 106 109 1012 1015 1018
Thus millimeter is abbreviated mm; kilometer km; and megameter Mm. Abbreviations are usually not used for deci, deka, and hecto.
chapter (answers are provided in the back of the text) that offer students a chance to assess their understanding of that chapter’s mathematical ideas. Chapter Review: Putting It All Together contains problems that apply all of the
basic concepts in the chapter. The answers to the odd-numbered problems are in the back of the text. 60-Second Summaries are short writing assignments found in the exercises and the
Explore & Extend problems that ask students to succinctly summarize their findings. Readings
are related to topics covered in the text and are available at www.wiley.com/college/kimeclark and at www.wileyplus.com.
Using Technology Technology is not required to teach this course. However, we provide the following online resources on the course website www.wiley.com/college/kimeclark and at www.wileyplus.com. Interactive Course Software
provides illustrations of the properties of each function, simulations of concepts, and practice in skill building. Their modules can be used in the classroom or computer lab, or downloaded for student use. DA T A
Excel and TI Connect™ Graph Link Files contain all the major data sets used in
the text and are available in Excel or TI Connect™ program formats. Graphing Calculator Manual is coordinated with the chapters in the text and offers
step-by-step instructions for using the TI-83/TI-84 family of calculators. NEWLY UPDATED! WileyPLUS is an online course management and assessment
system that provides resources for student learning, including short instructional videos. See Wiley’s description at the end of the Preface.
1
The manual is available for free to adopters online at www.wiley.com/college/kimeclark or at www.wileyplus.com. You can also contact your local Wiley sales representative to obtain a printed version of the manual.
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F I F T H
E D I T I O N
EXPLORATIONS IN COLLEGE ALGEBRA LINDA ALMGREN KIME JUDITH CLARK University of Massachusetts, Boston, Retired
B E V E R LY K . M I C H A E L University of Pittsburgh in collaboration with Norma M. Agras Miami Dade College Meg Hickey Massachusetts College of Art Sarah Hoffman University of Arizona John A. Lutts University of Massachusetts, Boston Peg Kem McPartland Golden Gate University, Retired Software developed by Hubert Hohn Massachusetts College of Art Funded by a National Science Foundation Grant
JOHN WILEY & SONS, INC.
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Vice President and Publisher
Laurie Rosatone
Acquisitions Editor
Joanna Dingle
Project Editor
Ellen Keohane
Editorial Program Assistant
Beth Pearson
Production Services Manager
Dorothy Sinclair
Senior Production Editor
Janet Foxman
Marketing Manager
Jonathan Cottrell
Creative Director
Harry Nolan
Photo Manager
Hilary Newman
Senior Designer
Madelyn Lesure
Media Editor
Melissa Edwards
Media Assistant
Laura Abrams
Production Services
Ingrao Associates
Cover and chapter opener photo composite: (underwater scene in the tropics) © Gray Hardel/Corbis; (divers) Comstock/Getty Images, Inc. Explorations photo feature: Stephen Frink/Stone/Getty Images, Inc. This book was set in Times 10/12 by Aptara® Inc., and printed and bound by Courier/Westford. The cover was printed by Courier/Westford. This book is printed on acid-free paper. ` Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. Copyright © 2011, 2008, 2005, 2001 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, website www.wiley.com/go/permissions. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. Library of Congress Cataloging in Publication Data: Kime, Linda Almgren. Explorations in College Algebra/Linda Almgren Kime, Judy Clark, Beverly K. Michael. – 5th ed. p. cm. Includes index. ISBN 978-0-470-46644-5 (pbk.) 1. Algebra—Textbooks. I. Kime, Linda Almgren. II. Clark, Judy. III. Michael, Beverly K. IV. Title. QA152.3.K56 2011 512.9—dc22 2010045241 Main Book ISBN 978-0-470-46644-5 Binder-Ready Version ISBN 978-0-470-91761-9 Printed in the United States of America 10
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To our students, who inspired us.
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Letter from a Student
My name is Lexi Fo urnier and semester I am enro I am a fre ll shman he e d in the Ap in Colleg re at Pitt. plied Alg e Algebra This ebra cour .” Before courses v s e c u o sing “Exp ming to P arying fro lorations itt, I had ta m algebra frustratio ken nume to calculu n, stress, r o u s, all of w s math and a dete told that I hich prod station fo u was requir c e d r m ath as a su ed to take law and c bject. Wh a math co reative w en I was u rse here, I riting majo me by inf was livid. r; why do orming m I am a pre I need ma e of this n math/scie th? My ad ew math nce majo v is e c r calmed la ss aimed rs the bas at teachin ic skills th At first I g nonwas skep ey will ne ti e cal, but I’ recomme d in every m writing nd this co day life. urse. Wh to you no been reali at I have w to emp stic math le hatically a r n e d s k thus far in ills presen confidenc th te is d e and mo in a “left course ha tivation. F brain” me ve relatable. or once in thod that The conc m f o y s ters epts are c career as amorphou lear and r a student, s topics a e m a li a s th ti d c is dressed in (as oppos to this cla ed to the my earlie ss. I enjoy a r b m s tr a d act, th oing my h classes). the lesson omework I look for s are appli and proje ward cable to m empower cts becau y life and ed by my s e I f m e e y understan l that future and This cour ding. because I se is a vit f e e l al additio my view n to the m on the su ath depar bject and call “ever tment. It stimulate yday math has altere d a n appreciati .” d It is my b o n f or what I elief that li k e to many stu and helpf dents wil ul as I ha l find the ve. Thank class as e you for y ncouragin our attenti g o n . Sincerely , Lexi Fou rnier Student, U niversity of Pittsbu rgh
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P R E FA C E This text was born from a desire to reshape the college algebra course, to make it relevant and accessible to all of our students. Our goal is to shift the focus from learning a set of discrete mechanical rules to exploring how algebra is used in the social and physical sciences, and in the world around you. By connecting mathematics to reallife situations, we hope students come to appreciate its power and beauty.
Guiding Principles The following principles guided our work. • Develop mathematical concepts using real-world data. • Pose a wide variety of problems designed to promote mathematical reasoning in different contexts. • Make connections among the multiple representations of functions. • Emphasize communication skills, both written and oral. • Facilitate the use of technology. • Provide sufficient practice in skill building to enhance problem solving.
Evolution of “Explorations in College Algebra” The fifth edition of Explorations is the result of an 18-year long process. Funding by the National Science Foundation enabled us to develop and publish the first edition, and to work collaboratively with a nationwide consortium of schools. Faculty from selected schools continued to work with us on the second, third, fourth, and now the fifth editions. During each stage of revision we solicited extensive feedback from our colleagues, reviewers and students. Throughout the text, families of functions are used to model real-world phenomena. After an introductory chapter on data and functions, we first focus on linear and exponential functions, since these are the two most commonly used mathematical models. We then discuss logarithmic, power, quadratic, polynomial and rational functions. Finally we look at ways to extend and combine all these functions to create new functions and apply them in more complex situations. The text adopts a problem-solving approach, where examples and exercises lie on a continuum from open-ended, non-routine questions to problems on algebraic skills. The materials are designed for flexibility of use and offer multiple options for a wide range of skill levels and departmental needs. The text is currently used in a variety of instructional settings including small classes, laboratory settings and large lectures, and in both two- and four-year institutions.
v
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Special Features and Supplements An instructor is free to choose among a number of special features. The Instructor’s Teaching and Solutions Manual provides extensive teaching ideas and support for using these features. The manual also contains solutions to the even-numbered problems.1
Exploring Mathematical Ideas and Skill Building NEW! Explore & Extend are short explorations that provide students and instructors with ideas for going deeper into topics or previewing new concepts. They can be found in almost every section. Explorations are extensive problem-solving situations at the end of each chapter that can be used for small group or individual projects. Algebra Aerobics are collections of skill-building practice problems found in each section. All of the answers are in the back of the text. Check Your Understanding is a set of mostly true/false questions at the end of each chapter (answers are provided in the back of the text) that offer students a chance to assess their understanding of that chapter’s mathematical ideas. Chapter Review: Putting It All Together contains problems that apply all of the
basic concepts in the chapter. The answers to the odd-numbered problems are in the back of the text. 60-Second Summaries are short writing assignments found in the exercises and the
Explore & Extend problems that ask students to succinctly summarize their findings. are related to topics covered in the text and are available at www.wiley.com/college/kimeclark and at www.wileyplus.com. Readings
Using Technology Technology is not required to teach this course. However, we provide the following online resources on the course website www.wiley.com/college/kimeclark and at www.wileyplus.com. Interactive Course Software provides illustrations of the properties of each
function, simulations of concepts, and practice in skill building. Their modules can be used in the classroom or computer lab, or downloaded for student use. DA T A
Excel and TI Connect™ Graph Link Files contain all the major data sets used in
the text and are available in Excel or TI Connect™ program formats. Graphing Calculator Manual is coordinated with the chapters in the text and offers
step-by-step instructions for using the TI-83/TI-84 family of calculators. NEWLY UPDATED! WileyPLUS is an online course management and assessment system that provides resources for student learning, including short instructional videos. See Wiley’s description at the end of the Preface.
1
The manual is available for free to adopters online at www.wiley.com/college/kimeclark or at www.wileyplus.com. You can also contact your local Wiley sales representative to obtain a printed version of the manual.
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The Fifth Edition Overall Changes Extensive faculty reviews guided our work on the fifth edition. • The new Explore & Extend problems are short explorations imbedded within almost every section. • The new Chapter 9: Creating New Functions from Old shows how to transform and combine functions to form new functions. It covers polynomials, rational functions, composition and inverse functions, and also provides more complex explorations. • Chapter 8 now covers quadratics and the mathematics of motion. • Data sets were created or updated throughout the text. • Revisions were made to many chapters for greater clarity. • Many new problems and exercises were created, ranging from basic algebraic manipulations to real-world applications. • Extended Explorations were integrated into chapters. The two Extended Explorations in the fourth edition have become Sections in Chapters 2 and 8.
Detailed Changes CHAPTER 1: An Introduction to Data and Functions has more emphasis on the concepts of input and output, and a different presentation of domain and range. These topics are now more fully integrated throughout text. CHAPTER 2: Rates of Change and Linear Functions now includes a discussion of the FAM1000 data set, regression lines and correlation coefficients (from 4th edition Extended Exploration: Looking for Links between Education and Earnings). CHAPTER 3: When Lines Meet: Linear Systems now starts with graphs of nonlinear systems to introduce intersection points in a real context and includes more real life examples of piecewise linear systems. CHAPTER 4: The Laws of Exponents and Logarithms: Measuring the Universe merges two sections to generalize the properties of exponents sooner.
CHAPTER 5: Growth and Decay: An Introduction to Exponential Functions has earlier and more extensive coverage of doubling times and half-lives and now introduces e through continuous compounding. CHAPTER 6: Logarithmic Links: Logarithmic and Exponential Functions The old section here on e and continuous compounding was moved to the end of Chapter 5. CHAPTER 8: Quadratics and the Mathematics of Motion is dedicated to quadratic functions and now includes a discussion of freely falling bodies (from 4th edition Extended Exploration: The Mathematics of Motion).
CHAPTER 9: New Functions from Old discusses ways of combining and transforming all the functions we studied and contains sections on polynomial, rational and inverse functions, and composition of functions. It concludes with a more complex collection of Explore & Extend type problems.
Acknowledgments We wish to express our appreciation to all those who helped and supported us during this extensive collaborative endeavor. We are grateful for the support of the National Science Foundation, whose funding made this project possible, and for the generous help of our program officers then, Elizabeth Teles and Marjorie Enneking. Our original Advisory Board, especially Deborah Hughes-Hallett and Philip Morrison, and our original editor, Ruth Baruth provided invaluable advice and encouragement.
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Over the last 18 years, through seven versions (including a rough draft and preliminary and 1st through 5th editions), we worked with more faculty, students, teaching assistants, staff, and administrators than we can possibly list here. We are deeply grateful for supportive colleagues at our own universities. The generous support we received from Theresa Mortimer, Patricia Davidson, Mark Pawlak, Maura Mast, Dick Cluster, Anthony Beckwith, Bob Seeley, Randy Albelda, Art MacEwan, Rachel Skvirsky, and Brian Butler, among many others, helped to make this a successful project. We are deeply indebted to Jennifer Blue, Celeste Hernandez, Paul Lorczak, Georgia Mederer, Ann Ostberg, and Sandra Zirkes for their dedicated search for mathematical errors in the text and solutions, and finding (we hope) all of them. A text designed around the application of real-world data would have been impossible without the time-consuming and exacting research done by Patrick Jarrett and Jie Chen. Edmond Tomastik, George Colpitts and Karl Schaffer were gracious enough to let us adapt some of their real-world examples in the text. One of the joys of this project has been working with so many dedicated faculty who are searching for new ways to reach out to students. These faculty, their teaching assistants and students all offered incredible support, encouragement, and a wealth of helpful suggestions. In particular, our heartfelt thanks goes to members of our original consortium: Sandi Athanassiou and all the wonderful teaching assistants at the University of Missouri, Columbia; Natalie Leone, University of Pittsburgh; Peggy Tibbs and John Watson, Arkansas Technical University; Josie Hamer, Robert Hoburg, and Bruce King, all past and present faculty at Western Connecticut State University; Judy Stubblefield, Garden City Community College; Lida McDowell, Jan Davis, and Jeff Stuart, University of Southern Mississippi; Ann Steen, Santa Fe Community College; Leah Griffith, Rio Hondo College; Mark Mills, Central College; Tina Bond, Pensacola Junior College; and Curtis Card, Black Hills State University. The following reviewers’ thoughtful comments helped shape the fifth edition: Wendy Ahrendsen, South Dakota State University; Shemsi Alhaddad, University of South Carolina, Lancaster; Mathai Augustine, Cleveland State Community College; Said Bagherieh, Georgia Perimeter College, Dunwoody Campus; Teri Barnes, McLennan Community College; Nicoleta Virginia Bila, Fayetteville State University; Steven Brownstein, University of Arizona; Linda Buckwalter, Harrisburg Area Community College; Elizabeth Burns, Bowling Green State University; Rose Cavin, Chipola Junior College; Daniel P. Fahringer, Harrisburg Area Community College; Kenneth J. Frerichs, Columbus State University; Mark H. Goadrich, Centenary College of Louisiana; Linda Green, Santa Fe College; Lorraine Gregory, Lake Superior State University; Johanna Halsey, Dutchess Community College; Donald Harden, Georgia State University; Erick Hofacker, University of Wisconsin, River Falls; Heather Holley, Santa Fe College; Michael J. Johnson, Meredith College; Vicky Klima, Appalachian State University; Naomi Landau, Pima Community College, Downtown Campus; Xuhui Li, California State University, Long Beach; Gretchen H. Lynn, West Virginia Wesleyan College; Samuel Ofori, Cleveland State Community College; Mary Pearce, Wake Technical Community College; Timothy A. Redl, University of HoustonDowntown; Randy Scott, Santiago Canyon College; Niandong Shi, East Stroudsburg University; Brian A. Snyder, Lake Superior State University; Gilfred B. Swartz, Monmouth University; David E. Thomas, Centenary College; Sherri Wilson, Fort Lewis College; Christopher Yarrish, Harrisburg Area Community College; and Changyong Zhong, Georgia State University. We are indebted to Laurie Rosatone at Wiley, whose gracious oversight helped to keep this project on track. Particular thanks goes to our new editors at Wiley, Joanna Dingle and especially to Ellen Keohane who, with the able editorial project assistant Beth Pearson, kept us on schedule. It has been a great pleasure, both professionally and personally, to work with Maddy Lesure on her creative cover designs and layouts of the text through multiple editions. Kudos to Sandra Dumas, Janet Foxman, Dorothy Sinclair, and Suzanne Ingrao in production for all their help in getting the
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text out. The accompanying media for Explorations would never have been produced without the experienced help from Melissa Edwards and Laura Abrams. Over the years many others at Wiley have been extraordinarily helpful in dealing with the myriad of endless details in producing a mathematics textbook. Our thanks goes to all of them. Our families couldn’t help but become caught up in this time-consuming endeavor. Judy’s husband, Gerry, became our consortium lawyer, and her daughters, Rachel and Caroline were there when needed for support and to mail packages. Kristin and her husband John provided editorial help and more importantly produced two grandchildren: Nola, who asks “why” and Cordelia, who sings numbers. Beverly’s husband, Dan, was patient and understanding about the amount of time this edition took. He tolerated Beverly working on Sundays and delaying their trip to Ireland. Beverly felt that without the support and encouragement from Dan, daughters Bridget and Megan, and new son-in-law Felipe Palamo, she couldn’t have made it through this edition. Linda’s husband, Milford, and her son Kristian provided invaluable scientific and, more importantly, emotional resources. Kristian and his wife Amy Mertl have just produced a marvelous grandchild, Evy, whom Linda believes already shows signs of mathematical acuity. All our family members ran errands, cooked meals, listened to our concerns, and gave us the time and space to work on the text. We offer our love and thanks to them. Finally, we wish to thank all of our students. It is for them that this book was written. Judy, Bev, and Linda P.S. We’ve tried hard to write an error-free text, but we know that’s impossible. You can alert us to any errors by sending an email to
[email protected]. Be sure to reference Explorations in College Algebra. We would very much appreciate your input. Since this text is a collaboration between authors and instructors, we encourage instructors to send new ideas and examples for the new Explore & Extend feature for possible future use. We’ll put the best ideas on our website. And last, but not least, we especially want to thank Dr. John Saber from Central Lake College for his kind and encouraging email: “. . . just wanted to thank you for writing this truly wonderful text.” It made our day.
WileyPLUS WileyPLUS is an innovative, research-based, online environment for effective teaching and learning.
What Do Students Receive with WileyPLUS? A Research-Based Design. WileyPLUS provides an online environment that integrates relevant resources, including the entire digital textbook, in an easy-to-navigate framework that helps students study more effectively. • WileyPLUS adds structure by organizing textbook content into smaller, more manageable “chunks.” • Related media, examples, and sample practice items reinforce the learning objectives. • Innovative features such as calendars, visual progress tracking and self-evaluation tools improve time management and strengthen areas of weakness. One-on-One Engagement. With WileyPLUS for Explorations in College Algebra, Fifth Edition students receive 24/7 access to resources that promote positive learning
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PREFACE
outcomes. Students engage with related examples (in various media) and sample practice items, including: • • • • •
Software Videos Readings Excel and TI-83/TI-84 Graph Link Files Guided Online (GO) Tutorial problems
Measurable Outcomes. Throughout each study session, students can assess their progress and gain immediate feedback. WileyPLUS provides precise reporting of strengths and weaknesses, as well as individualized quizzes, so that students are confident they are spending their time on the right things. With WileyPLUS, students always know the exact outcome of their efforts.
What Do Instructors Receive with WileyPLUS? WileyPLUS provides reliable, customizable resources that reinforce course goals inside and outside of the classroom as well as visibility into individual student progress. Precreated materials and activities help instructors optimize their time: Customizable Course Plan: WileyPLUS comes with a pre-created Course Plan designed by a subject matter expert uniquely for this course. Simple drag-and-drop tools make it easy to assign the course plan as-is or modify it to reflect your course syllabus. Pre-Created Activity Types Include: • Questions • Readings and Resources • Presentation • Print Tests • Concept Mastery Course Materials and Assessment Content: • PowerPoint Slides • Instructor’s Solutions and Teaching Manual • Readings • Question Assignments: selected end-of-section and Chapter Review problems coded algorithmically with hints, links to text, whiteboard/show work feature and instructor controlled problem solving help. • Computerized Test Bank • Printable Test Bank Gradebook: WileyPLUS provides instant access to reports on trends in class performance, student use of course materials, and progress toward learning objectives, helping inform decisions and drive classroom discussions. WileyPLUS. Learn More. www.wileyplus.com. Powered by proven technology and built on a foundation of cognitive research, WileyPLUS has enriched the education of millions of students, in over 20 countries around the world.
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accessible, affordable, active learning
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TA B L E
O F
C O N T E N T S
CHAPTER 1
AN INTRODUCTION TO DATA AND FUNCTIONS
1.1 Describing Single-Variable Data
2
Visualizing Single-Variable Data 2 Numerical Descriptors: What Is “Average” Anyway?
4
1.2 Describing Relationships between Two Variables Visualizing Two-Variable Data 10 Constructing a “60-Second Summary” Using Equations to Describe Change
11 13
1.3 An Introduction to Functions
19
What Is a Function? 19 Representing Functions: Words, Tables, Graphs, and Equations Input and Output: Independent and Dependent Variables 21 When Is a Relationship Not a Function? 21
1.4 The Language of Functions
28
36
Is There a Maximum or Minimum Value? 36 When is the Output of the Function Positive, Negative, or Zero? Is the Function Increasing or Decreasing? 37 Is the Graph Concave Up or Concave Down? 38 Getting the Big Idea 39 CHAPTER SUMMARY
20
26
Function Notation 26 Finding Output Values: Evaluating a Function 27 Finding Input Values: Solving Equations 27 Finding Input and Output Values from Tables and Graphs Rewriting Equations Using Function Notation 28 Domain and Range 31
1.5 Visualizing Functions
10
37
48
C H E C K Y O U R U N D E R S TA N D I N G
49
C H A P T E R 1 R E V I E W: P U T T I N G I T A L L T O G E T H E R
51
E X P L O R AT I O N 1 . 1 C O L L E C T I N G , R E P R E S E N T I N G , A N D A N A L Y Z I N G D ATA
CHAPTER 2
RATES OF CHANGE AND 2.1 Average Rates of Change 60 Describing Change in the U.S. Population over Time LINEAR FUNCTIONS Defining the Average Rate of Change 61 Limitations of the Average Rate of Change
62
2.2 Change in the Average Rate of Change 2.3 The Average Rate of Change Is a Slope Calculating Slopes
72
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60
67 72
56
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TABLE OF CONTENTS
2.4 Putting a Slant on Data
78
Slanting the Slope: Choosing Different End Points Slanting the Data with Words and Graphs 79
78
2.5 Linear Functions: When Rates of Change Are Constant
85
What If the U.S. Population Had Grown at a Constant Rate? A Hypothetical Example 85 Real Examples of a Constant Rate of Change 86 The General Equation for a Linear Function 88
2.6 Visualizing Linear Functions The Effect of b The Effect of m
92
93 93
2.7 Constructing Graphs and Equations of Linear Functions
99
Finding the Graph 99 Finding the Equation 101
2.8 Special Cases
107
Direct Proportionality 107 Horizontal and Vertical Lines 110 Parallel and Perpendicular Lines 112
2.9 Breaking the Line: Piecewise Linear Functions Piecewise Linear Functions
117
117
The absolute value and absolute value function Step functions 120
2.10 Constructing Linear Models of Data
118
124
Fitting a Line to Data: The Kalama Study 124 Reinitializing the Independent Variable 127 Interpolation and Extrapolation: Making Predictions
128
2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines 134 Using U.S. Census Data 134 Summarizing the Data: Regression Lines Regression lines: How good a fit?
135
137
Interpreting Regression Lines: Correlation vs. Causation Raising More Questions: Going Deeper 139 CHAPTER SUMMARY
138
146
C H E C K Y O U R U N D E R S TA N D I N G
147
C H A P T E R 2 R E V I E W: P U T T I N G I T A L L T O G E T H E R E X P L O R AT I O N 2 . 1 H AV I N G I T Y O U R WAY
149
154
E X P L O R AT I O N 2 . 2 A C A S E S T U D Y O N E D U C AT I O N A N D EARNINGS IN THE U.S.
156
CHAPTER 3
WHEN LINES MEET: LINEAR SYSTEMS
3.1 Interpreting Intersection Points: Linear and Nonlinear Systems When Curves Collide: Nonlinear Systems When Lines Meet: Linear Systems 163
160
160
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171
Visualizing Linear Systems 171 Strategies for Solving Linear Systems 171 Systems with No Solution or Infinitely Many Solutions 174 Linear Systems in Economics: Supply and Demand 175
3.3 Reading between the Lines: Linear Inequalities Above and Below the Line 181 Reading between the Lines 182 Manipulating Inequalities 184 Breakeven Points: Regions of Profit or Loss
181
185
3.4 Systems with Piecewise Linear Functions: Tax Plans Graduated vs. Flat Income Tax 193 Comparing the Flat and Graduated Tax Plans CHAPTER SUMMARY
193
195
199
C H E C K Y O U R U N D E R S TA N D I N G
200
C H A P T E R 3 R E V I E W: P U T T I N G I T A L L T O G E T H E R
202
E X P L O R AT I O N 3 . 1 F L AT V S . G R A D U AT E D I N C O M E TA X : W H O B E N E F I T S ? E X P L O R AT I O N 3 . 2 A C O M PA R I S O N O F H Y B R I D A N D C O N V E N T I O N A L AUTOMOBILES
209
CHAPTER 4
THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE
4.1 The Numbers of Science: Measuring Time and Space Powers of 10 and the Metric System Scientific Notation 214
4.2 Positive Integer Exponents
212
218
Exponent Rules 219 Common Errors 221 Estimating Answers 222
4.3 Zero, Negative, and Fractional Exponents
226
Zero and Negative Exponents 226 2n Evaluating A abB 227 Fractional Exponents 228 1 Expressions of the Form a 2: Square Roots 228 1 nth Roots: Expressions of the Form a n 229 Rules for Radicals 230 m Expressions of the Form a n 232
4.4 Converting Units
237
Converting Units within the Metric System 237 Converting between the Metric and English Systems Using Multiple Conversion Factors 238
4.5 Orders of Magnitude
238
242
Comparing Numbers of Widely Differing Sizes 242 Orders of Magnitude 242 Graphing Numbers of Widely Differing Sizes: Log Scales
243
212
207
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4.6 Logarithms as Numbers
247
Finding the Logarithms of Powers of 10 247 Finding the Logarithm of Any Positive Number Plotting Numbers on a Logarithmic Scale 250 CHAPTER SUMMARY
249
255
C H E C K Y O U R U N D E R S TA N D I N G
256
C H A P T E R 4 R E V I E W: P U T T I N G I T A L L T O G E T H E R
256
E X P L O R AT I O N 4 . 1 T H E S C A L E A N D T H E TA L E O F T H E U N I V E R S E
260
CHAPTER 5
GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS
5.1 Exponential Growth
264
The Growth of E. coli Bacteria 264 The General Exponential Growth Function 265 Doubling Time 266 Looking at Real Growth Data for E. coli Bacteria
5.2 Exponential Decay
268
271
The Decay of Iodine-131 271 The General Exponential Decay Function Half-Life 273
272
5.3 Comparing Linear and Exponential Functions
278
Linear Functions 278 Exponential Functions 278 Identifying Exponential Functions in a Data Table 279 A Linear vs. an Exponential Model through Two Points 280 Comparing the Average Rates of Change 282 In the Long Run, Exponential Growth Will Always Outpace Linear Growth
5.4 Visualizing Exponential Functions The Graphs of Exponential Functions The Effect of the Base a 286 The Effect of the Initial Value C 287 Horizontal Asymptotes 289
286
286
5.5 Exponential Functions: A Constant Percent Change Exponential Growth: Increasing by a Constant Percent Exponential Decay: Decreasing by a Constant Percent Revisiting Linear vs. Exponential Functions 295
292 293
5.6 More Examples of Exponential Growth and Decay Returning to Doubling Times and Half-Lives The Malthusian Dilemma 309 Forming a Fractal Tree 311
5.7
302
Compound Interest and the Number e
Compounding at Different Intervals Continuous Compounding Using e Continuous Compounding Formula Exponential Functions Base e 324 Converting ek into a 325
319 321 323
292
318
301
283
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5.8 Semi-Log Plots of Exponential Functions CHAPTER SUMMARY
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331
335
C H E C K Y O U R U N D E R S TA N D I N G
336
C H A P T E R 5 R E V I E W: P U T T I N G I T A L L T O G E T H E R E X P L O R AT I O N 5 . 1 : C O M P U T E R V I R U S E S
338
342
CHAPTER 6
LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
6.1 Using Logarithms to Solve Exponential Equations
346
Estimating Solutions to Exponential Equations 346 Rules for Logarithms 347 Solving Exponential Equations Using Logarithms 352 Solving for Doubling Times and Half-Lives 353
6.2 Using Natural Logarithms to Solve Exponential Equations Base e The Natural Logarithm 357 Returning to Doubling Times and Half-Lives 359 Converting Exponential Functions from Base a to Base e
361
6.3 Visualizing and Applying Logarithmic Functions Logarithmic Growth 367 Inverse Functions: Logarithmic vs. Exponential Applications of Logarithmic Functions 372
357
366
370
Measuring acidity: The pH scale 372 Measuring noise: The decibel scale 374
6.4 Using Semi-Log Plots to Construct Exponential Models for Data Why Do Semi-Log Plots of Exponential Functions Produce Straight Lines? CHAPTER SUMMARY
379
379
384
C H E C K Y O U R U N D E R S TA N D I N G
385
C H A P T E R 6 R E V I E W: P U T T I N G I T A L L T O G E T H E R E X P L O R AT I O N 6 . 1 : C H A N G I N G B A S E S
386
390
CHAPTER 7
POWER FUNCTIONS
7.1 The Tension between Surface Area and Volume
394
Scaling Up a Cube 394 Size and Shape 396
7.2 Direct Proportionality: Power Functions with Positive Powers Direct Proportionality 400 Properties of Direct Proportionality 401 Direct Proportionality with More Than One Variable
404
7.3 Visualizing Positive Integer Power Functions The Graphs of ƒ(x) ⫽ x and g(x) ⫽ x Odd vs. Even Positive Integer Powers The Effect of the Coefficient k 411 2
3
399
408
408 410
7.4 Comparing Power and Exponential Functions
416
Which Eventually Grows Faster, a Power Function or an Exponential Function?
416
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7.5 Inverse Proportionality: Power Functions with Negative Powers 421 Inverse Proportionality 421 Properties of Inverse Proportionality Inverse Square Laws 426
423
7.6 Visualizing Negative Integer Power Functions The Graphs of f (x) ⫽ x and g(x) ⫽ x Odd vs. Even Negative Integer Powers The Effect of the Coefficient k 436 ⫺1
⫺2
432
433
434
7.7 Using Logarithmic Scales to Find the Best Functional Model Looking for Lines 443 Why Is a Log-Log Plot of a Power Function a Straight Line? 444 Translating Power Functions into Equivalent Logarithmic Functions Analyzing Weight and Height Data 447 Allometry: The Effect of Scale 450 CHAPTER SUMMARY
445
458
C H E C K Y O U R U N D E R S TA N D I N G
459
C H A P T E R 7 R E V I E W: P U T T I N G I T A L L T O G T H E R E X P L O R AT I O N 7 . 1 : S C A L I N G O B J E C T S
460
464
CHAPTER 8
QUADRATICS AND THE MATHEMATICS OF MOTION
8.1 An Introduction to Quadratic Functions: The Standard Form 468 The Simplest Quadratic 468 Designing Parabolic Devices 469 The Standard Form of a Quadratic 470 Properties of Quadratic Functions 471 Estimating the Vertex and Horizontal Intercepts
473
8.2 Visualizing Quadratics: The Vertex Form Stretching and Compressing Vertically 478 Reflecting across the Horizontal Axis 478 Shifting Vertically and Horizontally 480 Using Transformations to Get the Vertex Form
478
483
8.3 The Standard Form vs. the Vertex Form Finding the Vertex from the Standard Form 487 Converting between Standard and Vertex Forms
487
489
8.4 Finding the Horizontal Intercepts: The Factored Form Using Factoring to Find the Horizontal Intercepts 496 Factoring Quadratics 497 Using the Quadratic Formula to Find the Horizontal Intercepts The discriminant 501 Imaginary and complex numbers
503
The Factored Form 504 Standard, Factored and Vertex Forms
506
500
496
443
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8.5 The Average Rate of Change of a Quadratic Function 8.6 The Mathematics of Motion
515
The Scientific Method 516 The Free Fall Experiment 516 Deriving an Equation Relating Distance and Time 516 Velocity: Change in Distance over Time 518 Acceleration: Change in Velocity over Time 520 Deriving an Equation for the Height of an Object in Free Fall Working with an Initial Upward Velocity 525 CHAPTER SUMMARY
510
522
531
C H E C K Y O U R U N D E R S TA N D I N G
532
C H A P T E R 8 R E V I E W: P U T T I N G I T A L L T O G E T H E R
533
E X P L O R AT I O N 8 . 1 : H O W F A S T A R E Y O U ? U S I N G A R U L E R T O M A K E A REACTION TIMER
536
CHAPTER 9
NEW FUNCTIONS FROM OLD
9.1 Transformations
540
Transforming a Function
540
9.2 The Algebra of Functions
553
9.3 Polynomials: The Sum of Power Functions
561
Defining a Polynomial Function 562 Visualizing Polynomial Functions 564 Finding the Vertical Intercept 567 Finding the Horizontal Intercepts 567
9.4 Rational Functions: The Quotient of Polynomials
575
Building a Rational Function: Finding the Average Cost of an MRI Machine Defining a Rational Function 576 Visualizing Rational Functions 577
9.5 Composition and Inverse Functions Composing Two Functions 585 Composing More Than Two Functions 588 Inverse Functions: Returning the Original Value
9.6 Exploring, Extending & Expanding CHAPTER SUMMARY
575
585
589 599
610
C H E C K Y O U R U N D E R S TA N D I N G
611
C H A P T E R 9 R E V I E W: P U T T I N G I T A L L T O G E T H E R
612
APPENDIX
Student Data Tables for Exploration 2.1 Data Dictionary for FAM1000 Data 621
SOLUTIONS
For all Algebra Aerobics and Check Your Understanding problems; for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1
INDEX
I-1
617
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A N I N T R O D U C T I O N T O D ATA AND FUNCTIONS
OVERVIEW How can you describe patterns in data? In this chapter we explore how to use graphs to visualize the shape of single-variable data and to show changes in two-variable data. Functions, a fundamental concept in mathematics, are introduced and used to model change.
After reading this chapter, you should be able to • describe patterns in single- and two-variable data • construct a “60-second summary” • define a function and represent it with words, tables, graphs, and equations • identify properties of functions • use the language of functions to describe and create graphs
1
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1.1 Describing Single-Variable Data
Visualizing Single-Variable Data Humans are visual creatures. Converting data to an image can make it much easier to recognize patterns.
Educational Attainment (2008) Americans 25 Years or Older 70,000,000 60,000,000
30%
50,000,000 40,000,000
20%
30,000,000 20,000,000
10%
Doctoral degree
Professional degree
Master’s degree
Bachelor’s degree
Associate’s degree
Some college (no degree)
High school degree
Grades 9–11
0
Grades K–8
10,000,000 0%
Percentage of Americans 25 yrs. or older
Bar charts: How well educated are Americans? Categorical data are usually displayed with a bar chart. The length of the bar for a single category tells you either the frequency count (the number of observations that fall into that category) or the relative frequency (the percentage of the total observations). Since the relative size of the bars is the same using either frequency or relative frequency counts, the two scales are often put on different vertical axes of the same chart. For example as shown in Figure 1.1, a bar chart of the educational attainment of Americans in 2008, approximately 60 million Americans had a high school degree but never went to college, and this represents approximately 31% of Americans 25 years or older.
Number of Americans 25 yrs. or older
Exploration 1.1 on p. 56 provides an opportunity to collect your own data and to think about issues related to classifying and interpreting data.
This course starts with you. How would you describe yourself to others? Are you a 5-foot 6-inch, black, 26-year-old female studying biology? Or perhaps you are a 5-foot 10-inch, Chinese, 18-year-old male English major. In statistical terms, characteristics such as height, race, age, and major that vary from person to person are called variables. Information collected about a variable is called data.1 Some variables, such as age, height, or number of people in your household, can be represented by a number and a unit of measure (such as 18 years, 6 feet, or 3 people). These are called quantitative variables. For other variables, such as gender or college major, we use categories (such as male and female or biology and English) to classify information. These are called categorical (or qualitative) data. The dividing line between classifying a variable as categorical or quantitative is not always clear-cut. For example, you could ask individuals to list their years of education (making education a quantitative variable) or ask for their highest educational category, such as college or graduate school (making education a categorical variable). Many of the controversies in the social sciences have centered on how particular variables are defined and measured. For nearly two centuries, the categories used by the U.S. Census Bureau to classify race and ethnicity have been the subject of debate.
Education level
Figure 1.1 Bar chart showing the education levels for Americans age 25 or older. Source: U.S. Bureau of the Census, www.census.gov. 1
Data is the plural of the Latin word datum (meaning “something given”)—hence one datum, two data.
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3
Always, remember to ask: Percentage of what? The bar chart in Figure 1.1 represents the percentage of Americans with high school degrees relative to Americans 25 years or older, not the percentage of all Americans. 1
What does the bar chart tell us? a. Using Figure 1.1, estimate the number and percentage of Americans age 25 years or older who have bachelor’s degrees, but no further advanced education. b. Estimate the percentage of Americans 25 years or older who have at least a high school education. c. What doesn’t the bar chart tell us? d. Write a brief summary of educational attainment in the United States.
S O L U T I O N
a. Those with bachelor’s degrees but no further education number about 38 million, or 19%. b. Those who have completed a high school education include everyone with a high school degree up to a Ph.D. We could add up all the percentages for each of those seven categories. But it’s easier to subtract from the whole those who do not meet the conditions; that is, subtract the percentage with grades K–8 and 9–11 education from 100%.
E X A M P L E
Grade School 1 Only Some 5 Total without High High School School Degree 6% 1
8%
5 14%
The percentage of Americans (age 25 or over) with a high school degree is about 100% 14% 86%. So more than four out of five Americans 25 years or older have completed high school. c. The bar chart does not tell us the total size of the population or the total number (or percentage) of Americans who have a high school degree. For example, if we include younger Americans between age 18 and age 25, we would expect the percentage with a high school degree to be higher. d. About 86% of adult Americans (age 25 or older) have at least a high school education. The breakdown for the 86% includes 31% who completed high school but did not go on, 45% who have some college (up to a bachelor’s degree), and about 10% who have graduate degrees. When compared with other countries, the United States population ranks among the most highly educated in the world.
An important aside: What a good graph should contain When you encounter a graph in an article or you produce one for a class, there are three elements that should always be present: 1. An informative title that succinctly describes the graph 2. Clearly labeled axes (or a legend) including the units of measurement (e.g., indicating whether age is measured in months or years) 3. The source of the data cited in the data table, in the text, or on the graph
See course software “F1: Histograms” on the course website.
Histograms: What is the distribution of ages in the U.S. population? A histogram is a specialized form of a bar chart that is used to visualize single-variable quantitative data. The horizontal axis on a histogram is a subset of the real numbers with the unit (representing, for example, number of years) and the size of each interval marked. The intervals are usually evenly spaced to facilitate comparisons (e.g., placed every 10 years). The size of the interval can reveal or obscure patterns in the data. As with a bar chart, the vertical axis can be labeled with a frequency or a relative frequency count. For example, the histogram in Figure 1.2 shows the distribution of ages in the United States as projected for 2010.
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U.S. Population Age Distribution (2010 projection) 25,000,000 20,000,000 15,000,000 10,000,000
95 to 99 years
100 years and over
Age interval
90 to 94 years
85 to 89 years
80 to 84 years
75 to 79 years
70 to 74 years
65 to 69 years
60 to 64 years
55 to 59 years
50 to 54 years
45 to 49 years
40 to 44 years
35 to 39 years
30 to 34 years
25 to 29 years
20 to 24 years
15 to 19 years
5 to 9 years
0
10 to 14 years
5,000,000
Under 5 years
Number of people
4
11/20/10
Figure 1.2 Histogram of the U.S. population in 5-year intervals. Source: U.S. Bureau of the Census, www.census.gov.
2
What does the histogram tell us? a. What 5-year age interval contains the most Americans? Roughly how many are in that interval? (Refer to Figure 1.2.) b. Estimate the number of people under age 20. c. Construct a topic sentence for a report about the U.S. population.
S O L U T I O N
a. The interval from 45 to 49 years contains the largest number of Americans, about 23 million. b. The sum of the frequency counts for the four intervals below age 20 is about 84 million. c. According to projections by the U.S. Census Bureau, in 2010 the number of Americans in each 5-year age interval will be fairly flat up to age 45, will peak between ages 45 and 54, and then fall in a gradual decline.
E X A M P L E
Numerical Descriptors: What Is “Average” Anyway? In 2010 the U.S. Bureau of the Census reported that the mean age for Americans was 37.8 and the median age was 36.9 years.
The Mean and Median The mean is the sum of a list of numbers divided by the number of terms in the list. The median is the middle value of a list of numbers ordered from the smallest to the largest; half the numbers are less than or equal to the median and half are greater than or equal to the median. If the number of observations is odd, then the median is the middle number in the list. If the number of observations is even, then the median is the mean of the two middle numbers in the list.
The mean age of 37.8 represents the sum of the ages of every American divided by the total number of Americans. The median age of 36.9 means that if you placed all the ages in order, 36.9 would lie right in the middle; that is, half of Americans are younger than or equal to 36.9 and half are 36.9 or older.
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In the press you will most likely encounter the word “average” rather than the term “mean” or “median.”2 The term “average” is used very loosely. It usually represents the mean, but it could also represent the median or something much more vague, such as the “average” American household. For example, the media reported that: The average American home in 2008 had more television sets than people. . . . There were 2.86 TV sets in the typical home and 2.5 people.3 The average American family owed more than $8,000 in credit debt in 2008 . . . and is averaging about 5.4 credit cards.4
See the reading “The Median Isn’t the Message,” on the course website, to find out how an understanding of the median gave renewed hope to the renowned scientist Stephen J. Gould when he was diagnosed with cancer.
The significance of the mean and median The median divides the number of entries in a data set into two equal halves and is unchanged by changes in values above and below it. For example, as long as the median income is larger than the poverty level, it will remain the same even if all poor people suddenly increase their incomes up to that level and everyone else’s income remains the same. The mean is the most commonly used statistic in the news media and, unlike the median, can be affected by a few extreme values called outliers. For example, suppose Bill Gates, founder of Microsoft and the richest man in the world, were to move into a town of 10,000 people, all of whom earned nothing. The median income would be $0, but the mean income would be in the millions. That’s why income studies usually use the median.
3
“Million-dollar Manhattan apartment? Just about average” According to a report cited on therealdeal.com, in November 2009 the median price of purchasing an apartment in Manhattan was $850,000 and the mean price was $1.32 million. How could there be such a difference in price? Which value do you think better represents apartment prices in Manhattan?
S O L U T I O N
Apartments that sold for exorbitant prices (in the millions) could raise the mean above the median. If you want to buy an apartment in Manhattan, the median price is probably more important because it tells you that half of the apartments cost $850,000 or less.
E X A M P L E
An Introduction to Explore & Extend Throughout the text there are Explore & Extend activities that provide you with ideas for going deeper or previewing what comes next. While these activities are optional, it is hoped they will be engaging, and doing them will increase your understanding. Enjoy!
EXPLORE & EXTEND
1.1
Could This Be True? Lake Wobegon is a mythical town where “all the women are strong, all the men are good-looking, and all the children are above average.”—Garrison Keillor Could this be true? If you consider only the children in Lake Wobegon, could all the children be “above average” if you use the median? What if you use the mean? If you consider all the children in the United States, could it be possible that all the children in Lake Wobegon are “above average”? Why or why not?
2
The word “average” has an interesting derivation according to Klein’s etymological dictionary. It comes from the Arabic word awariyan, which means “merchandise damaged by seawater.” The idea being debated was that if your ships arrived with water-damaged merchandise, should you have to bear all the losses yourself or should they be spread around, or “averaged,” among all the other merchants? The words averia in Spanish, avaria in Italian, and avarie in French still mean “damage.” 3 Source: The Nielson Co., 2009. 4 Source: www.creditcards.com.
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An Introduction to Algebra Aerobics In each section of the text there are “Algebra Aerobics” with answers in the back of the book. They are intended to give you practice in the algebraic skills introduced in the section and to review skills we assume you have learned in previous courses. These skills should provide a good foundation for doing the exercises at the end of each section.
Algebra Aerobics 1.1 1. Fill in the following table. Round decimals to the nearest thousandth. Fraction
Decimal
Percent
7 12
b. Calculate the percentage of the population who are over 40 years old. 7. In the chart below, which job type had the largest number of jobs in 2008? The smallest? Which job type is projected to increase the most? The least? What should you consider in interpreting these percentages?
0.025 2% 1 200
Jobs of the Future Occupations that will add jobs the fastest from 2008 to 2018, according to Labor Department projections.
0.35 0.8%
Jobs in 2008, in thousands
Occupation Biomedical engineers
Age
Frequency Count
1–20 21–40 41–60 61–80
35 28
Total
137
Relative Frequency (%) 38
Network-systems and datacommunications analysts
292.0
Home health aides
921.7
Personal and home-care aides
817.2
Financial examiners
72% 53.4% 50.0% 46.0%
27.0
41.2%
Source: Wall Street Journal, December 11, 2009.
8. From the histogram below, create a frequency distribution table. Assume that the total number of people represented by the histogram is 1352. (Hint: Estimate the relative frequencies from the graph and then calculate the frequency count in each interval.) Distribution of Ages (in years)
Percentage
2. Calculate the following: a. A survey reported that 80 people, or 16% of the group, were smokers. How many people were surveyed? b. Of the 236 students who took a test, 16.5% received a B grade. How many students received a B grade? c. Six of the 16 people present were from foreign countries. What percent were foreigners? 3. Given the list of numbers, find the mean and the median. a. 9, 2, 2, 6, 5 b. 2, 2, 5, 6 4. The mean hourly wage of six convenience store employees is $7.50. The hourly wages of five of the employees are $4.75, $5.50, $5.75, $8.00, and $9.50. Find the hourly wage of the sixth. 5. When looking through the classified ads, you found that 16 jobs had a starting salary of $20,000, 8 had a starting salary of $32,000, and 1 had a starting salary of $50,000. Find the mean and median starting salary for these jobs. 6. a. Fill in the table below. Round your answers to the nearest whole number.
Projected change, 2008–18
16.0
35 30 25 20 15 10 5 0
1–20
21–40 41–60 Years
61–80
9. Calculate the mean and median for the following data: a. $475, $250, $300, $450, $275, $300, $6000, $400, $300 b. 0.4, 0.3, 0.3, 0.7, 1.2, 0.5, 0.9, 0.4 10. Explain why the mean may be a misleading numerical summary of the data in Problem 9(a).
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1.1
7
Exercises for Section 1.1 Course software required for Exercise 22 and recommended for Exercise 21. 1. The following graph shows the amount of venture capital investment in different types of “clean technology.” a. Which type of “clean technology” had the largest amount of venture capital investment in 2009? About how much was this? b. Which type of “clean technology” had the greatest amount of money lost from venture capital investment from 2008 to 2009? About how much was this? c. What types of “clean technology” gained money from venture capital investments from 2008 to 2009?
3. The bar graph below was published in a 2009 report by the Organization for Economic Cooperation and Development (OECD). a. Summarize what you think this graph tells you about unmet health care needs. b. For which country was the disparity between below and above average incomes the largest? The smallest? c. What additional information would you need in order to calculate the number of people with unmet health care needs due to costs in each country? Health Care Gap Persons reporting an unmet health care need* due to costs in seven OECD countries, by income group, 2007
Green for Green
60
Venture capital investment in selected ‘clean technology’ sectors
52
Above average income Below average income
50
Solar
Percent
40 Transportation
2008
10
2009
3
Billions of Dollars
Includes electric vehicles, advanced batteries and fuel cells Source: Cleantech Group.
2. The accompanying bar chart shows the five countries with the largest populations in 2010 (estimates). The Five Most Populous Countries (2010 est.) 1600 1400 1200 1000 800 600
US
Australia
Germany
UK
New Zealand
2
18
7
Canada
Wind 1
8 9
Netherlands
0
0
6
22
3
Smart grid
*Did not get medical care, missed medical test, treatment or follow-up, or did not fill prescription or missed doses.
Source: “Measuring disparities in health status and in access and use of health care in OECD countries”, Michael de Looper and Gaetan Lafortune, OECD Health Working Paper No. 43, 2009, available at www.oecd.org/health.
4. The point spread in a football game is the difference between the winning team’s score and the losing team’s score. For example, in the 2010 Super Bowl game, the New Orleans Saints won with 31 points versus the Indianapolis Colts’ 17 points. So the point spread was 14 points. a. In the accompanying histogram, what is the interval with the most likely point spread in a Super Bowl? The least likely?
400
Point Spreads in 44 Super Bowls
200
12
China
India
United States
Indonesia
10
Brazil
Source: CIA Factbook, www.cia.gov/cia.
a. What country has the largest population, and approximately what is its population size? b. The population of India is projected in the near future to exceed the population of China. Given the current data, what is the minimum number of additional persons needed to make India’s population larger than China’s? c. What additional information would you need in order to calculate the percentage of the 2010 world population for each of these five countries?
Frequency
Population (in millions)
18
20
Biofuels
0
25
24 21
Energy efficiency
32
30
30
8 6 4 2 0
0–4
5–9
10–14 15–19 20–24 25–29 30–34 35–39 40–45 Point spread
Source: www.docsports.com/point-spreads-for-every-super-bowl.html.
b. What percentage of these 44 Super Bowl games had a point spread of 9 or less? Of 30 or more?
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5. Given here is a table of salaries taken from a survey of some recent graduates (with bachelor degrees) from a well-known university in Pittsburgh. Salary (in thousands)
Number of Graduates Receiving Salary
21–25 26–30 31–35 36–40 41–45 46–50
2 3 10 20 9 1
a. How many graduates were surveyed? b. Is this quantitative or qualitative data? Explain. c. What is the relative frequency of people having a salary between $26,000 and $30,000? d. Create a histogram of the data. 6. The accompanying bar chart shows the predictions of the Pew Research Center about the future racial and ethnic composition of American society. American Indian/Alaskan American comprised less than 1% of the population for each of the years on the chart and thus is not represented. U.S. Population 1960–2050 Share of total, by racial and ethnic groups 0.6 11 3.5
5
9
13 13 14 29
85 67 47
1960 White
2005 Hispanic
2050 projected Black
Asian
Source: Immigration to play lead role in future U.S. growth. Pew Research Center, 2008. http://pewresearch.org/ pubs/729/united-states-population-projections.
b. Describe three changes in the racial composition of the United States since 1960. c. Write a topic sentence describing an overall trend. 7. a. Compute the mean and median for the list: 5, 18, 22, 46, 80, 105, 110. b. Change one of the entries in the list in part (a) so that the median stays the same but the mean increases. 8. Suppose that a church congregation has 100 members, each of whom donates 10% of his or her income to the church. The church collected $250,000 last year from its members. a. What was the mean contribution of its members? b. What was the mean income of its members? c. Can you predict from the given data, the median income of its members? Explain your answer. 9. Suppose that annual salaries in a certain corporation are as follows: Level I (30 employees) Level II (8 employees) Level III (2 employees)
$18,000 $36,000 $80,000
Find the mean and median annual salary. Suppose that an advertisement is placed in the newspaper giving the mean annual salary of employees in this corporation as a way to attract applicants. Why would this be a misleading indicator of salary expectations? 10. Suppose the grades on your first four exams were 78%, 92%, 60%, and 85%. What would be the lowest possible average (using the mean) that your last two exams could have so that your grade in the class, based on the mean of the six exams, is at least 82%? 11. Read Stephen Jay Gould’s article “The Median Isn’t the Message” and explain how an understanding of statistics brought hope to a cancer victim. 12. a. On the first quiz (worth 25 points) given in a section of college algebra, one person received a score of 16, two people got 18, one got 21, three got 22, one got 23, and one got 25. What were the mean and median of the quiz scores for this group of students? b. On the second quiz (again worth 25 points), the scores for eight students were 16, 17, 18, 20, 22, 23, 25, and 25. i. If the mean of the scores for the nine students was 21, then what was the missing score? ii. If the median of the scores was 22, then what are possible scores for the missing ninth student?
a. The U.S. Bureau of the Census projected that there will be approximately 438 million people in the United States in the year 2050. Approximately how many people will be of Hispanic origin in 2050?
13. Why is the mean age larger than the median age in the United States? What prediction would you make for your State? What predictions would you make for other countries? You can check your predictions with data from the U.S. Census Bureau at www.census.gov.
Hispanic used to be considered a racial classification. It is now considered an ethnic classification, since Hispanics can be black, white, or any other race. For this survey, Hispanics formed one category and did not overlap with other categories.
14. Up to and including Barack Obama, the ages of the last 15 presidents when they first took office were 55, 51, 54, 51, 60, 62, 43, 55, 56, 52, 69, 64, 46, 54, and 48. a. Find the mean and median ages of the past 15 presidents when they took office.
5
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1.1
14. (continued) b. If the mean age of the past 16 presidents is 54.75, at what age did the missing president take office? c. Beginning with age 40 and using 5-year intervals, find the frequency count for each age interval. d. Create a frequency histogram using your results from part (c).
United States–2010
Male
16. Why do you think most researchers use median rather than mean income when studying “typical” households? 17. According to the Federal Reserve Board, in 2007, the median net worth of American families was $120,300 and the mean net worth was $556,300. How could there be such a wide discrepancy? 15
12
9
6
3
0
Internet Users in the World by Geographic Regions—2009
425.8 259.6
Latin America/ Caribbean
186.9 86.2
Africa
58.3
Middle East Oceania/ Australia
0 50 10 0 15 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0 55 0 60 0 65 0 70 0 75 0 80 0 85 0
21.1
Millions of users
World Internet Penetration Rates by Geographic Regions—2009 2
North America
1.6 1.2
76.2%
Oceania/Australia
60.8%
Europe
53.0%
Latin America/ Caribbean
31.9% 28.8%
Middle East Asia Africa
20.1%
6
9
12
15
Female
100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0
764.4
North America
3
Ghana–2010
Male
Europe
0
Population (in millions)
19. Use the following graphs to describe three interesting facts about Internet use worldwide in 2009.
Asia
Female
100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0
15. Herb Caen, a Pulitzer Prize–winning columnist for the San Francisco Chronicle, remarked that a person moving from state A to state B could raise the average IQ in both states. Is he right? Explain.
18. Read the CHANCE News article and explain why the author was concerned.
9
0.8
0.4
0
0
0.4 0.8 1.2 1.6
2
Population (in millions)
Source: U.S. Bureau of Census, International Data Base, www.census.gov.
21. Estimate the mean and median from the given histogram. (Hint: Replace each dollar interval with a dollar amount approximately in the middle of the interval.)
8.7%
Monthly Allowance of Junior High School Students 26.6% 0
10% 20% 30% 40% 50% 60% 70% 80% 90% Penetration Rate
Source: Internet World States, www.internetworldstats.com/stats.htm Estimated Internet users are 1,802,330,457 for December 31, 2009. Copyright © 2020, Miniwatts Marketing Group.
20. Population pyramid charts are used to depict the overall age structure of a society. The accompanying pyramids show the age structure in 2010 for Ghana, a developing country, and for the United States, an industrialized nation. Describe three major differences in the distribution of ages in the these two countries in 2010.
Number of students
World, Avg.
14 12 10 8 6 4 2 0
0–9
10–19
20–29 Dollars
30–39
40–49
Note: The course software “F4: Measures of Central Tendency” in FAM1000 Census Graphs can help you understand the mean and median and their relationship to histograms.
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headline for these data. Describe three key features that support your headline.
22. (Computer and course software required.) Open up the course software “F1: Histograms” in FAM1000 Census Graphs in the course software. The 2009 U.S. Census data on 1000 randomly selected U.S. individuals and their families are imbedded in this program. You can use it to create histograms for education, age, and different measures of income. Try using different interval sizes to see what patterns emerge. Decide on one variable (say education) and compare the histograms of this variable for different groups of people. For example, you could compare education histograms for men and women or for people living in two different regions of the country. Pick a comparison that you think is interesting. Create a possible
23. In January 2010, a New York Times article6, reported that outside of school, 8- to 18-year-old Americans spend more than seven and a half hours a day using electronic devices such as smart phones, computers or televisions, not counting the hour and a half spent on texting, or the half-hour spent on talking on their cell phones. The following graph shows the average (mean) amount of time spent with selected media in 2009. The total time using these selected media was a little over seven and a half hours. What can you conclude about how 8- to 18-year-olds were using these media?
Average Hours and Minutes Spent with each Medium in a Typical Day Among 8- to 18-years-olds TV content
2009
Music/audio
4:29
Computer
2:31
Video games
1:29
1:13
Print
Movies
0:38 0:25 10:45
New York Times Graphics
1.2 Describing Relationships between Two Variables Exploration 1.1 Homework provides an opportunity to collect and analyze two-variable data.
By looking at two-variable data, we can learn how change in one variable affects change in another. How does the weight of a child determine the amount of medication prescribed by a pediatrician? How does median age or income change over time? In this section we examine how to describe these changes with graphs, data tables, words, and equations.
Visualizing Two-Variable Data E X A M P L E
1
Scatter plots Table 1.1 shows data for two variables, the year and the median age of the U.S. population. Plot the data in Table 1.1 and then use your graph to describe the changes in the U.S. median age over time.
Median Age of the U.S. Population, 1850–2050* Year
Median Age
Year
Median Age
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940
18.9 19.4 20.2 20.9 22.0 22.9 24.1 25.3 26.4 29.0
1950 1960 1970 1980 1990 2000 2005 2010 2025 2050
30.2 29.5 28.0 30.0 32.8 35.3 36.7 36.9 38.2 39.0
Table 1.1 *Data for 2010–2050 are projected. Source: U.S., Bureau of the Census, www.census.gov.
6
DA T A
MEDAGE
There are Excel and graph link files for the median age data called MEDAGE on the course website.
The article by Tamar Lewin, “If your kids are awake they’re probably online,” cited the Kaiser Foundation Study, Generation M2: Media in the Lives of 8- to 18-year-olds, 2010, as its source for data used in article and graph.
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Describing Relationships between Two Variables
11
In Table 1.1 we can think of a year and its associated median age as an ordered pair of the form (year, median age). For example, the first two elements correspond to the ordered pair (1850, 18.9) and the second corresponds to (1860, 19.4). Figure 1.3 shows a scatter plot of the data. The graph is called a time series because it shows changes over time. In newspapers and magazines, the time series is the most frequently used form of data graphic.7 Median age of U.S. population over time: A time series 40 Median age (in years)
35 30 25 20 15 10 5 0 1850
1890
1930
1970
2010
2050
Year
Figure 1.3
Our graph shows that the median age of the U.S. population grew quite steadily for one hundred years, from 1850 to 1950. Although the median age decreased between 1950 and 1970, since 1970 it has continued to increase. From 1850 to the present, the median age nearly doubled, and projections for 2025 and 2050 indicate continued increases, though at a slower pace.
Constructing a “60-Second Summary” To communicate effectively, you need to describe your ideas succinctly and clearly. One tool for doing this is a “60-second summary”—a brief synthesis of your thoughts that could be presented in one minute. Quantitative summaries should be straightforward and concise. They often start with a topic sentence that summarizes the key idea, followed by supporting quantitative evidence. After you have identified a topic you wish to write about or present orally, some recommended steps for constructing a 60-second summary are: • Collect relevant information (possibly from multiple sources, including the Internet). • Search for patterns, taking notes. • Identify a key idea (out of possibly many) that could provide a topic sentence. • Select evidence and arguments that support your key idea. • Examine counterevidence and arguments and decide if they should be included. • Construct a 60-second summary, starting with your topic sentence. To help your ideas take shape, put them down on paper, then refine and modify. Quantitative reports should not be written in the first person. For example, you might say something like “The data suggest that . . .” rather than “I found that the data . . .” E X A M P L E
2
A 60-Second Summary The annual federal surplus () or deficit () since World War II is shown in Table 1.2 and Figure 1.4 (a scatter plot where the points have been connected). Construct a 60second summary describing the changes over time. 7
Edward Tufte in The Visual Display of Information (Cheshire, Conn.: Graphics Press, 2001, p. 28) reported on a study that found that more than 75% of all graphics published were time series.
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Fiscal year 8 Federal Budget: Surplus (1) or Deficit (2) Billions of Dollars ⫺48 ⫺3 ⫺3 ⫺0 ⫺1 ⫺3 ⫺53 ⫺74 ⫺212 ⫺221 ⫺269 ⫺290 ⫺255 ⫺203
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1991 1992 1993 1994
Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
⫺164 ⫺107 ⫺22 69 126 236 128 ⫺158 ⫺378 ⫺413 ⫺318 ⫺248 ⫺161 ⫺459 ⫺1,413
Table 1.2 Source: U.S. Office of Management and Budget.
S O L U T I O N
Federal Budget Surplus or Deficit (in billions $)
Billions of Dollars 300 200 Federal budget (billions)
Year
100 0 –100 –200 –300 –400 –500 1940
1950
1960
1970
1980
1990
2000
2010
Year
Figure 1.4 Note that the data point (2009, 21,413) is not included on the graph.
Between 1945 and 2009 the annual U.S. federal deficit moved from a 30-year stable period, with as little as $0 deficit, to a period of oscillations, leading in 2009 to the largest deficit ever recorded. From the 1970s to 1992, the federal budget ran an annual deficit, which generally was getting larger until it reached almost $300 billion in 1992. From 1992 to 1997, the deficit steadily decreased, and from 1998 to 2001 there were relatively large surpluses. The maximum surplus occurred in 2000, when it reached $236 billion. But by 2002 the federal government was again running large deficits. In 2009 the deficit reached $1,413 billion, the largest recorded up to that time.
Algebra Aerobics 1.2a 1. The net worth of a family at any given time is the difference between assets (what is owned) and liabilities (what is owed). The table and graph below show the median net worth of U.S. families, adjusted for inflation.
Median Family Net Worth (in 2007 dollars) $130,000 $120,000 $110,000 $100,000 $90,000 $80,000
$30,000 $20,000
2008
2007
2006
2005
$0
2004
$10,000 2003
Source: The Federal Reserve Board, www.federalreserve.gov.
$40,000
2002
91,300 101,200 102,200 120,300
2001
1998 2001 2004 2007
$50,000
2000
Median Net Worth ($)
$60,000
1999
Year
$70,000
1998
Median Family Net Worth (adjusted for inflation using 2007 dollars)9
Year
8
The government’s fiscal year runs from October 1 through September 30 and is designated by the year in which it ends. For example, the fiscal year for 2010 is from October 1, 2009 to September 30, 2010. 9
“Constant dollars” is a measure used by economists for comparing the value of money over time, eliminating the effects of inflation or deflation. It is based on the buying power of the dollar in a certain base year. For example, the net worth in 1998 of $91,300 “in constant 2007 dollars” was equal to the amount of goods and services that cost $91,300 in 2007. The actual net worth in 1998, measured in what is called “current dollars,” was much lower.
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4. Use the World Population graph to answer the following: a. The world population in 2000 was how many times greater than the world population in 1900? What was the difference in population size? b. The world population in 2100 is projected to be how many times greater than the world population in 2000? What is the difference in population size? c. Describe the difference in the growth in world population from 1900 to 2000 versus the projected growth from 2000 to 2100.
10 Less developed countries
4 2
2150
2100
1950
1900
1850
1800
1750
2050
More developed countries
0
2000
Population (in billions)
World Population with Projections to 2150
6
13
3. Use the World Population graph to estimate the following projections for the year 2150. a. The total world population. b. The total populations of all the more developed countries. c. The total populations of all the less developed countries. d. Write a topic sentence about the estimated world population in 2150.
1. (continued) a. Write a few sentences about the trend in U.S. median family net worth. b. What additional information might be useful in describing the trend in median net worth? 2. Use the World Population graph to estimate: a. The year when the world population reached 4 billion. b. The year that it is projected to reach 8 billion. c. The number of years it will take to grow from 4 to 8 billion.
8
Describing Relationships between Two Variables
Source: Population Reference Bureau, www.prb.org.
Using Equations to Describe Change Sometimes the relationship between two variables can also be described with an equation. An equation gives a rule on how change in the value of one variable affects change in the value of the other. If the variable n represents the number of years of education beyond eighth grade and E represents yearly median earnings (in dollars) for people living in the United States, then the following equation and graph in Figure 1.5 model the relationship between E and n:
Median earnings (dollars)
E 5 1140 1 6250 n
125,000
Earnings vs. Education
E
100,000 75,000
E = 1140 + 6250 n
50,000 25,000
n 0
2
4
6
8
10 12 14 16
Education (years past grade 8)
Figure 1.5
This equation provides a useful tool for describing how earnings and education are linked and for making predictions.10 For example, to predict the median earnings, E, for 10 In “Looking for Links between Education and Earnings,” in Section 2.11, we show how such equations are derived and how they are used to analyze the relationship between education and earnings.
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those with a high school education, we replace n with 4 (representing 4 years beyond eighth grade, or a high school education) in our equation to get E 5 1140 1 6250 ? 4 5 $26,140 Our equation predicts that for those with a high school education, median earnings will be about $26,140. An equation that is used to describe a real-world situation is called a mathematical model. Such models offer compact, often simplified descriptions of what may be a complex situation. The accuracy of the predictions made with such models can be questioned and disciplines outside of mathematics may be needed to help answer such questions. Yet these models are valuable guides in our quest to understand social and physical phenomena in our world.
Describing the relationship between abstract variables Variables can represent quantities that are not associated with real objects or events. The following equation or mathematical sentence defines a relationship between two quantities, which are named by the abstract variables x and y: y 5 x2 1 2x 2 3 Solutions to the equation are pairs of values for x and y that make the sentence true. By convention, we express these solutions as ordered pairs of the form (x, y). 3
Identifying solutions to an equation Given y x2 2x 3 a. Are (1, 0) and (0, 1) solutions to y x 2 2x 3? b. How many solutions are there? c. Create a table with several solutions for this equation. d. Use technology to graph the equation. State the relationship between points on the graph and solutions to the equation. Label one solution on your graph.
S O L U T I O N
a. (1, 0) would be a solution to y x2 2x 3, since 0 12 2(1) 3, whereas (0, 1) would not be a solution, since 1 2 02 1 2(0) 2 3. b. There are infinitely many possible solutions to the equation y x2 2x 3, since we could substitute any real number for x and find a corresponding y. c. Table 1.3 lists a few solutions. d. The graph of the equation is shown in Figure 1.6. All the points on the graph represent solutions to the equation, and every solution is a point on the graph of the equation.
E X A M P L E
10
x
y
–4 –3 –2 –1 0 1 2 3
5 0 –3 –4 –3 0 5 12
Table 1.3
y
Coordinates of point P are: (horizontal coordinate, vertical coordinate) (x, y) (2, 5)
P(2, 5) x
–10
10
–10
Figure 1.6 Graph of y 5 x2 1 2x 2 3 where one solution of infinitely many solutions is labeled.
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15
Note that sometimes an arrow is used to show that a graph extends indefinitely in the indicated direction. In Figure 1.6, the arrows show that both arms of the graph extend indefinitely upward. Solutions of an Equation The solutions of an equation in two variables x and y are the ordered pairs (x, y) that make the equation a true statement. Graph of an Equation The graph of an equation in two variables displays the set of points that are solutions to the equation.
E X A M P L E
4
Solutions for equations in one or two variables Describe how the solutions for the following equations are similar and how they differ. 3x 1 5 5 11 x125x12 31x5y15
S O L U T I O N
The solutions are similar in the sense that each solution for each particular statement makes the statement true. They are different because: There is only one solution (x 2) of the single-variable equation 3x 5 11. There are an infinite number of solutions for x of the single-variable equation x 2 x 2, since any real number will make the statement a true statement. There are infinitely many solutions, in the form of ordered pairs (x, y), of the two-variable equation 3 x y 5. y 2
E X A M P L E
5
S O L U T I O N
Estimating solutions from a graph The graph of the equation x2 4y2 4 is shown in Figure 1.7. a. From the graph, estimate three solutions of the equation. b. Check your solutions using the equation.
x –3
3 –2
Figure 1.7
a. The coordinates (0, 1), (2, 0), and (1, 0.8) appear to lie on the ellipse, which is the graph of the equation x2 4y2 4. b. If substituting the ordered pair (0, 1) into the equation makes it a true statement, then (0, 1) is a solution. Given substitute x 5 0 and y 5 1 evaluate
x2 1 4y2 5 4 (0) 2 1 4(1) 2 5 4 454
We get a true statement, so (0, 1) is a solution to the equation. For the ordered pair (22, 0): Given substitute x 5 22 and y 5 0 evaluate
x2 1 4y2 5 4 (22) 2 1 4(0) 2 5 4 4105 4 45 4
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Again we get a true statement, so (22, 0) is a solution to the equation. For the ordered pair (1, 0.8): x2 1 4y2 5 4
Given
(1) 2 1 4(0.8) 2 5 4
substitute x 5 1 and y 5 0.8 evaluate
1 1 4(0.64) 5 4 3.56 2 4
We get a false statement, so (1, 0.8) is not a solution, although it is close to a solution. EXPLORE & EXTEND
1.2
Finding Patterns In each data table (a) to (e), look for a pattern in terms of how y changes when x changes. Describe with words how to find y in terms of x. Assume the pattern continues indefinitely and extend the data table to include a few negative values for x. Describe the pattern that you found with a formula. a.
x
y
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0
b.
x
y
0 1 2 3 4
5 8 11 14 17
c.
x
y
0 1 2 3 4
0 1 4 9 16
d.
x
y
0 1 2 3 4
0 1 8 27 64
e.
x
y
0 1 2 3 4
0 2 12 36 80
Create a formula for y in terms of x and generate a data table. Show that data table to your class or a fellow student and see if they can describe your formula with words and an equation.
Algebra Aerobics 1.2b 1. a. Describe in your own words how to compute the value for y, given a value for x, using the following equation: y 5 3x2 2 x 1 1 b. Which of the following ordered pairs represent solutions to the equation? (0, 0), (0, 1), (1, 0), (21, 2), (22, 3), (21, 0) c. Use x 5 0, 61, 62, 63 to generate a small table of values that represent solutions to the equation. 2. Repeat the directions in Problem 1(a), (b), and (c) using the equation y 5 (x 2 1) 2. 3. Given the equations y1 5 4 2 3x and y2 5 22x2 2 3x 1 5, fill in the following table. x y1 y2
24
22
21
0
1
2
4
a. Use the table to create two scatter plots, one for the ordered pairs (x, y1) and the other for (x, y2). b. Draw a smooth curve through the points on each graph. c. Is (1, 1) a solution for equation y1? For y2? d. Is (21, 6) a solution for equation y1? For y2? e. Look at the graphs. Is the ordered pair (23, 2) a solution for either equation? Verify your answer by substituting the values into each equation. 4. Given the equation y 5 x2 2 3x 1 2, a. If x 5 3/2, find y. b. Find two points that are not solutions to this equation. c. If available, use technology to graph the equation and then confirm your results for parts (a) and (b).
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17
Exercises for Section 1.2 1. In June 1981, the first cases of AIDS were reported in the United States. In 2008, it was estimated that more than one million people were living with HIV in the United States and that more than half a million died after developing AIDS. The following graph shows AIDS diagnoses and deaths from 1981 to 2007. Assume you work for a newspaper and are asked to report on the following data.
3. The accompanying table shows the number of personal and property crimes in the United States from 1980 to 2007.
Year
Personal Crimes (in thousands)
Property Crimes (in thousands)
40,000
1980 1985 1990 1995 2000 2005 2006 2007
1,345 1,328 1,820 1,799 1,425 1,391 1,418 1,408
12,064 11,103 12,655 12,064 10,183 10,175 9,984 9,843
30,000
Source: U.S. Bureau of the Census, www.census.gov.
AIDS Diagnoses and Deaths 80,000 70,000 60,000 50,000
20,000
a. Create a scatter plot of the personal crimes over time. Connect the points with line segments. b. Approximately how many times more property crimes than personal crimes were committed in 1980? In 2007? c. Write a topic sentence about property and personal crime from 1980 to 2007.
2007
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1983
1981
0
1985
10,000
Year AIDS diagnoses
Deaths from AIDS
Source: www.avert.org. Avert is an international AIDS charity.
a. What are three important facts that emerge from this graph? b. Construct a 60-second summary that could accompany the graph in the newspaper article. 2. The following graph shows changes in the Dow Jones Industrial Average, which is based on 30 stocks that trade on the New York Stock Exchange and is the best-known index of U.S. stocks.
4. The following graph shows the amount of revenues from newspaper ad sales over time. a. What are the ordered pairs associated with the highest and lowest revenues? b. What are the units for the ordered pair (1995, 13742)? Interpret the meaning of this ordered pair. c. Write a 60-second summary about these changes. Can you think of any reasons to explain changes in revenue over time? Newspaper Classified Ads Revenue has Plummeted in the Past Few Years
Dow Jones Industrial Average 13k
Newspaper revenue from classified ads, 1980–2008
12k
$20,000
10k 9k 8k
2008
Jan.
Feb.
Dec.
Nov.
Oct.
Sep.
Aug.
Jul.
May
Jun.
Apr.
Feb.
2009
Mar.
Jan.
Dec.
Nov.
Oct.
Sep.
Aug.
Jul.
Jun.
May
Apr.
Feb.
Mar.
7k
2010
Note that the symbol on the lower left corner of the graph indicates that the vertical axis does not start at 0. Source: http://finance.yahoo.com.
Newspaper annual classified ads revenue, in millions
11k
$19,508
$11,500
$13,742
$10,000
$9,975 $8,375
$4,222 $0 1980
a. What time period does the graph cover? b. Estimate the lowest Dow Jones Industrial Average. During what month and year did it occur? c. Estimate the highest value for the Dow Jones Industrial Average during that period. When did it occur? d. Write a topic sentence describing the change in the Dow Jones Industrial Average over the given time period.
$17,342
1985
1990
1995
2000
2005 2008
Source: Business Analysis and Research Newspaper Association of America, “Advertising Expenditures.” Available at http//www.naaorg/ TrendsandNumbers /Advertising-Expenditures-aspx.
5. The National Center for Chronic Disease Prevention and Health Promotion published the following data on the chances that a man has had prostate cancer at different ages.
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5. (continued)
d. The less expensive car carries more passengers. e. State two other facts you can derive from the graphs. f. Which car would you buy? Why?
Lifetime Risk of Developing Prostate Cancer Age
Risk
Percent Risk
45 50 55 60 65 70 75 80
1 in 25,000 1 in 476 1 in 120 1 in 43 1 in 21 1 in 13 1 in 9 1 in 6
0.004% 0.21% 0.83% 2.3% 4.8% 7.7% 11.1% 16.7%
8. a. Which (if any) of the following ordered pairs (x, y) is a solution to the equation y x2 2x 1? Show how you came to your conclusion. (2, 7)
6. Use the following graph to write a 60-second summary of the attitudes of Americans about the role of women in society.
9. Consider the equation R 2 5T. a. Determine which, if any, of the following points (T, R) satisfy this equation. (0, 4)
(1, 3)
(2, 0)
b. Find two additional ordered pairs that are solutions to the equation. c. Make a scatter plot of the solution points found. d. What does the scatter plot suggest about where more solutions could be found? Check your predictions. 10. Use the accompanying graph to estimate the missing values for x or y in the table. x
y
15
y
23
The Changing Role of Women
2
Women should return to their traditional role in society 80 75
66 Percent of Population
(2, 1)
b. Find one additional ordered pair that is a solution to the equation above. Show how you found your solution.
a. What is the relationship between age and getting prostate cancer? b. Make a scatter plot of the percent risk for men of the ages given. c. Use the “percent risk” data to find how much more likely men 50 years old will develop prostate cancer than men who are 45. How much more likely are men 55 years old to develop prostate cancer than men who are 50? d. Looking at these data, when would you recommend annual prostate checkups to begin for men? Explain your answer in terms of the interests of the patient and of the insurance company.
21 22 21
60
x
2
4
–4
7
40
20
(1, 0)
–5
30 19
0 1988
1992
1996
2000
2004
2008
11. The graph of the equation (x 2)2 3 y is shown below. Estimate three solutions from the graph and then check your solutions using the equation. y
Disagree
Agree
10
8
7. The following three graphs describe two cars, A and B.
6
B
Size
Age
A A B Cost
Cruising speed
Range (miles)
Source: Pew Research Center for the People & the Press May 21, 2009 survey report “Trends in Political Values and Core Attitudes: 1987–2009.”
A
4
B 2
Passenger capacity
For parts (a)–(d), decide whether the statement is true or false. Explain your reasoning. a. The newer car is more expensive. b. The slower car is larger. c. The larger car is newer.
2
–3 –2
–2
–4
4
6
x
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12. The graph of the equation x(x ⫺ 3)2 ⫽ y is shown below. Estimate three solutions from the graph and then check your solutions using the equation. y
8
4
2
2
4
6
15. Given the four ordered pairs (21, 3), (1, 0), (2, 3), and (1, 2), for each of the following equations, identify which points (if any) are solutions for that equation. a. y 5 2x 1 5 c. y 5 x2 2 x 1 1 4 2 b. y 5 x 2 1 d. y 5 x 1 1 16. Yale economist Robert J. Shiller and Wellesley economist Karl Case created the Case-Shiller index of American housing prices that tracks the value of housing back to 1890 in consistent terms, factoring out the effects of inflation. The 1890 benchmark is 100 on the chart. If a standard house sold in 1890 for $100,000 (inflation-adjusted to today’s dollars), an equivalent house would have sold for $66,000 in 1920 (66 on the index scale) and $220,000 in 2006 (220 on the index scale). Use the following graph, based on the Case-Shiller index, to write a 60-second summary about home values in the United States since 1890. You can use the Internet to find the current value of homes and see if these projections are accurate.
6
–2
19
x
A History of Home Values
–2
CURRENT BOOM
JULY 2006
200 PROJECTION
–4
190 180
DECLINE AND RUN-UP Prices dropped as mass production techniques appeared early in the 20th century. Prices spiked with post-war housing demand.
13. For parts (a)–(d) use the following equation: y ⫽ ⫺2x . a. If x ⫽ 0, find the value of y. b. If x is greater than zero, what can you say about the value of y? c. If x is a negative number, what can you say about the value of y? d. Can you find an ordered pair that represents a solution to the equation when y is greater than zero? If so, find it; if not, explain why. 2
150 140
1980’S BOOM
130
120
120
110
110
100
100
80
2010
2000
1990
1980
1970
1960
1950
1940
1930
1920
1910
60
1900
70 1890
c. y 5 x3 1 x2 d. y 5 3(x 2 2)(x 2 1)
1970’S BOOM
160
90
14. Find the ordered pairs that represent solutions to each of the following equations when x ⫽ 0, when x ⫽ 3, and when x ⫽ ⫺2. a. y 5 2x2 1 5x b. y 5 2x2 1 1
WORLD GREAT WORLD WAR I DEPRESSION WAR II
130
170
BOOM TIMES Two gains in recent decades were followed by returns to levels consistent since the late 1950’s. Since 1997, the index has risen about 83 percent.
Sources: Irrational Exuberance, 2nd ed., 2006, by Robert Shiller; The New York Times, August 26, 2006. Updates and projections: www.ritholz.com. New York Times Graphics
1.3 An Introduction to Functions What Is a Function? When we speak informally of one quantity being a function of some other quantity, we mean that one depends on the other. For example, someone may say that what they wear is a function of where they are going, or what they weigh is a function of what they eat, or how well a car runs is a function of how well it is maintained. In mathematics, the word “function” has a precise meaning. A function is a special type of relationship between two quantities. If the value of one quantity uniquely determines the value of a second quantity, then the second quantity is a function of the first. For example, median age (as shown in Example 1 on p. 10) is a function of time, since each input of a year determines a unique (one and only one) output value of a median age. A function is a rule that assigns to each input quantity exactly one output quantity. The output is a function of the input.
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Representing Functions: Words, Tables, Graphs, and Equations A function rule can be described using words, data tables, graphs, or equations. The median age example uses data tables and graphs. When a function is described in these various ways, each description provides the same information but with a different emphasis. 1
Sales tax Twelve states have a sales tax of 6%; that is, for each dollar spent in a store in these states, the law says that you must pay a tax of 6 cents, or $0.06. Represent the sales tax as a function of purchase price using words, an equation, table, and graph.11
S O L U T I O N
Using words The sales tax is 6% of the purchase price, which means you multiply the purchase price by six hundredths to get the sales tax.
E X A M P L E
Using an equation We can write this relationship as an equation where T represents the amount of sales tax and P represents the price of the purchase (both measured in dollars): amount of sales tax 5 0.06 ? price of purchase T 5 0.06P Our function rule says: “Take the given value of P and multiply it by 0.06; the result is the corresponding value of T.” The equation represents T as a function of P, since for each value of P the equation determines a unique (one and only one) value of T. The purchase price, P, is restricted to dollar amounts greater than or equal to zero. Using a table We can use this formula to make a table of values for T determined by the different values of P (see Table 1.4). Such tables are sometimes seen beside cash registers. P (purchase price in $) T (sales tax in $)
0
1
2
3
4
5
6
7
8
9
10
0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60
Table 1.4
Using a Graph The points in Table 1.4 were used to create a graph of the function (Figure 1.8). The table shows the sales tax only for selected purchase prices, but we could have used any positive dollar amount for P. We connected the points on the scatter plot to suggest the many possible intermediate values for price. For example, if P $2.50, then T $0.15. Graph of 6% Sales Tax 1.00 Sales tax (dollars)
20
10/27/10
T
0.50
0
2
4
6
8
10
P
Purchase price (dollars)
Figure 1.8 11 In 2009, a sales tax of 6% was the most common rate for a sales tax in the United States. See www.taxadmin.org for a listing of the sales tax rates for all of the states.
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Input and Output: Independent and Dependent Variables Since a function is a rule that assigns to each input a unique output, we think of the output as being dependent on the input. We call the input of a function the independent variable and the output the dependent variable. When a set of ordered pairs represents a function, then each ordered pair is written in the form
y Dependent variable (output)
(input, output) (independent variable, dependent variable)
or equivalently,
If x is the independent and y the dependent variable, then the ordered pairs would be of the form (x, y)
x Independent variable (input)
The mathematical convention is for the first variable, or input of a function, to be represented on the horizontal axis and the second variable, or output, on the vertical axis (see Figure 1.9). Sometimes the choice of the independent variable is arbitrary or not obvious. For example, economists argue as to whether wealth is a function of education or education is a function of wealth. As seen in the next example, there may be more than one correct choice.
Figure 1.9
E X A M P L E
2
Identifying independent and dependent variables In the sales tax example, the equation T 0.06P gives the sales tax, T, as a function of purchase price, P. In this case T is the dependent variable, or output, and P is the independent variable, or input. But for this equation we can see that P is also a function of T; that is, each value of T corresponds to one and only one value of P. It is easier to see the relationship if we solve for P in terms of T, to get P5
T 0.06
Now we are thinking of the purchase price, P, as the dependent variable, or output, and the sales tax, T, as the independent variable, or input. So, if you tell me how much tax you paid, I can find the purchase price.
When Is a Relationship Not a Function? Not all relationships define functions. A function is a special type of relationship, one where for each input, the rule specifies one and only one output. Examine the following examples. Function
Not a Function
Function
Input Output
Input Output
Input Output
1 2 3
1
1 2 3
6 7 8
Each input has only one output.
E X A M P L E
3
2 3
6 7 8 9
The input of 1 gives two different outputs, 6 and 7, so this relationship is not a function.
6
Each input has only one output. Note that a function may have identical outputs for different inputs.
Does the table represent a function? Consider the data in Table 1.5 on the rate of unemployment in the United States. Which variable/s could be used as the input of a function? What about the output?
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Unemployment in the United States Year, Y
Unemployment Rate, U
1975 1980 1985 1990 1995 2000 2005 2008 2009 2010
8.5 7.1 7.2 5.6 5.6 4.0 5.1 5.8 9.3 10.0 (est)
Table 1.5 Source: Bureau of Labor Statistics, 2010
http://www.bls.gov/cps/prev_yrs.htm. S O L U T I O N
1990 1995
5.6
U is a Function of Y The year, Y, can be used as the input of a function where the unemployment rate, U, is the output. For each year there is one and only one output; therefore, U is a function of Y. Note that different inputs (such as 1990 and 1995) can have the same output (5.6) and still satisfy the conditions of a function.
5.6
Y is NOT a Function of U The unemployment rate, U, can NOT be used as the input of a function where the year, Y, is the output. When U 5.6, there are two corresponding values for Y, 1990 and 1995, and this would violate the condition for a function of a unique (one and only one) output for each input. Therefore Y is NOT a function of U.
1990 1995
Remember that each input can have only one output to satisfy the conditions of a function.
S O L U T I O N
The input and output of a function on a graph How would the axes be labeled for each graph of the following functions? a. Density of water is a function of temperature. b. Radiation intensity is a function of wavelength. c. A quantity Q is a function of time t. Figure 1.10 shows how to label the input and output of these functions on a graph. a.
c. Q
b.
Temperature
Quantity
4
Radiation intensity
E X A M P L E
Density of water
22
10/27/10
t Wavelength
Time
Figure 1.10 Various labels for axes, depending on the context.
How to tell if a graph represents a function: The vertical line test For a graph to represent a function, each value of the input on the horizontal axis must be associated with one and only one value of the output on the vertical axis. If you can draw
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a vertical line that intersects a graph in more than one point, then at least one input is associated with two or more outputs, and the graph does not represent a function. Vertical Line Test If there is a vertical line that intersects a graph more than once, the graph does not represent a function. The graph in Figure 1.11 represents y as a function of x. For each value of x, there is only one corresponding value of y. No vertical line intersects the curve in more than one point. The graph in Figure 1.12 does not represent a function. One can draw a vertical line (an infinite number, in fact) that intersects the graph in more than one point. Figure 1.12 shows a vertical line that intersects the graph at both (4, 2) and (4, 2). That means that the value x 4 does not determine one and only one value of y. It corresponds to y values of both 2 and 2. y
y
5
5
(4, 2)
–5 5
x –5
5
x
(4, –2)
–5
–5
Figure 1.11 The graph represents y as a function of x since there is no vertical line that intersects the curve at more than one point.
Figure 1.12 The graph does not represent y as a function of x since there is at least one vertical line that intersects this curve at more than one point.
EXPLORE & EXTEND
1.3
Is It a Function? Consider the set of data on the National Collegiate Athletic Association (NCAA) championship basketball games in the table below. Which variables could be used as the input of a function? What about the output? What must be true of the scatter plots of the functions? Of the scatter plots that are not functions? Year
Number of Wins
University of Arizona Tournament Seed
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
27 28 24 28 20 30 20 20 19 21
1 2 3 1 9 3 8 8 10 12
Note: A tournament seed is an assigned number between 1 and 16 that ranks a team’s potential opponents at every future stage of the NCAA competition.
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Algebra Aerobics 1.3 1. Which of the following tables represent functions? Justify your answer. Input
Output
Input
Output
1 2 3 4
5 8 8 10
1 2 2 4
5 7 8 10
Table A
Table B
Graph A
Graph B
Graph C
5. Consider the scatter plot in the graph below. Is weight a function of height? Is height a function of weight? Explain your answer. Graph of Weight versus Height
2. Does the following table represent a function? If so, why? If not, how could you change the values in the table so that it represents a function? Output
1 1 3 4
5 7 8 10
10
x –10
150 140 130 120 110 100 46
3. Refer to the following graph. Is y a function of x? y
Weight (pounds)
Input
160
10
4. Which of the graphs in the next column represent functions and which do not? Why?
50 54 58 Height (inches)
62
6. Consider the following table. a. Is D a function of Y? b. Is Y a function of D? Y
1992
1993
1994
1995
1996
1997
D
$2.50
$2.70
$2.40
2$0.50
$0.70
$2.70
7. a. Write an equation for computing a 15% tip in a restaurant. Does your equation represent a function? If so, what are your choices for the independent and dependent variables? b. How much would the equation suggest you tip for an $8 meal? c. Compute a 15% tip on a total check of $26.42.
Exercises for Section 1.3 A graphing program is required for Exercise 12. 1. The following table gives the high temperature in Santa Cruz, California, for each of five days in January, 2010. a. Is the temperature a function of the date? b. Is the date a function of the temperature? Date
High Temperature Santa Cruz, California
Jan. 1 Jan. 2 Jan. 3 Jan. 4 Jan. 5
62F 65F 64F 66F 64F
Source: www.weather.com.
2. Which of the following tables describe functions? Explain your answer. a. Input value Output value
22 28
21 21
0 0
1 1
2 8
b. Input value Output value
0 24
1 22
2 0
1 2
0 4
c. Input value Output value
10 3
7 6
4 9
7 12
10 15
d. Input value Output value
0 3
3 3
9 3
12 3
15 3
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3. Determine whether each set of points represents a function. (Hint: It may be helpful to plot the points.) a. (2, 6), (4, 6), (1, 3), (4, 3) b. (2, 2), (3, 2), (4, 2), (6, 2) c. (2, 3), (2, 3), (–2, 3), (2, 3) d. (1, 2), (1, 0), (1, 1), (1, 2) 4. Consider the accompanying table, listing the weights (W) and heights (H) of five individuals. Based on this table, is height a function of weight? Is weight a function of height? Justify your answers. Weight W (lb)
Height H (in)
120 120 125 130 135
54 55 58 60 56
i.
x
y
0 1 2 3 4
5 6 7 8 9
ii.
x
y
0 1 2 3 4
1 2 5 10 17
iii.
x
y
0 1 2 3 4
3 3 3 3 3
b. Which of your equations represents y as function of x? Justify your answer.
a.
x
y
0 1 2 3 4
0 3 6 9 12
b.
x
y
0 1 2 3 4
–2 1 4 7 10
c.
x
y
0 1 2 3 4
0 –1 –4 –9 –16
9. The basement of a large department store features discounted merchandise. Their policy is to reduce the previous month’s price of the item by 10% each month for 5 months, and then give the unsold items to charity.
Urbanization Effects on Streams Biotic integrity (B-IBI)
7. a. Find an equation that represents the relationship between x and y in each of the accompanying tables.
8. For each of the accompanying tables find a function that takes the x values and produces the given y values.
5. The accompanying graph uses a measure of urbanization as the percent of impervious area (area covered by materials such as asphalt, concrete, and so on that prevent the absorption of water) and shows the relationship to biotic integrity (the capability of supporting and maintaining a natural habitat).
50 40
a. Let S1 be the sale price for the first month and P the original price. Express S1 as a function of P. What is the price of a $100 garment on sale for the first month?
30 20 10 0
25
0
10
20 30 40 50 Percent impervious area
60
Source: http://www.seattlecentral.edu/qelp/sets/077/077.html.
Use the graph to determine if the level of biotic integrity is a function of the percent of imperious area. Explain your answer. 6. Consider the accompanying graph describing the sales of organic products from 2005 to 2008. Use the vertical line test to determine if the percent change in sales of organic products is a function of time. Explain your answer. Organic Product Sales 30
Percent Change
d. Let S5 be the sale price for the fifth month and P the original price. Express S5 as a function of P. What is the final price of a $100 garment on sale for the fifth month? By what total percentage has the garment now been reduced from its original price? 10. Write a formula to express each of the following sentences:
b. The time in Paris is 6 hours ahead of New York. Use P for Paris time and N for New York time to express P as a function of N. (Represent your answer in terms of a 12-hour clock.) How would you adjust your formula if P comes out greater than 12?
20 15 Year-over-year change in unit sales volume of organic products
5 0 '05
c. Let S3 be the sale price for the third month and P the original price. Express S3 as a function of P. What is the price of a $100 garment on sale for the third month?
a. The sale price is 20% off the original price. Use S for sale price and P for original price to express S as a function of P.
25
10
b. Let S2 be the sale price for the second month and P the original price. Express S2 as a function of P. What is the price of a $100 garment on sale for the second month?
'06
'07
Source: The Nielsen Company.
'08
c. For temperatures above 08F the wind chill effect can be estimated by subtracting two-thirds of the wind speed (in miles per hour) from the outdoor temperature. Use C for the effective wind chill temperature, W for wind speed, and T for the actual outdoor temperature to write an equation expressing C in terms of W and T.
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11. Determine whether y is a function of x in each of the following equations. If the equation does not define a function, find a value of x that is associated with two different y values. a. y 5 x2 1 1 b. y 5 3x 2 2
13. If we let D stand for ampicillin dosage expressed in milligrams and W stand for a child’s weight in kilograms, then the equation
c. y 5 5 d. y2 5 x
D 5 50W
12. (Graphing program required.) For each equation below, write an equivalent equation that expresses z in terms of t. Use technology to sketch the graph of each equation. Is z a function of t? Why or why not? a. 3t 2 5z 5 10 b. 12t2 2 4z 5 0
c. 2(t 2 4) 2 (z 1 1) 5 0
gives a rule for finding the safe maximum daily drug dosage of ampicillin (used to treat respiratory infections) for children who weigh less than 10 kilograms (about 22 pounds).12 a. What are logical choices for the independent and dependent variables? b. Does the equation represent a function? Why? c. Generate a small table and graph of the function.
1.4 The Language of Functions Not all equations represent functions. (See p. 23, Figure 1.12 where the graph of the equation does not pass the vertical line test.) But functions have important qualities, so it is useful to have a way to indicate when a relationship is a function.
Function Notation When a quantity y is a function of x, we can write y is a function of x or in abbreviated form, we say: y equals “f of x” or using function notation, y 5 f (x) The expression y 5 f (x) means that the rule f is applied to the input value x to give the output value, f(x): output 5 f(input) or
dependent variable 5 f(independent variable) The letter f is often used to denote the function, but we could use any letter, not just f.
Understanding the symbols Suppose we create a function where the input is multiplied by three and 1 is added to the product. We could write this function as y 5 3x 1 1
(1)
or with function notation as R(x) 5 3x 1 1
where y 5 R (x)
(2)
Equations (1) and (2) represent the same function, but with function notation we name the function—in this case R—and identify the input, x, and output, 3x 1 1. Function notation can provide considerable economy in writing and reading, by using R(x) instead of the full expression to represent the function. 12 Information extracted from Anna M. Curren and Laurie D. Muntlay, Math for Meds; Dosages and Solutions, 6th ed., San Diego: W. I. Publications, 1990.
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The Language of Functions If y is a function, f, of x, then y f(x) where
f is the name of the function, y is the output or dependent variable, x is the input or independent variable. f (input) output f (independent) dependent
Finding Output Values: Evaluating a Function Function notation is particularly useful when a function is being evaluated at a specific input value. Suppose we want to find the value of the previous function R(x) when our input value is 10. Using Equation (1) we would say, “find the value of y when x 10.” With function notation, we simply write R(10). To evaluate the function R at 10 means calculating the value of the output when the value of the input is 10: R (input) 5 output R(x) 5 3x 1 1 Substitute 10 for x R(10) 5 3(10) 1 1 5 31 So, applying the function rule R to the input value of 10 gives an output value of 31. Common Error The expression f(x) does not mean “f times x.” It means the function f evaluated at x.
E X A M P L E
1
S O L U T I O N
Using function notation with equations Evaluate f(5), f(0), and f(22) for the function f(x) 5 2x2 1 3. To evaluate f(5), we replace every x in the formula with 5. Given
f (x) 5 2x2 1 3
substitute 5 for x
f(5) 5 2(5) 2 1 3 5 2(5)(5) 1 3 5 53
Similarly, if x 0, if x 2,
f(0) 5 2(0) 2 1 3 5 3 f (22) 5 2(22) 2 1 3 5 (2 ? 4) 1 3 5 11
Finding Input Values: Solving Equations In Example 1 above, we found output values when we knew the input. Sometimes the situation is reversed and we know the output and want to find the corresponding input.
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2
What is the value of the input? a. Use the sales tax function T 0.06P to find the purchase price P when the sales tax T on a purchase is $8.40. b. Given f (x) 5x 2.5, find x when f (x) 21.
S O L U T I O N
a. We want to find P when T $8.40, so substitute $8.40 for T and then solve the equation:
E X A M P L E
T 0.06P substitute $8.40 for T
$8.40 0.06P
divide both sides by 0.06
$140 P
The purchase price is $140 when the sales tax is $8.40. b. To find an input value for x that results in f (x) 21, solve the equation 21 5x – 2.5 add 2.5 to each side divide by 5
23.5 5x 4.7 x
When the input is x 4.7, then f(x) 21 or equivalently f (4.7) 21.
Finding Input and Output Values From Tables and Graphs E X A M P L E
3
S O L U T I O N
E X A M P L E
4
S O L U T I O N
Using function notation with data tables Use Table 1.6 to fill in the missing values: a. S(0) ? b. S(1) ? c. S(?) 4
Input, x Output, S(x)
21
–2
0
1
2
1
4
0
1
4
Table 1.6
a. S(0) means to evaluate S when the input x 0. The table says that the corresponding output is also 0, so S(0) 0. b. S(1) 1. c. S(?) 4 means to find the input when the output is 4. When the output is 4, the input is 2 or 2, so S(2) 4 and S(2) 4. Using function notation with graphs Use the graph in Figure 1.13 to estimate the missing values: a. f (0) ? b. f (5) ? c. f (?) 0 Remember that by convention the horizontal axis represents the input (or independent variable) and the vertical axis represents the output (or dependent variable). a. f (0) 2 b. f (5) 1.5 c. f (10) 0 and f (10) 0
3
f (x)
–15
0
15
Figure 1.13 Graph of a function.
Rewriting Equations Using Function Notation In order to use function notation, an equation needs to be in the form output 5 some rule applied to input or equivalently dependent variable 5 some rule applied to independent variable
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Translating an equation into this format is called putting the equation in function form. Many graphing calculators and computer graphing programs accept only equations in function form. To put an equation into function form, we first need to identify the independent and the dependent variables. The choice is sometimes obvious, at other times arbitrary. If we use the mathematical convention that x represents the input or independent variable and y the output or dependent variable, when we put equations into function form, we want to solve for y or y 5 some rule applied to x E X A M P L E
5
S O L U T I O N
Does the equation represent a function? Analyze the equation 4x 3y 6. Decide whether or not the equation represents y as a function of x. If it does, write the relationship using function notation. First, put the equation into function form. Since y is the output, then given the equation
4x 2 3y 5 6
subtract 4x from both sides
23y 5 6 2 4x
divide both sides by 23
23y 6 2 4x 5 23 23 6 24x 1 23 23
simplify
y5
simplify and rearrange terms
4 y5 x22 3
We now have an expression for y in terms of x. Using technology or by hand, we can generate a table and graph of the equation (see Table 1.7 and Figure 1.14). Since the graph passes the vertical line test, y is a function of x. 6
x
y
3 0 3 6
6 2 2 6
y
x –6
6
–6
Figure 1.14 Graph of y 5 43x 2 2.
Table 1.7
If we name our function f, then using function notation, we have y 5 f(x) E X A M P L E
6
S O L U T I O N
where f (x) 5 43x 2 2
An equation where the input gives more than one output Analyze the equation y 2 x 0. Generate a graph of the equation. Decide whether or not the equation represents a function. If the equation represents a function, write the relationship using function notation. Assume y is the output. Put the equation in function form: Given the equation
y2 2 x 5 0
add x to both sides
y2 5 x
To solve this equation, we take the square root of both sides of the equation and we get y 5 6 !x
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This gives us two solutions for any value of x 0 as shown in Table 1.8. For example, if x 4, then y can either be 2 or 2 since both 22 4 and (2)2 4. 5
y
(4, 2)
x 0 1 2 4 9
y
x –1
0 1 or 21
!2 or 2!2 2 or 22 3 or 23
0
5
(4, –2)
–5
Figure 1.15 Graph of the equation y2 5 x is not a function.
Table 1.8
The graph of the equation in Figure 1.15 does not pass the vertical line test. In particular, the solutions (4, 2) and (4, 2) lie on the same vertical line. So y is not a function of x and we cannot use function notation to represent this relationship.
Algebra Aerobics 1.4a 1. Given g(x) 5 3x, evaluate g(0), g(21), g(1), g(20), and g(100). 2. Consider the function f (x) 5 x2 2 5x 1 6. Find f(0), f (1), and f (23). 2 3. Given the function f(x) 5 , evaluate f (0), x21 f(21), and f(23).
9. From the table below, find f(0) and f(20). Find two values of x for which f (x) 10. Explain why f (x) is a function.
4. Determine the value of t for which each of the functions has a value of 3. r(t) 5 5 2 2t
p(t) 5 3t 2 9
m(t) 5 5t 2 12
In Problems 5–7 solve for y in terms of x. Determine if y is a function of x. If it is, rewrite using f(x) notation. 5. 2(x 2 1) 2 3(y 1 5) 5 10 6. x2 1 2x 2 y 1 4 5 0 7. 7x 2 2y 5 5 8. From the accompanying graph, estimate f (4), f (1), f (0), and f (3). Find two approximate values for x such that f(x) 0.
x
f(x)
0 10 20 30 40
20 10 0 10 20
10. From the accompanying graph, estimate f (3), f (0), f (1), and f (3). Find two approximate values of x for which f (x) 0 and f (x) 1.5.
4
y f (x)
Graph of f (x). y 2 –4 –4
4
4
x
–4
–4
x
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Domain and Range A function is often defined only for certain values of the input (or independent variable) and corresponding values of the output (or dependent variable).
Domain and Range of a Function The domain of a function is the set of possible values of the input. The range is the set of corresponding values of the output.
Many times, especially under real-world conditions, there are restrictions on the domain and the range. The restrictions are often implied, as in Example 7 below. As we study each family of functions, we return to the question of whether there are restrictions on the domain and range. E X A M P L E
7
Finding a reasonable domain and range In the sales tax example at the beginning of this section, we used the equation T 5 0.06P to represent the sales tax, T, as a function of the purchase price, P (where all units are in dollars). What are the domain and range of this function?
S O L U T I O N
Since negative values for P are meaningless, P is restricted to dollar amounts greater than or equal to zero. In theory there is no upper limit on prices, so we assume P has no maximum amount. In this example, the domain is all dollar values of P greater than or equal to 0 We can express this more compactly as domain ⫽ dollar values of P $ 0 What are the corresponding values for the tax T? The values for T in our model cannot be negative. As long as there is no maximum value for P, there will be no maximum value for T. So, the range is all dollar values of T greater than or equal to 0 or
range ⫽ dollar values of T $ 0
(Note: The symbol ⱖ means “greater than or equal to and the symbol ⱕ means “less than or equal to”)
Representing the domain and range with interval notation Interval notation is often used to represent the domain and range of a function.
Interval Notation A closed interval [a, b] indicates all real numbers x for which a ⱕ x ⱕ b. Closed intervals include their endpoints. An open interval (a, b) indicates all real numbers x for which a ⬍ x ⬍ b. Open intervals exclude their endpoints. Half-open (or equivalently half-closed) intervals are represented by [a, b) which indicates all real numbers x for which a ⱕ x ⬍ b or (a, b] which indicates all real numbers x for which a ⬍ x ⱕ b.
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For example, if the domain is values of x greater than or equal to 50 and less than or equal to 100, then domain 5 all x values with 50 # x # 100 5 [50, 100] If the domain is values of x greater than 50 and less than 100, then domain 5 all x values with 50 , x , 100 5 interval (50, 100) If we want to exclude 50 but include 100 as part of the domain, we would represent the interval as (50, 100]. The interval can be displayed on the real number line as:
50
100
In general, a hollow dot indicates exclusion and a solid dot inclusion. Note: Since the notation (a, b) can also mean the coordinates of a point, we will say the interval (a, b) when we want to refer to an interval.
Visualizing the domain and range Recall that by convention, the input of a function is represented on the horizontal axis and the output on the vertical axis. Figure 1.16 illustrates the set of inputs or domain [4, 7] and the outputs or range of f (x) [2, 6].
8
y
f (x) range
32
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8
–8
x
domain –8
Figure 1.16
E X A M P L E
8
Finding the domain and range from a graph The graph in Figure 1.17 shows the water level of the tides in Pensacola, Florida, over a 24-hour period. Are the Pensacola tides a function of the time of day? If so, identify the independent and dependent variables. Use interval notation to describe the domain and range of this function.
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Water level (in centimeters)
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33
+10
0
–10
0
4
8 12 16 Time (in hours)
20
24
Figure 1.17 Diurnal tides in a 24-hour period in Pensacola, Florida. Source: Adapted from Fig. 8.2 in Oceanography: An Introduction to the Planet Oceanus, by Paul R. Pinet. Copyright © 1992 by West Publishing Company, St. Paul, MN. All rights reserved.
S O L U T I O N
The Pensacola tides are a function of the time of day since the graph passes the vertical line test. The independent variable is time, and the dependent variable is water level. The domain is from 0 to 24 hours, and the range is from about 210 to 110 centimeters. Using interval notation: domain 5 [0, 24] range 5 [210, 10]
9
Summarizing music cassette sales The following graph in Figure 1.18 shows music cassette tape sales, S(t) over time, t. Music Cassette Sales in U.S. from 1975 to 2005 480,000,000
Number cassettes sold
E X A M P L E
360,000,000
240,000,000 S(t) 120,000,000
t 0 19751977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 Year
Figure 1.18 Source: Support Team, www.swivel.com.
Find the following: a. ________ is a function of __________. b. Input (with units): _______ Output (with units): ______ c. Estimate: Domain: _______ Range: ________ d. Estimate: S (1983 ) < __________ t such that S(t) < 120,000,000 What are the practical implications of these statements? e. Estimate the highest and lowest values of S(t). What are the practical implications of these statements?
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S O L U T I O N
a. Cassette sales S(t) is a function of time, t. For each input t, there is one and only one output, S(t). b. Input: time in years Output: number of cassettes sold c. Domain: 1975 to 2005 Range: approx. 1 million to 450 million d. S (1983) < 240,000,000, which means that in 1983 there were approximately 240 million cassettes sold. S (1980) < 120,000,000 and S (1999) < 120,000,000, which means that in both 1980 and in 1999 there were approximately 120 million cassettes sold. e. The highest value of S(t) is approximately 450 million in 1988. The lowest value is approximately 1 million in 2005. Cassette sales soared from the mid-1970s up until 1988, reaching 450 million. But with CD sales and the Internet market, cassette sales declined rapidly to about 1 million in 2005.
Algebra Aerobics 1.4b 1. Express each of the following using interval notation. a. x . 2 b. 4 # x , 20 2. Express the given interval as an inequality. a. [23, 10) b. (22.5, 6.8] 3. Express each of the following statements in interval notation. a. Harry’s GPA is at least 2.5 but at most 3.6. b. A good hitter has a batting average of at least 0.333. c. Starting annual salary at a position is anything from $35,000 to $50,000 depending upon experience. 4. You invest $1,500 in a 10-year bond that gives you 4% interest on $1,500 each year for 10 years. a. Construct a function that represents the amount of interest, I(n), earned after n years. What is the domain and range in this context? What is the input and output? b. Find I(6). c. Find n if I(n) 240.
5. Determine the domain and the range of the functions whose graphs are given. 5
y
–5
5
x
–5
Graph A 5
–5
y
5
x
–5
Graph B
Exercises for Section 1.4 1. Given T(x) 5 x2 2 3x 1 2, evaluate T(0), T(21), T(1), and T(25). 2. Given f (x) 5
x , evaluate f(0), f(21), f(1), f(20), and x21
f(100). 3. Assume that for persons who earn less than $20,000 a year, income tax is 16% of their income. a. Generate a formula that describes income tax in terms of income for people earning less than $20,000 a year.
b. What are you treating as the independent variable? The dependent variable? c. Does your formula represent a function? Explain. d. If it is a function, what is the domain? The range? 4. Suppose that the price of gasoline is $3.09 per gallon. a. Generate a formula that describes the cost, C, of buying gas as a function of the number of gallons of gasoline, G, purchased. b. What is the independent variable? The dependent variable? c. Does your formula represent a function? Explain.
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4. (continued) d. If it is a function, what is the domain? The range? e. Generate a small table of values and a graph. 5. The cost of driving a car to work is estimated to be $2.00 in tolls plus 32 cents per mile. Write an equation for computing the total cost C of driving M miles to work. Does your equation represent a function? What is the independent variable? What is the dependent variable? Generate a table of values and then graph the equation. 6. For each equation, write the equivalent equation that expresses y in terms of x. If the equation represents a function, use function notation to express the relationship. a. 3x 1 5x 2 y 5 3y c. x(x 2 1) 1 y 5 2x 2 5 b. 3x(5 2 x) 5 x 2 y d. 2(y 2 1) 5 y 1 5x(x 1 1) 7. If f (x) 5 x2 2 x 1 2, find: a. f(2) b. f(21)
c. f(0)
9. Look at the accompanying table. a. Find p(24), p(5), and p(1). b. For what value(s) of n does p(n) 5 2?
p(n)
a. Which variable(s) could be used as the input of a function? Explain your answer. b. What is the input (with units)? Output (with units)? c. What is the domain? The range? d. If g represents this function, complete the following for one pair of values: g(________) ________. 12. Pennsylvania has a flat-rate personal income tax. It can be represented as T(I) 0.0307I, where I is the income in dollars and T(I) is the tax owed in dollars. a. What is the income tax rate in Pennsylvania? b. What is the input? The output? c. Estimate T(35,000). What is its meaning? d. Estimate I if T(I) 1535. What is its meaning? 13. From the accompanying graph of y f(x): a. Find f(2), f(1), f(0), and f(1). b. Find two values of x for which f (x) 3. c. Estimate the range of f. Assume that the arms of the graph extend upward indefinitely.
d. f(25)
8. If g(x) 5 2x 1 3, evaluate g(0), g(1), and g(21).
n
5
24
23
22
21
0
1
2
3
4
0.063
0.125
0.25
0.5
1
2
4
8 16
35
y f (x)
5 32 –5
5
x
10. The data in the table below show the median weight for various ages of Caucasian men. (1 kg < 2.2 lbs). –5
20 25 30 35 40 45 50
55
60
65 70 75
Weight (kg) 72 76 78 80 82 83 83.5 83.5 82.5 82 80 78 Source: www.halls.md/chart/men-weight-w.htm.
a. Which variable(s) could be used as the input of a function? Explain your answer. b. What is the input (with units)? Output (with units)? c. What is the domain? The range? d. If f represents this function, complete the following for one pair of values: f (________ ) ________. 11. The data in the table below show the depth and corresponding velocity of the Columbia River in the state of Washington. Depth (ft)
Velocity (ft/sec)
Depth (ft)
Velocity (ft/sec)
0.7 2.0 2.6 3.3 4.6 5.9
1.55 1.11 1.42 1.39 1.39 1.14
7.3 8.6 9.9 10.6 11.2
0.91 0.59 0.59 0.41 0.22
Source: Savini, J., and Bodhaine, G. L. (1971), Analysis of current meter data at Columbia River gauging stations, Washington and Oregon; USGS Water Supply Paper 1869-F.
14. Given f (x) 1 0.5x and g(x) 5 x2 1 1, evaluate: a. f(0), g(0) c. f(2), g(1) b. f(–2), g(–3) d. f(3), g(3) 15. If f(x) (2x 1)2, evaluate f(0), f(1), and f(2). 16. The following graph shows live births, B(t), over time, t, for the years 1955–2007. Live Births in the USA 1955–2007
Births (in thousands)
Age(years)
4500 4250 4000 B(t) 3750 3500 3250 3000 2750 2500 2250 t 2000 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year
Source: U.S. Statistical Abstract, 2010.
Determine the following: a. ________ is a function of __________. b. Input (with units): ________ Output (with units): ________ c. Estimate Domain: ________ Range: ________ What is the practical meaning of these values?
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16. (continued) d. Estimate B(2000 ) < ________ What is the practical meaning of this value? e. Estimate t if B(t) 3750 What is the practical meaning of these values? 17. The following graph shows U.S. deaths, D(t), over time, t, for the years 1995–2006 in 1-year intervals. Deaths (in thousands)
Deaths in USA 1995–2006 2,460 2,440 2,420 2,400 D(t) 2,380 2,360 2,340 2,320 t 2,300 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Determine the following: a. ________ is a function of ________. b. Input (with units): ________ Output (with units): ________ c. Domain: ________ Range: ________ What is the practical meaning of these values? d. Estimate D(2000) < ________ What is the practical meaning of this value? e. Estimate t if D(t) 2390 What is the practical meaning of this value? f. Using Exercise 16, find B(2000) D(2000) and interpret its meaning.
Years
Source: U.S. Statistical Abstract, 2010.
1.5 Visualizing Functions In this section we return to the question: How does change in one variable affect change in another variable? Graphs are one of the easiest ways to recognize change. We start with four basic questions:
Is There a Maximum or Minimum Value? If a function has a maximum (or minimum) value, then it appears as the highest point (or lowest point) on its graph. E X A M P L E
1
Estimating the maximum and minimum from a graph Determine if each function in Figure 1.19 has a maximum or minimum, then estimate its value. y
y 100
g (x)
h(x)
100
x
x –20
200
20
–25
f (x)
–100
25
–100
–200
Figure 1.19 Graphs of ƒ(x), g(x), and h(x). S O L U T I O N
The function f (x) in Figure 1.19 appears to have a maximum value of 40 when x 5 25 but has no minimum value since both arms of the function extend indefinitely downward. The function g(x) appears to have a minimum value of 20 when x 5 5, but no maximum value since both arms of the function extend indefinitely upward. The function h(x) appears to have a maximum value of 200, which occurs when x 5 25, and a minimum value of 250, when x 5 5.
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When Is the Output of the Function Positive, Negative, or Zero? From a graph we can estimate over what intervals a function f (x) ⬎ 0, f (x) ⬍ 0 and when f (x) ⫽ 0. E X A M P L E
2
S O L U T I O N
Estimating the output of a function from a graph In Figure 1.19 on prior page, estimate the x interval when h(x) is positive, negative, or zero. Use interval notation when appropriate. h(x) ⬎ 0 over x intervals (⫺25, 0) and (10, 25] h(x) ⬍ 0 over x interval (0, 10) h(x) ⫽ 0 when x ⫽ 0 and 10
Is the Function Increasing or Decreasing?
Decreasing
Increasing
E X A M P L E
3
A function f is decreasing over a specified interval if the values of f (x) decrease as x increases over the interval. A function f is increasing over a specified interval if the values of f (x) increase as x increases over the interval. The graph of an increasing function rises as we move from left to right. The graph of a decreasing function falls as we move from left to right. Increasing and decreasing production Figure 1.20 shows over 100 years of annual natural gas production in the United States. Create a 60-second summary about gas production from 1900 to 2009 in the U.S.
Annual U.S. Natural Gas Marketed Production 25000
Billion cubic feet
20000
15000
10000
5000
0 1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Year
Figure 1.20 Source: U.S. Energy Information Administration.
S O L U T I O N
Natural gas production in the United States overall showed a steady increase from 1900 to the early 1970s, increasing from a relatively small amount in 1900 to a peak of over 22,000 billion cubic feet in 1973. For the next 10 years production generally decreased, with a few exceptions, to about 16,000 billion cubic feet in the early 1980s, reaching a low not seen since the mid-1960s. From the mid-1980s to the beginning of the twentyfirst century there again was a steady increase in production. In the twenty-first century production at first decreased and then from 2006 to 2009 increased steadily, reaching 21,000 billion cubic feet, coming close to its all-time high in 1973.
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Is the Graph Concave Up or Concave Down? What does the concavity of a graph mean? The graph of a function is concave up if it bends upward and it is concave down if it bends downward.
Concave up
Concave down
Concavity is independent of whether the function is increasing or decreasing. Decreasing
Increasing Concave down
4
Concave up
Graphs are not necessarily pictures of events13 The graph in Figure 1.21 shows the speed of a roller coaster car as a function of time.
Speed
E X A M P L E
Concave down
Concave up
t0
t1
t2
t3
t4
Time
Figure 1.21
a. Describe how the speed of the roller coaster car changes over time. Describe the changes in the graph as the speed changes over time. b. Draw a picture of a possible track for this roller coaster. S O L U T I O N
a. The speed of the roller coaster car increases from t0 to t2, reaching a maximum for this part of the ride at t2. The speed decreases from t2 to t4. The graph of speed versus time is concave up and increasing from t0 to t1 and then concave down and increasing from t1 to t2. From t2 to t3 the graph is concave down and decreasing, and from t3 to t4 it is concave up and decreasing. b. A picture for a possible track of the roller coaster is shown in Figure 1.22. Notice how the track is an upside-down picture of the graph of speed versus time. (When the roller coaster car goes down the speed increases, and when the roller coaster car goes up, the speed decreases.) Height of track
38
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t0
t1
t2
t3
t4
Time
Figure 1.22 13 Example 4 is adapted from Shell Centre for Mathematical Education, The Language of Functions and Graphs. Manchester, England: University of Nottingham, 1985.
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Getting the Big Idea We now have the basic vocabulary for describing a function’s behavior. Think of a function and its graph as telling a story. We want to decipher not just the individual words, but the basic plot. In each case we should ask: Is there an overall pattern? Are there significant deviations from that pattern? E X A M P L E
5
Generating a rough graph Generate a rough graph of each of the following situations. a. A cup of hot coffee cooling. b. U.S. venture capital (money provided by investment companies to business start-ups) increased modestly but steadily in the mid-1990s, soared during the “dotcom bubble” (in the late 1990s), with a high in 2000, and then suffered a drastic decrease in 2001 back to pre-dotcom levels. After a few years of growth another sharp decline started in 2008. c. A simple predator-prey model: initially as the number of lions (the predators) increases, the number of gazelles (their prey) decreases. When there are not enough gazelles to feed all the lions, the number of lions decreases and the number of gazelles starts to increase. b.
Room temp.
U.S. Venture Capital 1995–2010
1995
2000
Time (minutes)
E X A M P L E
6
c.
Predator-Prey Model of Lions vs. Gazelles Number of animals
A Cup of Hot Coffee Cooling over Time
Venture capital ($)
a. Temperature (°F)
S O L U T I O N
2005
2010
Lions
Gazelles Time
Year
Looking for patterns You are a TV journalist. Summarize for your viewers the essence of the graph in Figure 1.23. The poverty rate is the percentage of the population living in poverty. Note: The vertical scale is used in two different ways on this graph. Number in Poverty and Poverty Rate: 1959 to 2007 Numbers in millions, rates in percent 50 45
Number in poverty
40
37.3 million
35 30 25 20
Poverty rate
15
12.5 percent
10 5 0 1959
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure 1.23 Source: U.S. Census Bureau, Current Population Survey, 1960 to 2008. Annual Social and Economic Supplements.
S O L U T I O N
Not surprisingly there is a general relationship between the number of people in poverty and the poverty rate, mirrored in the shapes of their graphs. In 1959 both hit a record high over the 48-year span from 1959 to 2007. Both poverty measures declined
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after that until about 1967. The poverty rate then continued to hover around 12.5%, close to a record low. However the number of people in poverty fluctuated, reaching 37.3 million in 2007, almost the 1959 high. E X A M P L E
7
Looking for deviations from a pattern The following graph on household net worth in the United States appeared in The New York Times in March 2009. Household net worth encompasses all of a household’s assets (such as housing, stocks, personal property) minus their total debts (such as mortgages, other unpaid loans, credit card debt). a. Describe how household net worth has changed since 1955. b. What title would you give to the graph for a newspaper article about household net worth in the United States? $70 trillion 60 50 40
As a result of the current economic downturn, the net worth of U.S. households fell at least $12.9 trillion since June 2007. United States household net worth Quarterly, adjusted for inflation
30 20 Not adjusted for inflation
10 0
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Source: Federal Reserve Board, via Have Analytics. The New York Times, March 12, 2009. New York Times Graphics
S O L U T I O N
a. From 1955 to 2000, household net worth in the United States (both adjusted and not adjusted for inflation) generally increased, with only a few relatively slight decreases at the end of the 1960s and from 1973 to 1975. The more dramatic decreases were in the twenty-first century. After falling for a few years at the beginning of the twenty-first century, household net worth began a steep climb to an all-time high of about $65 trillion dollars in 2007, which was followed by its steepest decline, losing almost $13 trillion. At end of the first decade of the twentyfirst century, household net worth had declined to slightly over $50 trillion, and if adjusted for inflation, was close to where it started at the beginning of the century. b. The title of the graph in The New York Times was “A Sudden Decline.” Another title could be “A Dramatic Loss after Growth for Years.”
EXPLORE & EXTEND
Bridget in the Bathtub Bridget, the 6-year-old daughter of a professor at the University of Pittsburgh, loved playing with her rubber duckie in the bath at night. Her mother drew the accompanying graph for her math class. It shows the water level (measured directly over the drain) in Bridget’s tub as a function of time.
Water level
1.5
0
t1
t2
t3
t4 Time
t5
t6
t7
t8
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Pick out the time period during which: • • • • •
The tub is being filled Bridget is entering the tub She is playing with her rubber duckie She leaves the tub The tub is being drained
Water in Containers: Sketch the Flow Assume that water is pouring at a constant rate into each of the containers shown. The height of water in the container is a function of the volume of liquid. Sketch a graph of this function for each container.
(a)
(b)
(c)
The rest of this text examines patterns in functions and their graphs. Families of functions—linear, exponential, logarithmic, power, polynomial, quadratic, and rational—provide useful mathematical models for describing the world around us.
Algebra Aerobics 1.5 1. Create a topic sentence for each of the following graphs for a newspaper article. a.
b. Accelerating Slide Quarterly newspaper print ad sales
5% (in 2007 dollars)
0% $50,233 –5%
–10%
0.4% 0.3% QI- QII- QIII- QIV- QI- QII- QIII- QIV- QI- QII- QIII- QIV- QI06 06 06 06 07 07 07 07 08 08 08 08 09 –2.6% –3.7% –6.4% –9.0% –10.2%
–15%
–11.6% –14.4% –16.1% –19.3%
2005 2007
2000
1995
1990
1985
–20% 1980
1975
1970
55,000 50,000 45,000 40,000 35,000 $38,771 30,000 25,000 20,000 15,000 10,000 5,000 0 1965
dollars
Real Median Household Income: 1967 to 2007
Note: Median household income data are not available before 1967. Source: U.S. Census Bureau, Current Population Survey, 1968 to 2008 Annual Social and Economic Supplements.
–20.6%
–25% –29.7% –30%
Source: Newspaper Association of America.
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2. Use the accompanying graph showing mean contract mortgage rates, D(t), over time, t, to estimate the following: a. Domain: Range: b. D(2008) < c. t if D(t) < 7.00 d. The maximum value for mortgage rates during the time period represented on the graph. In what year does the maximum occur? What are the approximate coordinates at the maximum point? e. The minimum value for mortgage rates. In what year does this occur? What are the coordinates of this point? Mean Contract Mortgage Rates in United States 1963 to 2009 15.00 14.00
National Average Contract Mortgage Rate
13.00 12.00 11.00
D(t )
10.00 9.00 8.00
S
S
t
t
Graph C
Graph D
4. Generate a rough sketch of the following situation. U.S. AIDS cases increased dramatically, reaching an all-time high for a relatively short period, and then consistently decreased, until a recent small increase. 5. Sketch a graph for each of the following characteristics, and then indicate with arrows where the functions are increasing and where they are decreasing. a. Concave up with a minimum point at (2, 1). b. Concave down with a maximum point at (3, 2). 6. Estimate the interval for x when f(x) 0, when f (x) 0, and when f (x) 0 for each function in Graphs A and B.
7.00 6.00 5.00 2009
2007
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
1963
4.00
4
y
Source: www.Mortgage-X.com. Note: The contract mortgage rate is based on conventional fixed and adjustable rate mortgages for a given time period. –4
3. Choose which of the graphs is the “best” graph to describe the following situation. Speed (S) is on the vertical axis and time (t) is on the horizontal axis. A child in a playground tentatively climbs the steps of a large slide, first at a steady pace, then gradually slowing down until she reaches the top, where she stops to rest before sliding down.
4
f (x) –4
Graph A
4
y
f (x)
S
S
–4
4
–4
t
t Graph A
x
Graph B
Graph B
x
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43
Exercises for Section 1.5 Graphing program required for Exercise 27.
3. The following graph shows the daily mean Arctic temperature T(d), in kelvins, over d days of the year. (Note: Kelvin is a unit of temperature 273.15 K ⫽ 0° Celsius and 1 kelvin ⫽ ⫺272.15⬚ Celsius.)
1. Identify the graph (A or B) that a. Increases for 1 , x , 3 b. Increases for 2 , x , 5 c. Decreases for 22 , x , 2 d. Decreases for 3 , x , 5
Mean Arctic temperature (in kelvins)
Mean Arctic Temperature
y 6 4 2 –4
–2
2
4
6
x
–2 –4 –6
Graph A y
280 275 273.15 K = 0° Celsius (freezing point) 270 265 260
T (d )
255 250 245 240 235
0
50
100
150
200
250
300
350
d
Days of year
6
Source: http://ocean.dmi.dk/arctic/meant80n.uk.php.
4 2 –4
–2
2
4
6
Find the following: a. ________ is a function of ________. b. Input (with units): ________ Output (with units): ________ c. Estimate Domain: ________ Range: ________ d. Estimate T(100) < ________ Estimate the interval for d over which T(d ) ⬎ 273.15 (the melting point of ice). What are the practical implications of these statements? e. Estimate the maximum and minimum values of T(d). What are the practical implications of these statements? f. Describe the overall shape of the graph of T(d).
x
–2 –4 –6
Graph B
2. The following graph shows the number of square kilometers of deforestation, D(t), in the Brazilian Amazon during the years, t, from 1988 to 2009.
30000 25000 20000 15000 10000
t
2008 2009
2006
2004
2002
2000
1998
1996
1994
1992
0
1990
5000 1988
Sq. km deforested land per year
Deforestation in the Brazilian Amazon, 1988–2009 35000
4. For each of the following functions, a. Over which interval is the function decreasing? b. Over which interval is the function increasing? c. Does the function appear to have a minimum? If so, where? d. Does the function appear to have a maximum? If so, where? e. Describe the concavity.
Year
Source: ©luoman/iStockphoto
Find the following: a. ________ is a function of ________. b. Input (with units): ________ Output (with units): ________ c. Estimate the Domain: ________ Range: ________ d. Estimate D(1990) < D(2008) < Estimate t so that D(t) < 22,000 e. From the graph, estimate the maximum amount of land deforested in a particular year. In what year did that occur? In what year was the least amount of land deforested?
4
y
y
12
x –12
4
x
10
–4
–12
Graph A
–4
Graph B
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8. Look at the graph of y f(x) in the accompanying figure.
5. For the following function, 12
y
y
8
f(x) –6
6
x
x –8
12
–6
–12
a. b. c. d. e. f.
Over which interval(s) is the function positive? Over which interval(s) is the function negative? Over which interval(s) is the function decreasing? Over which interval(s) is the function increasing? Does the function appear to have a minimum? If so, where? Does the function appear to have a maximum? If so, where?
6. Choose which graph(s) A, B, C, D, match the description: As x increases, the graph is: a. Increasing and concave up b. Increasing and concave down c. Concave up and appears to have a minimum value at (3, 2) d. Concave down and appears to have a maximum value at (3, 2) y 5
4
y
x 2
–8
x –3
3 –1
–10
Graph A
Graph C
y 6 10
y
9. Use the graph of Exercise 8 to answer the following questions about f (x). a. Over which interval(s) is f(x) 0? b. Over which interval(s) is f(x) 0? c. Over which interval(s) is f(x) increasing? d. Over which interval(s) is f(x) decreasing? e. How would you describe the concavity of f (x) over the interval (0, 5) for x? Over (5, 8) for x? f. Find a value for x when f (x) 4. g. f(8) ? 10. Match each graph with the best description of the function. Assume that the horizontal axis represents time, t. i. The height of a ball thrown straight up is a function of time. ii. The distance a truck travels at a constant speed is a function of time. iii. The number of daylight hours is a function of the day of the year. iv. The temperature of a pie baking in an oven is a function of time.
x –8
–2
t
2
–2
Graph B
Q
Q
x 6
–2
Find f (6), f(2), and f(12). Find f (0). For what values of x is f(x) 0? Is f(8) 0 or is f(8) 0? How many times would the line y 1 intersect the graph of f (x)? f. What are the domain and range of f (x)? g. What is the maximum? The minimum? a. b. c. d. e.
Graph D
7. Examine each of the graphs in Exercise 6. Assume each graph describes a function f(x). The arrows indicate that the graph extends indefinitely in the direction shown. a. For each function estimate the domain and range. b. For each function estimate the x interval(s) where f(x) 0. c. For each function estimate the x interval(s) where f(x) 0.
t
Graph A
Graph C Q
Q
t
t Graph B
Graph D
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11. Choose the best graph to describe the situation. a. A student in a large urban area takes a local bus whose route ends at the college. Time, t, is on the horizontal axis and speed, s, is on the vertical axis. s
t
t
Graph A
Unemployment Rate in the United States
s
t Graph B
45
13. The accompanying graph shows the unemployment rate for those people 16 years and older from January 1999 to January 2010.
Percent of unemployed
s
Visualizing Functions
Graph C
8
6
4
01 /9 9 01 /0 0 01 /0 1 01 /0 2 01 /0 3 01 /0 4 01 /0 5 01 /0 6 01 /0 7 01 /0 8 01 /0 9 01 /1 0
b. What graph depicts the total distance the student traveled in the bus? Time, t, is on the horizontal axis and distance, d, is on the vertical axis.
10
Month/Year
Source: Bureau of Labor Statistics, www.bls.gov. d
d
d
t Graph D
t
t Graph F
Graph E
12. The accompanying graph shows the changes in the height of tides in San Francisco Bay, h(t), over time, t, between January 27, 2010, and January 29, 2010. Height of Tides in San Francisco 7.00 Height (feet relative to MLLW)
a. Estimate the maximum and the minimum unemployment rate between January 1999 and January 2010. b. Estimate the time frame when the unemployment rate was experiencing an overall decrease. c. When does the unemployment rate seem to be rising the fastest? d. Write a 60-second summary about the unemployment rate from January 1999 to January 2010. 14. a. In the accompanying graphs, estimate the coordinates of the maximum and minimum points (if any) of the function. b. Specify the interval(s) over which each function is increasing. c. Estimate when f (x) 0, f (x) 0, and f (x) 0.
6.00 5
5.00 4.00
y
5
y
h(t)
3.00 2.00
–5
5
x
–4
4
1.00 t
0.00 –1.00
–5
–2.00 01/27 01/27 01/27 01/27 01/28 01/28 01/28 01/28 01/29 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am Date/Time
Source: tidesandcurrents.hoaa.gov. Note: MLLW (mean lower low water level) is the mean lowest tide.
Estimate the following: a. The t interval(s) when h(t) 0. Explain what this means about the tide level. b. The t interval(s) when h(t) 0. Explain what this means about the tide level. c. The value(s) of t when h (t) 0. d. One interval when h(t) is decreasing. Explain what this means about the tide level. e. The value of h(t) when t is 6 p.m., 1/27. f. The t interval(s) when the graph is concave down. g. The maximum and minimum values of h(t). Explain these results in terms of tide levels.
–5
Graph A
Graph B
15. Use the graph to estimate: a. When f(x) 0, when f(x) 0, and when f(x) 0 b. The coordinates of the maximum and minimum for f(x) c. When f (x) is increasing and decreasing d. The domain and range of f(x) y 10 8
f(x)
6 4 2
x –10 –8
–6
–4
–2
2 –2 –4
4
6
8
10
x
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16. Sketch a plausible graph for each of the following and label the axes. a. The amount of snow in your backyard each day from December 1 to March 1. b. The temperature during a 24-hour period in your home town during one day in July. c. The amount of water inside your fishing boat if your boat leaks a little and your fishing partner bails out water every once in a while. d. The total hours of daylight each day of the year. e. The temperature of an ice-cold drink left to stand. 17. Examine the accompanying graph, which shows the populations of two towns.
19. The accompanying graph shows the world population over time and future predictions. Historical World Population with Predictions 12,000 10,000 Population (millions)
CHAPTER 1
46
10/27/10
8,000 6,000 4,000 2,000
Population Trends: Erie, PA, and Trenton, NJ
0 0
Erie Trenton
140 120
250
500
750 1000 1250 1500 1750 2000 2250 Year
Source: www.worldonline.n1/invd/world/whist2.html.
100
a. Over what interval does the world population show dramatic growth? b. Does the dramatic growth slow down? c. Write a topic sentence for a report for the United Nations.
80 60 40 20 0 1940
1950
1960
1970
1980
1990
2000
2010
Year
Source: www.demographics.com.
a. What is the range of population size for Erie? For Trenton? b. During what years did the population of Erie increase? c. During what years are both populations relatively stable? d. When were the populations approximately equal? 18. The graph of the following function f(x) has a domain of ( ` , ` ).
5
y f (x)
20. Make a graph showing what you expect would be the relative ups and downs throughout the year of sales (in dollars) of a. Turkeys c. Bathing suits in your state b. Candy d. Textbooks at your school bookstore 21. Examine the graphs of military reserve enlisted personnel over the years 1990 to 2007. Female Enlisted Reserve Personnel 220 Number (thousands)
Number in thousands
160
210 200 190 180 170 160 150 1990 1992 1994 1996 1998 2000 2002 2004 2006
–5
5
Year
x
Male Enlisted Reserve Personnel 1,500
–5
When is f(x) 0? Over what x interval(s) is f(x) 0? Over what x interval(s) is f(x) 0? Estimate the maximum and minimum value for f (x) over the domain of the function. e. Over what x interval(s) is f(x) concave up and increasing? Concave up and decreasing?
a. b. c. d.
Number (thousands)
1,400 1,300 1,200 1,100 1,000 900 800 700 1990 1992 1994 1996 1998 2000 2002 2004 2006 Year
Source: U.S. Dept. of Defense, Official Guard and Reserve Manpower Strengths and Statistics, annual.
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21. (continued) a. In what year(s) did female and male enlisted personnel reach a maximum? b. What was the maximum and minimum for both men and women over the time interval [1990, 2007]? c. Describe the trends in number of female and male reserve enlisted personnel. d. Given the current state of world affairs, what would you expect would happen to the enlistment numbers for 2007 to the present? 22. Generate a rough sketch of a graph of internal pressure vs. time for the following situation: When a soda is removed from the fridge, the internal pressure is slightly above the surrounding air pressure. With the can unopened, the internal pressure soon more than doubles, stabilizing at a level three times the surrounding air pressure. 23. A student breaks her ankle and is taken to a doctor, who puts a cast on her leg and tells her to keep the foot off the ground altogether. After 2 weeks she is given crutches and can begin to walk around more freely, but at the beginning of the third week she falls and is resigned to keeping stationary again for a while. After 6 weeks from her first fall, she is given a walking cast in which she can begin to put her foot on the ground again. She is now able to limp around using crutches. Her walking speed slowly progresses. At 12 weeks she hits a plateau and, seeing no increase in her mobility, starts physical therapy. She rapidly improves. At 16 weeks the cast is removed and she can walk freely. Make a graph of the student’s mobility level during her recovery. 24. Every January 1 a hardy group called the L Street Brownies celebrates the New Year by going for a swim at the L Street Beach in South Boston. The water is always very cold, and swimmers adopt a variety of strategies for getting into it. The graph shows the progress of three different friends who join in the event, with percentage of body submerged on the vertical axis and time on the horizontal axis. Match the graphs to the descriptions below of how each of the friends manages to get completely submerged in the icy ocean. Swimmers in South Boston
Percent submerged
100 75 50 25 0 0
1
2
3
4
5
Time
Minutes
a. Ali has done this before and confidently walks in until his head is underwater; then he puts his head out and swims around a few minutes; then he walks out. b. Ben dashes in until the water is up to his knee, trips over a hidden rock, and falls in completely; he stands up and, since he is now totally wet, runs back out of the water.
47
c. Cat puts one foot in, takes it out again, and shivers. She makes up her mind to get it over with, runs until she is up to her waist, dives in, swims back as close to the water’s edge as she can get, stands up, and steps out of the water. 25. Examine the graphs of the functions y f(x) and y g(x). Complete the following: a. As x approaches ` , f(x) approaches ________. As x approaches ` , f(x) approaches ________. b. As x approaches ` , g(x) approaches ________. As x approaches ` , g(x) approaches ________. y
y 30
30
f (x)
g(x)
2
–2
x
x 2
–2
Graph B
Graph A
26. Describe the behavior of ƒ (x) in the accompanying figure over the interval ( ` , ` ) for x, using such words as “increases,” “decreases,” “concavity,” “maximumminimum,” and “approaches infinity.” y 4
f(x)
x –3
3
27. (Graphing program required.) This exercise is to be done with a partner. Name the partners person #1 and person #2. a. Person #1, using technology, graphs the function f(x) 0.5(x 3)(x 2) 2, but does not show the graph to person #2. b. Person #1 describes to person #2 the behavior of the graph of ƒ (x) so that heshe can sketch it on a piece of paper. c. Switch roles; now person #2, using technology, graphs g(x) 0.5(x 3)(x 2)2, but does not show the graph to person #1. d. Person #2 describes to person #1 the behavior of the graph of g(x) so that heshe can sketch it on a piece of paper. e. Compare the accuracy of the graphs and compare the shapes of the two graphs. What do ƒ (x) and g(x) have in common? How do they differ? 28. In Section 1.2 we examined the annual federal budget surplus or deficit. The federal debt takes into account the cumulative effect of all the deficits and surpluses for each year together
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28. (continued) with any interest or payback of principal. The accompanying graph shows the accumulated gross federal debt as a percent of the gross domestic product from 1940 to 2010.
Environment Less of a Priority Percent rating the issue a “top priority ” 80%
The gross domestic product (GDP) is a measure of the total output of goods and services produced in a given country.
60
The National Debt as a Percent of Gross Domestic Product (Data through 2007 is from Bush’s whitehouse.gov)
20
120%
Through Feb. 27 2009
110% 100% 90%
Obama
80% Bush Reagan
70% Truman
50%
Clinton Kennedy LBJ
30% 20% 10% 0% 1940
1950
1960
Nixon Ford
1970
G.W. Bush zfacts.com
Ike
40%
Carter
60%
1980
1990
2000
2010
Year
Economy Jobs Energy
40
0 00
Environment
Global Environment Economy warming Energy Jobs Global warming 02
04
06
t 09
08
Source: Pew Research Center.
a. For what intervals of t did f(t) increase? Decrease? Explain the implications of your answers. b. Estimate the maximum values of f(t) and minimum values of f(t). Explain the implications of your answers. c. Construct a 60-second summary that could accompany this graph. 30. For the functions f (x) and g(x) shown on the accompanying graph, find the values of x that make the following true. a. f(x) ⫽ 0 b. g(x) ⫽ 0 c. f(x) ⫽ g(x)
Source: zfacts.com.
5
a. What are three interesting facts that emerge from this graph? b. Construct a 60-second summary that could accompany the graph.
y
f(x) g(x) –5
29. The accompanying graph shows the issues that people in the U.S. rated as “top priority” from 2000 to 2009. Let f(t) represent the percent choosing the environment as a top priority where t ⫽ year.
5
x
–5
CHAPTER SUMMARY Single-Variable Data
Equations in Two Variables
Single-variable data can be represented with graphs such as bar charts and histograms and with numerical descriptors such as mean and median.
The solutions of an equation in two variables x and y are the ordered pairs (x, y) that make the equation a true statement. For example, (3, ⫺5) is one solution of the equation 2x ⫹ y ⫽ 1.
Two-Variable Data
Water level (in centimeters)
Graphs of two-variable data show how change in one variable affects change in the other. The accompanying graph is called a time series because it shows changes over time.
The graph of an equation in two variables displays the set of points that are solutions to the equation.
Functions A variable y is a function of a variable x if each value of x determines a unique (one and only one) value of y. Functions can be represented with words, graphs, equations, and tables.
+10
When a set of ordered pairs represents a function, then each ordered pair is written in the form
0 –10
0
4
8 12 16 Time (in hours)
20
24
(independent variable, dependent variable) (input, output) (x, y)
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By convention, on the graph of a function, the input (or independent variable) is represented on the horizontal axis and the output (or dependent variable) on the vertical axis. The domain of a function is the set of all possible values of the input. The range is the set of corresponding values of the output. A graph does not represent a function if it fails the vertical line test. If you can draw a vertical line that crosses the graph two or more times, the graph does not represent a function.
49
Visualizing Functions Graphs of functions may demonstrate the following properties: Maximum
Minimum
Increasing
Decreasing
Concave up
Concave down
Function Notation The expression f (x) means the rule f is applied to the input value x to give the output value, f(x): f(input) 5 output
y
200
h(x) range
For example, f (x) ⫽ 2x ⫹ 5 tells us f is the name of a function where the input is x, the output is f (x), and the rule is to multiply the input by 2 and add 5. To evaluate this function when the input is 4, we write
x –25
f(4) ⫽ (2 ? 4) ⫹ 5 1 f(4) ⫽ 13
25
–200
domain
CHECK YOUR UNDERSTANDING I. Is each of the statements in Problems 1–23 true or false? 1. Histograms and bar charts are used to graph singlevariable data.
4. Tides are the rise and fall of the sea level. The graph shows the changes in the height of the tide in Kahului, Hawaii, on December 12 and 13, 2009.
2. Means and medians, both measures of central tendency, can be used interchangeably.
5. The maximum height of the tide, called high tide, occurred at about 1 a.m. on December 13, 2009.
3. A scatter plot is a plot of data points (x, y) for twovariable data.
6. The accompanying figure of the annual rainfall cycles at the Eke and Haleakala weather stations on Maui Island, Hawaii, show that the least rainfall at both stations occurs in June.
Problems 4 and 5 refer to the accompanying figure.
Annual Rainfall Cycles, at Two Sites on Maui Island, Hawaii
2.50
700
2.00
600
1.50
500
1.00
400
mm
0.50 0.00
Eke
Haleakala
300 200
Date/Time (in 2009)
Source: NOAA, Tides and Currents. http://tidesandcurrents.noaa.gov. Note: MLLW (mean lower low tide) is the average lowest tide.
100 0
Ju l. Au g. Se p. Oc t. N ov . De c.
–0.50 12/12 12/12 12/12 12/12 12/13 12/13 12/13 12/13 12/14 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am
Ja n. Fe b. M ar . Ap ril M ay Ju ne
Height (feet relative to MLLW)
Height of Tides in Kahului, Hawaii 3.00
Source: Rainfall Atlas of Hawaii. Water Resources Research Center, University of Hawaii, Manoa.
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7. The independent variable of a function is also called the output of the function.
18. The function f (x) has both a maximum and a minimum over the interval [1, 3].
8. The graph of the equation describing the sales revenue R ($ million) as a function of the amount spent on advertising A ($ thousand) is the set of ordered pairs (R, A) that satisfy the equation.
19. The function f(x) decreases over the interval (0, 2). Problems 20 and 21 refer to the accompanying figure. y
9. If sales revenue R is a function of advertising A, then R is the dependent variable and A is the independent variable.
9 (–1, 7)
10. If M F(q), then F is a function of q.
6
11. In the accompanying graph y is a function of x.
(2, 4)
y
3
g(x) x 0
–3
3
x
20. The domain of the function g(x) is the interval [1, 2]. 21. The range of the function g(x) is the interval [4, 7]. Problems 12–14 refer to the accompanying graph of h(x).
Problems 22 and 23 refer to the accompanying figure.
y
Demographics of Twitter Users 40%
(1, 3)
37%
Dec-08
35% 31%
x h(x)
12. The function h(x) has a maximum value at (1, 3). 13. The function h(x) is concave down. 14. The function h(x) increases to the right of x 1 and increases to the left of x 1. Problems 15 and 16 refer to the following table.
R
21
2
0
3
S
10
8
6
4
% of internet users
50
10/27/10
Sept-09
30% 25% 20% 19%
20%
19%
15% 10%
10% 5% 0%
10% 5%
10% 4% 2%
18–24
25–34
35–44
45–54
55–64
4%
65+
Source: http://pewresearch.org.
22. The histogram describes the number of Twitter users by age. 23. Internet users who are 18 to 34 years old are more likely to use Twitter than older users. II. In Problems 24–29, give an example as specified.
15. The variable S is a function of R. 16. The variable S is decreasing over the domain for R. 17. A set of ordered pairs of the form (M, C) could represent C as a function of M. Problems 18 and 19 refer to the accompanying graph of f (x). y 3
x 5
–3
f(x) –3
24. A relationship between two variables w and z described as a table where w is a function of z but z is not a function of w. 25. A relationship between two variables w and z represented by a graph where w is a function of z. 26. A graph of a function that is concave up and has a minimum value at the point (2, 0). 27. A graph of a function where the domain is the set of real numbers and that has no maximum or minimum values. 28. A graph of a function for which f(2) 0. 29. A topic sentence describing federal prosecution of crimes as shown in the graph on the top of the next page.
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51
34. The graph in the accompanying figure is increasing and concave up.
70,000 cases
y
60 IMMIGRATION 50 Bush administration
x
40
35. All functions have at least one minimum value. NARCOTICS/ DRUGS
36. Sometimes the choice as to which variable will be the independent variable for a function is arbitrary.
30
37. The graph in the accompanying figure represents P as a function of Q. P
20
WEAPONS 10 WHITECOLLAR CRIMES ’90
’95
’00
Q
0
’05 ’08*
*First 11 months of the fiscal year
Problems 38 to 40 refer to the accompanying table.
Source: Transactional Records Access Clearinghouse, Syracuse University.
III. Is each of the statements in Problems 30–40 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample or explain why it is false. 30. A function can have either a maximum or a minimum but not both. 31. A function is any relationship between two quantities. 32. If q 0, then f(q) 0. 33. The graph in the accompanying figure is decreasing and concave down. y
Hybrid Bicycle Sales by Year 2006–2008 as Percentage of Total Bicycle Sales Year
2006
2007
2008
Percent of Sales (%)
15.0
16.5
19.0
Source: U.S. Department of Commerce. Note: A hybrid bike blends the best characteristics of both road and mountain bikes.
38. If hybrid bicycle sales are a function of time, the input is time in years and the output is the sales of hybrid bicycles as a percent. 39. If percent of hybrid sales, S, is represented by the function S f (t) where t is time in years, then f (19.0) 2008.
x
40. Hybrid bicycle sales, in percent of total bicycle sales, increased from 2006 to 2008.
CHAPTER 1 REVIEW: PUTTING IT ALL TOGETHER 1. A man weighed 160 pounds at age 20. Now, at age 60, he weighs 200 pounds. a. What percent of his age 20 weight is his weight gain? b. If he went on a diet and got back to 160 pounds, what percent of his age 60 weight would be his weight loss?
c. Explain why the weight gain is a different percentage than the weight loss even though it is the same number of pounds. 2. Data from the World Health Organization Report 2009 shows vast differences in average spending per person on health for different countries. Data on health spending for selected countries is shown on the top of the next page.
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Health Care Spending in 2007 Selected Countries
Country China India Democratic People’s Republic of Korea (North) Norway United States
As % of Gross Domestic Product Expenditure/Capita (GDP) (in U.S. dollars) 4.5 3.6 3.5
112.00 35.00 0.30
8.9 15.8
7,376.00 7,287.00
a. What are the mean and median for this set of data? b. China caught what percent of the commercial fish among the leading 10 nations? Of the world? c. Will the mean and median for the 10 leading nations change if China substantially increases the amount of commercial fish it catches while other nations remain at the same level? 5. The following table shows the five leading causes of death around the world in 2007. The Five Leading Causes of Death Worldwide Cause
Source: Health expenditure series. World Health Organization, February 2009 (latest updates are available on www.who.int./nha/country/en/ ).
3. The Sacramento Bee newspaper had the following statement on January 20, 2010: The average teacher salary last year was $66,995, an increase of 1.8 percent from 2008, according to new state figures. Districts laying off less-experienced, lower-paid teachers accounted for almost all of that increase— cumulative school payroll in California was flat from 2008 to 2009. Teacher pay varies widely by district. Explain how laying off lower-paid teachers could account for an increase in average salaries. Do you think the “average” used in the statement was the mean or the median? Explain your answer. 4. In 2006 there were 143,648 thousand metric tons of commercial fish caught worldwide. The chart below lists the world’s 10 leading commercial fishing nations. The 10 Leading Commercial Fishing Nations in 2006 Fish Caught (in thousands of metric tons) China Peru India Indonesia United States Chile Japan Thailand Vietnam Russia
51,521 7,046 6,979 6,052 5,325 4,971 4,921 4,162 3,618 3,390
Source: U.S. Bureau of Census, International Statistics. www.census.gov.
Heart disease Stroke Lower respiratory infections (e.g., pneumonia, emphysema, bronchitis) Chronic Obstructive Pulmonary Disease Diarrhoeal diseases
12.2 9.7 7.1
5.1 3.7
Source: U.S. Bureau of Census, International Statistics. www.census.gov.
a. What is the leading cause of death worldwide? b. Create a bar chart using these data. c. What percentage of worldwide deaths is not accounted for in the table? List at least two other diseases or conditions that can cause death. Could either of them account for more that 5% of all deaths worldwide? 6. Given g(x) 2x2 3x 1, evaluate g(0) and g(1). 7. In a certain state, car buyers pay an excise tax of 2.5%. This means that someone who buys a car must pay the state a onetime fee of 2.5% of the car’s value. Use P to represent the price of the car and E to represent excise tax, then express E as a function of P. 8. A waitress makes $2.50 an hour in wages and receives tips totaling 18% of the price of each meal she serves. Let H represent the hours she works, P represent the total cost of the meals she served, and E represent what she earns in a week before taxes. Write an equation expressing E in terms of H and P. 9. The following graph shows the number of babies per million babies that were named Emma in the United States between Usage of ‘Emma’ per million babies
a. China had the largest population in the world in 2007 with 1.3 billion people. Estimate how much China spent on health care (in U.S. dollars) in 2007. b. India and North Korea spent close to the same percentage of their gross domestic product on health care, but a very different amount per person (per capita) in 2007. How can this be? c. Norway and the United States spent close to the same amount per person but very different percentages of their gross domestic product on health care. How can this be? d. If Norway spent about the same percent of its GDP on health care as the United States, about how much would Norway be spending per person?
% of All Deaths
8,000 7,000 6,000 5,000 4,000
Emma
3,000 2,000 1,000 1880s 1900s 1920s 1940s 1960s 1980s 2000s
Source: www.babynamewizard.com.
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9. (continued) 1880 and 2008. Create a title for the graph and explain in a few sentences what is happening over time to the name Emma.
Poll Responses in 2008 and 2009 to “What do You Expect the Economy to be in 6 Months?” 60 50 Percentage
10. The graph below shows, by age group, the percentage of persons with and without health insurance in the United States in 2007.
40 30 20
Expects economy to improve
Health Insurance in the United States, 2007
Expects economy to remain the same
10 Uninsured at the time of interview
8.9
Uninsured at least part of the year
Expects economy to get worse
Children under 18 years Adults 18–64 years
19.4
0 Feb Mar Apr MayJun Jul AugSep Oct NovDec Jan Feb Mar Apr MayJun Jul AugSep Oct -08 -09 -09
12.6 23.7
Date
Source: www.HarrisInteractive.com.
5.0 14.3
b. What type of reversal in opinion happened over the time period covered by the graph? c. Write a topic sentence for a newspaper article about the shift in opinion over this time period.
32.7
Public coverage
12.3 59.9 69.6
Private coverage 0
20
40
60
80
100
Percent
Source: COC/NCHS/National Health Interview Surveys, 2007.
a. Compare the percentages of children and adults aged 18–64 who were covered by a public insurance plan in 2007. b. Is the following statement true or false? Justify your answer. “In 2007, children were less likely than adults aged 18–64 to be uninsured for more than one year.” c. Write a topic sentence for a newspaper article about health insurance in the United States in 2007. 11. a. Determine whether each set of points represents a function. i. (6, 21), (5, 21), (8, 3), (23, 2) ii. (5, 3), (7, 21), (5, 22), (4, 23) iii. (7, 23), (8, 23), (22, 23), (9, 23) iv. (4, 0), (4, 23), (4, 7), (4 2 1) b. Create a table of values for the variables A and B, where B is a function of A but A is not a function of B. c. Create two graphs, one that represents a function and one that does not represent a function. 12. Find an equation that represents the relationship between x and y in each of the following tables. Specify in each case if y is a function of x. a. x b. x c. x y y y 0 1 2 3 4
2 4 6 8 10
0 1 2 3 4
21 0 3 8 15
0 1 2 3 4
1 2 9 28 65
13. The following graph shows changes in U.S. opinion regarding expectations about the economy. a. At what date was the percentage of people who expected the economy to improve approximately the same as those who expected the economy to get worse? Estimate that percentage.
14. The following graph shows fuel economy in relation to speed. a. Does the graph represent a function? If not, why not? If so, what are the domain and range? b. Describe in words how speed relates to fuel economy. 35 30 Fuel economy (mpg)
Uninsured for more than a year
53
25 20 15 10 5 0
5
15
25
35 45 Speed (mph)
55
65
75
(mpg miles per gallon of gas, mph miles per hour) Source: www.fueleconomy.gov.
15. In January 2010 the exchange rate for the Chinese yuan and U.S. dollar was about 7 to 1; that is, you could exchange 7 yuan for 1 U.S. dollar. a. Let Y number of Chinese yuan and D number of U.S. dollars. Complete the following table. D Y
1
2
3
4
20
100
b. Find an equation that expresses Y as a function of D. What is the independent variable? The dependent variable? c. Complete the following table: Y D
7
14
21
35
70
700
d. Find an equation that expresses D as a function of Y. What is the independent variable? The dependent variable? 16. The average number of calories burned with exercising varies by the weight of the person. The table on the next page gives the approximate calories burned per hour from disco dancing.
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16. (continued)
18. Below is the graph of f(x). y
Calories Burned per Hour from Disco Dancing Weight calories burned per hour
5
100 lb. person
125 lb. person
150 lb. person
175 lb. person
200 lb. person
264
330
396
462
528 –4
4
x
Source: http://www.fitresource.com/Fitness.
Let t be the number of hours spent exercising and C the total number of calories burned. a. Write an equation expressing C1 in terms of t for a 125 lb. person disco dancing and an equation for C2 for a 175 lb. person disco dancing. b. Do your equations in part (a) represent functions? If not, why not? c. For the equations in part (a) that represent functions, what is the independent variable? The dependent variable? What is a reasonable domain? What is a reasonable range? d. Generate small tables of values and graphs for C1 and C2. How do the two graphs compare?
–5
a. Determine the values for f(⫺1), f(0), f(2). b. For what value(s) of x does f(x) ⫽ ⫺2 ? c. Specify the interval(s) over which the function is i. Increasing ii. Decreasing iii. Concave up iv. Concave down v. f(x) ⫽ 0, f(x) ⬎ 0, f(x) ⬍ 0. 19. Below is a graph of data collected by the FBI.
e. How do you think these graphs would compare to the corresponding graphs for a 200 lb. person? Explain.
Life Expectancy, World and Botswana, 1970–2008 Life expectancy (years)
80
3500 Number of arrests
17. The following graph shows the life expectancy in Botswana compared with world life expectancy. The dotted lines on the graph represent projections.
Juvenile Arrests for Murder, 1990 to 2007 4000 3000 2500 2000 1500 1000 500
75
0 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Year
70 65 60
World
55
Botswana
Source: U.S. Bureau of the Census, Statistical Abstract of the United States: 2009.
50
If we let N(x) ⫽ number of juvenile arrests for murder in year x,
45
a. Estimate N(1993). What are the coordinates of this point? What does this point represent?
40 1970
1980
1990 Year
2000
2010
Source: World Bank, 2010.
b. Estimate the coordinates of the minimum point. What does this point represent?
a. Does the graph for Botswana represent a function? If not, why not? If so, what are the domain and range?
c. Over what interval is the function increasing? Over what interval is the function decreasing?
b. Over the given time period, when did life expectancy in Botswana reach its maximum? Estimate this maximum value.
d. What is the domain of N(x)? What is the range?
c. When was life expectancy in Botswana projected to reach its minimum? Estimate this minimum value. What might have caused the sudden drop in life expectancy? d. Over what interval(s) was life expectancy (actual and projected) in Botswana increasing? Decreasing? e. Over what interval(s) is Botswana’s graph concave down? Concave up? f. Write a short description of life expectancy in Botswana compared with world life expectancy.
e. Write a brief summary about juvenile arrests between 1990 and 2007. 20. From the graph of y ⫽ f(x) on the following page, estimate the following: a. Domain and range of f(x) b. When f(x) ⫽ 0 c. Interval(s) where f(x) ⬎ 0 d. Interval(s) where f(x) ⬍ 0 e. Interval(s) where f(x) is increasing f. Interval(s) where f(x) is decreasing
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20. (continued)
55
Export Expertise Share of world exports, % of total 6
y
14 12
United States
*
10
–4
Percent
f (x)
4
x
–2
21. The accompanying graph shows the percent of total world exports for the United States and for China from 1970 to 2010. The dotted line represents projections for China to 2014. Write a 60-second summary comparing the difference between China and the United States in the share of total world exports for these countries between 1970 and 2010.
8 6 4
China
2 0 1970
80
90
2000 Year
10 14
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E X P L O R AT I O N 1 . 1 Collecting, Representing, and Analyzing Data Objectives • • • •
explore issues related to collecting and analyzing data. learn techniques for organizing and graphing data using technology. describe and analyze single-variable data using histograms and numerical descriptors. describe and analyze two-variable data using scatter plots and line graphs.
Material/Equipment • class questionnaire • measuring tapes in inches • optional measuring devices: small eye chart, flexibility tester, measuring device for blood pressure • computer with spreadsheet program or graphing calculator with statistical plotting features • data from class questionnaire or other small data set formatted either as spreadsheet or graph link file • overhead projector or projection panel for computer or graphing calculator • transparencies for printing or drawing graphs for overhead projector (optional) Related Readings (On the web at www.wiley.com/college/kimeclark) “U.S. Government Definitions of Census Terms” “Health Measurements” Related Software “F1: Histograms,” in FAM 1000 Census Graphs Procedure This exploration may take two class periods. It can easily be modified to meet time constraints. Day One In a Small Group or with a Partner 1. Pick (or your instructor will assign you) one of the undefined variables on the questionnaire. Spend about 15 minutes coming up with a workable definition and a strategy for measuring that variable. Be sure there is a way in which a number or single letter can be used to record each individual’s response on the questionnaire. 2. Consult the reading “Health Measurements” if you decide to collect health data. Class Discussion After all of the definitions are recorded on the board, discuss your definition with the class. Is it clear? Does everyone in the class fall into one of the categories of your definition? Can anyone think of someone who might not fit into any of the categories? Modify the definition until all can agree on some wording. As a class, decide on the final version of the questionnaire and record it in your class notebook. 56
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Exploration 1.1
57
In a Small Group or with a Partner Help each other when necessary to take measurements and fill out the entire questionnaire. Questionnaires remain anonymous, and you can leave blank any question you can’t or don’t want to answer. Hand in your questionnaire to your instructor by the end of class. Exploration-Linked Homework Read “U.S. Government Definitions of Census Terms” for a glimpse into the federal government’s definitions of the variables you defined in class. How do the “class” definitions and the “official” ones differ? Day Two Class Demonstration 1. You’ll need an electronic version of the data set from which to choose one variable for the whole class to study (e.g., age from the class data). If you haven’t used a spreadsheet or graphing calculator before, you’ll need a basic technical introduction. (Note: If you are using a TI-83 or TI-84 graphing calculator, there are basic instructions in the Graphing Calculator Manual on www.wiley.com/college/kimeclark.) 2. Select an interval size and then construct a frequency histogram and a relative frequency histogram. Construct another histogram using a different interval size. Are any new patterns revealed? 3. Calculate the mean and median using spreadsheet or graphing calculator functions. In a Small Group or with a Partner Choose another variable from your data. Pick an interval size, and then generate both a frequency histogram and a relative frequency histogram. Calculate the mean and median. If possible, make copies of the histograms for both your partner and yourself. Discussion/Analysis With your partner(s), analyze and jot down patterns that emerge from the data. How could you describe your results? What are some limitations of the data? What other questions are raised and how might they be resolved? Prepare a summary to give to the class. If possible, use an overhead projector with a transparency of your histogram or a projector linked to your graphing calculator. If not, use a paper copy of your histogram. Exploration-Linked Homework Read Section 1.2. For this assignment you can either create a two-variable data comparison from your class data or find published data and graphs to analyze. Creating a two-variable data comparison: 1. Examine the list of categories from the class survey. Choose two variables that may be related. 2. Create a scatter plot of the variables you choose. Describe any trends or patterns in the data. What new information do you get from a scatter plot that is not evident from the data list? 3. Are there any deviations from the pattern(s) you observed? Finding a two-variable comparison: 1. Find a published table or graph that shows a relationship between two variables. If you have access to the Internet, the following websites can be used to find data and graphs: www.census.gov www.newyorktimes.com www.wallstreetjournal.com 2. What are the two variables? How does change in one variable relate to change in the other variable? Are there any deviations from the pattern(s) you observed? Write a 60-second summary describing your results. Include questions raised by the data and your analysis.
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ALGEBRA CLASS QUESTIONNAIRE (You may leave any category blank) 1. Age (in years) 2. Sex (female 5 1, male 5 2) 3. Your height (in inches) 4. Distance from your navel to the floor (inches) 5. Estimate your average travel time to school (in minutes) 6. What is your most frequent mode of transportation to school? (F 5 by foot, C 5 car, P 5 public transportation, B 5 bike, O 5 other) 7. Arm span (inches), length from finger tip to tip with arms stretched against a wall 8. Foot length (inches) The following variables will be defined in class. We will discuss ways of coding possible responses and then use the results to record our personal data. 9. The number of people in your household 10. Your employment status 11. Your ethnic classification 12. Your attitude toward mathematics Health Data 13. Your pulse rate before jumping (beats per minute) 14. Your pulse rate after jumping for 1 minute (beats per minute) 15. Blood pressure: systolic (mm Hg) 16. Blood pressure: diastolic (mm Hg) 17. Flexibility (in inches) 18. Vision, left eye 19. Vision, right eye Other Data
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R AT E S O F C H A N G E A N D LINEAR FUNCTIONS
OVERVIEW How does the U.S. population change over time? How do children’s heights change as they age? Average rates of change provide a tool for measuring how change in one variable affects a second variable. When average rates of change are constant, the relationship is linear.
After reading this chapter, you should be able to • calculate and interpret average rates of change • understand how representations of data can be biased • recognize that a constant rate of change denotes a linear relationship • represent linear functions with equations, tables, graphs, or words • derive by hand a linear model for a set of data • use regression lines to summarize linear trends from scatter plots
59
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2.1 Average Rates of Change In Chapter 1 we looked at how change in one variable could affect change in a second variable. In this section we’ll examine how to measure that change.
Describing Change in the U.S. Population over Time
Year
Population in Millions
1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
3.9 5.3 7.2 9.6 12.9 17.1 23.2 31.4 39.8 50.2 63.0 76.2 92.2 106.0 123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4 309.2 (est.)
We can think of the U.S. population as a function of time. Table 2.1 and Figure 2.1 are two representations of that function. They show the changes in the size of the U.S. population since 1790, the year the U.S. government conducted its first decennial census. Time, as usual, is the independent variable and population size is the dependent variable. Population of the United States 350
300
250 Population in millions
Population of the United States: 1790–2010
200
150
100
50
0 1800
1840
1880
1920
1960
2000
Year
Table 2.1
Figure 2.1 Source: U.S. Bureau of the Census, Statistical Abstract of the United States: 2010.
DA T A
USPOP
Change in population Figure 2.1 clearly shows that the size of the U.S. population has been growing over the last two centuries, and growing at what looks like an increasingly rapid rate. How can the change in population over time be described quantitatively? One way is to pick two points on the graph of the data and calculate how much the population has changed during the time period between them. Suppose we look at the change in the population between 1900 and 1990. In 1900 the population was 76.2 million; by 1990 the population had grown to 248.7 million. How much did the population increase? change in population (248.7 76.2) million people 172.5 million people This difference is portrayed graphically in Figure 2.2, at the top of the next page.
Change in time Knowing that the population increased by 172.5 million tells us nothing about how rapid the change was; this change clearly represents much more dramatic growth if it
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2.1
Time Change: 90 Years
350
350
300
300
Population in millions
Population in millions
Population Change: 172.5 Million People
250 200 172.5 million people
150
61
100 50 0
250 200 150 100 90 years
50 0
1800
1840
1880 1920 Year
1960
2000
1800
Figure 2.2
1840
1880 1920 Year
1960
2000
Figure 2.3
happened over 20 years than if it happened over 200 years. In this case, the length of time over which the change in population occurred is change in years (1990 1900) years 90 years This interval is indicated in Figure 2.3 above.
Average rate of change To find the average rate of change in population per year from 1900 to 1990, divide the change in the population by the change in years: average rate of change 5 5
change in population change in years 172.5 million people 90 years
< 1.92 million people/year In the phrase “million people/year” the slash represents division and is read as “per.” So our calculation shows that “on average,” the population grew at a rate of 1.92 million people per year from 1900 to 1990. Figure 2.4 depicts the relationship between time and population increase. Average Rate of Change: 1900–1990
Population in millions
350 300 250 Average rate of change 1.92 million per year
200
172.5 million people
150 100 90 years
50 0 1800
1840
1880 1920 Year
1960
2000
Figure 2.4
Defining the Average Rate of Change The notion of average rate of change can be used to describe the change in any variable with respect to another. If you have a graph that represents a plot of data points of the form (x, y), then the average rate of change between any two points is the change in the y value divided by the change in the x value.
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The average rate of change of y with respect to x 5
change in y change in x
If the variables represent real-world quantities that have units of measure (e.g., millions of people or years), then the average rate of change should be represented in terms of the appropriate units: units of the average rate of change 5
units of y units of x
For example, the units might be dollars/year (read as “dollars per year”) or pounds/ person (read as “pounds per person”). E X A M P L E
1
Average rate of change in median age Between 1850 and 1950 the median age in the United States rose from 18.9 to 30.2, but by 1970 it had dropped to 28.0. a. Calculate the average rate of change in the median age between 1850 and 1950. b. Compare your answer in part (a) to the average rate of change between 1950 and 1970 in terms of numbers and societal context. c. Would you expect the median age to increase or decrease after 1970?
S O L U T I O N
a. Between 1850 and 1950, change in median age change in years (30.2 2 18.9) years 11.3 years (age) 5 5 (1950 2 1850) years 100 years (calendar) 0.113 years/year
average rate of change 5
The units are a little confusing. But the results mean that between 1850 and 1950 the median age increased an average of 0.113 years each calendar year. b. Between 1950 and 1970, average rate of change 5 5
change in median age change in years (28.0 2 30.2) years 22.2 years 5 (1970 2 1950) years 20 years
0.110 years/year Note that since the median age dropped in value between 1950 and 1970, the average rate is negative. So between 1850 and 1950 the median age increased (by 0.113 year each year), but between 1950 and 1970 the median age decreased (by 0.110 year each year). Why the decline? Americans started having babies after the end of World War II (in 1945) creating the “baby boom” lasting until 1964. So in our time period of 1950–1970, baby boomers lowered the overall median age. c. Given that all the baby boomers are now aging, we would expect the median age to increase. The U.S. Census gave the U.S. median age as 28.0 in 1970 but as 36.9 in 2010, the highest ever.
Limitations of the Average Rate of Change The average rate of change is an average. So it has the limitations of any average. For example, we calculated that the average rate of change of the U.S. population between
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Average Rates of Change
63
1900 and 1990 was 1.92 million people/year. But it is highly unlikely that each year the population grew by exactly 1.92 million people. Similarly, if the arithmetic average, or mean, height of students in your class is 67, you wouldn’t expect everyone (or possibly anyone) to be exactly 67 tall. The average rate of change depends upon the end points. If the data points do not all lie in a straight line, the average rate of change varies over different intervals. E X A M P L E
2
Average rate of change in population Calculate and graph the average rate of change of the U.S. population in these time periods: a. 1840 to 1940 b. 1880 to 1980
S O L U T I O N
Table 2.2 and Figures 2.5 and 2.6 show the average rate of change calculations and graphs for parts (a) and (b). Note that the average rate of change for 1880–1980 is larger, and the corresponding line segment on the graph is steeper, than that for 1840–1940. Time Interval
Change in Time
Change in Population
Average Rate of Change
1840–1940
100 yr
132.2 17.1 115.1 million
1880–1980
100 yr
226.5 50.2 176.3 million
115.1 million < 1.15 million/yr 100 yr 176.3 million < 1.76 million/yr 100 yr
Table 2.2 Average Rate of Change: 1880–1980 350
300
300
Population in millions
Population in millions
Average Rate of Change: 1840–1940 350
250 200 150 100
Average rate of change 1.15 million per year
50
250 200 150
Average rate of change 1.76 million per year
100 50 0
0 1800
Figure 2.5
1840 1880 1920 Year
1960 2000
1800
1840 1880 1920 Year
1960 2000
Figure 2.6
EXPLORE & EXTEND
2.1 DA T A
FEDDEBT
The Federal Debt The federal debt describes the total amount the government owes, accumulated over time (as opposed to the yearly federal surplus or deficit, see pages 11–12). The accompanying table and graph on the next page show the federal debt from 1945 to 2010. (Note: you can find the current debt at www.brillig.com/debt_clock.) a. Has the gross federal debt decreased over any five-year interval between 1945 and 2010? The federal debt has clearly been growing since 1950. But how quickly? b. What was the average rate of change in the federal debt between 1945 and 1980? Between 1980 and 2010? How could you interpret these results?
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Accumulated Gross Federal Debt Year
Debt (billions of $)
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
260 257 274 291 322 381 542 909 1,818 3,207 4,921 5,674 7,932 12,130 (est.)
Gross Federal Debt (billions of $)
14,000
Debt (billions of $)
12,000 10,000 8,000 6,000 4,000 2,000 0 1940 1950 1960 1970 1980 1990 2000 2010 Year
Source: www.census.gov.
c. Over what five-year interval does the largest rate of change occur? Calculate that rate of change. Show how you can use the debt at the starting point of this interval and the average rate of change to generate the debt at the end point. d. Create a newspaper title to describe these data and the graph.
Algebra Aerobics 2.1 1. Suppose your weight five years ago was 135 pounds and your weight today is 143 pounds. Find the average rate of change in your weight with respect to time. 2. The table below shows data on U.S. international trade as reported by the U.S. Bureau of the Census.
Year 2000 2008
U.S. Exports U.S. Imports (billions of $) (billions of $) 1,070.6 1,826.6
1,450.0 2,522.5
Annual Deaths in Motor Vehicle Accidents (thousands) 1980 52.1
U.S. Trade Balance Exports Imports (billions of $) 379.4 695.9
a. What is the average rate of change between 2000 and 2008 for: i. Exports? ii. Imports? iii. The trade balance, the difference between what we sell abroad (exports) and buy from abroad (imports)? b. What do these numbers tell us? 3. The following table indicates the number of deaths in motor vehicle accidents in the United States as listed by the U.S. Bureau of the Census.
4.
5.
6.
7.
1990 44.6
2000 41.9
2008 43.3
Find the average rate of change: a. From 1980 to 2000 b. From 2000 to 2008 Be sure to include units. A car is advertised to go from 0 to 60 mph in 5 seconds. Find the average rate of change (i.e., the average acceleration) over that time. According to the National Association of Insurance Commissioners, the mean cost for automobile insurance has gone from $689 in 2000 to $800 in 2009. What is the average rate of change from 2000 to 2009? A football player runs for 1056 yards in 2006 and for 978 yards in 2010. Find the average rate of change in his performance from 2006 to 2010? The African elephant is an endangered species, largely because poachers illegally kill elephants to sell the ivory from their tusks. In Kenya alone between 1972 and 1989, the elephant population fell from 140,000 to a mere 19,000. In 1989 the Convention on Trade in
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Endangered Species introduced a ban on all ivory trade. But the estimated number of elephants in Kenya in 2009 is still only 30,000. a. Calculate the average rate of change between 1972 and 1989 and then between 1989 and 2009. Describe what each rate means.
Average Rates of Change
65
b. Using the annual average rate of change between 1989 and 2009, in how many years after 1989 would the number of elephants return to the 1972 size of 140,000?
Exercises for Section 2.1 1. If r is measured in inches, s in pounds, and t in minutes, identify the units for the following average rates of change: change in r change in t change in s a. b. c. change in s change in r change in r
b. In 2007, aerospace industry net profits were $16.9 billion. Find the average rate of change in net profits: i. From 1992 to 2007 ii. From 2002 to 2007
2. Assume that R is measured in dollars, S in ounces, T in dollars per ounce, and V in ounces per dollar. Write a product of two of these terms whose resulting units will be: a. Dollars b. Ounces
9. The U.S. Bureau of the Census provided the following data concerning computer use in all U.S. elementary and secondary schools.
3. Your car’s gas tank is full and you take a trip. You travel 212 miles, then you fill your gas tank up again and it takes 10.8 gallons. If your change in distance is 212 miles and your change in gallons is 10.8, what is the average rate of change of gasoline used, measured in miles per gallon? 4. The gas gauge on your car is broken, but you know that the car averages 22 miles per gallon. You fill your 15.5-gallon gas tank and tell your friend, “I can travel 300 miles before I need to fill up the tank again.” Why is this true? 5. The consumption of margarine (in pounds per person) decreased from 10.9 in 1990 to 4.6 in 2006. What was the annual average rate of change? (Source: www.census.gov) 6. The percentage of people who own homes in the United States has gone from 65.5% in 1980 to 67.6% in 2009. What is the average rate of change in percentage points per year? 7. The accompanying table shows females’ SAT scores in 2000 and 2008.
Year
Average Female Verbal SAT
Average Female Math SAT
2000 2008
504 498
498 499
Source: www.collegeboard.com.
Find the average rate of change: a. In the math scores from 2000 to 2008 b. In the verbal scores from 2000 to 2008 8. a. In 1992 the aerospace industry showed a net loss (negative profit) of $1.84 billion. In 2002 the industry had a net profit of $8.97 billion. Find the average annual rate of change in net profits from 1992 to 2002.
Academic Year End
Total Number of Computers
Students per Computer
1985 2005 2006
630,000 13,600,000 14,165,000
84.1 4.0 3.9
a. Find the average rate of change from 1985 to 2005 in: i. The number of computers being used ii. The number of students per computer b. Find the average rate of change from 2005 to 2006 in: i. The number of computers being used ii. The number of students per computer c. Using the annual average rate of change between 2005 and 2006, in how many years after 2005 would there be a projection of two students per computer. d. Look online to see if you can find more up-to-date data. 10. According to the U.S. Bureau of the Census, the percentage of persons 25 years old and over completing 4 or more years of college was 4.6 in 1940 and 29.4 in 2008. a. Plot the data, labeling both axes and the coordinates of the points. b. Calculate the average rate of change in percentage points per year. c. Write a topic sentence summarizing what you think is the central idea to be drawn from these data. 11. According to a 2010 Kaiser Family Foundation report, during 2009 kids spent almost every waking hour outside of school using electronic devices such as iPhones, computers, or televisions. Those aged 8 to 18 spent an average seven and a half hours a day in 2009 versus six and a half hours in 2004. Calculate the annual average rate of change of electronic use for kids from 2004 to 2009.
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12. The following graph was produced by the U.S. Department of Transportation to report actual and predicted highway crash fatalities (up until 2020). The left-hand axis shows the number of highway fatalities and the right-hand axis the percent of all crashes that were fatal.
14. The data show U.S. consumption and exports of cigarettes.
Year 1960 1980 2000 2007
Highway Crash Fatalities 6%
60,000
y1
50,000 y2
40,000 30,000
5% 4% 3% 2% 1%
% of fatal crashes
No. of fatalities
70,000
20,000 0 ’65 ’70 ’75 ’80 ’85 ’90 ’95 ’00 ’05 ’10 ’15 ’20 Number (y1)
Rate (y2)
a. Estimate the number of fatalities (y1) in 1965 and 1995 and calculate the annual average rate of change. b. Estimate the number of fatalities (y1) in 1995 and those predicted to 2020. Calculate the annual average rate of change. c. The fatality rate (y2) between 1995 and 2020 is predicted to be flat ( just below 2%), while the number of fatalities is expected to increase. How could that be? 13. Use the information in the accompanying table to answer the following questions. Percentage of Persons 25 Years Old and Over Who Have Completed 4 Years of High School or More All White Black Asian/Pacific Islander
1940
2008
24.5 26.1 7.3 22.6
86.6 87.1 83.0 88.7
Source: U.S. Bureau of the Census, Statistical Abstract of the United States: 2009.
a. What was the average rate of change (in percentage points per year) of completion of 4 years of high school from 1940 to 2008 for whites? For blacks? For Asian/Pacific Islanders? For all? b. If these rates continue, what percentages of whites, of blacks, of Asian/Pacific Islanders, and of all will have finished 4 years of high school in the year 2010? Check the Internet to see if your predictions are accurate. c. If these rates continue, in what year will 100% of whites have completed 4 years of high school or more? In what year 100% of blacks? In what year 100% of Asian/Pacific Islanders? Do these projections make sense? d. Write a 60-second summary describing the key elements in the high school completion data. Include rates of change and possible projections for the near future.
U.S. Consumption (billions)
Exports (billions)
484 631 430 364
20 82 148 102
Source: U.S. Department of Agriculture.
a. Calculate the average rates of change in U.S. cigarette consumption from 1960 to 1980, from 1980 to 2007, and from 1960 to 2007. b. Compute the average rate of change for cigarette exports from 1960 to 2007. Does this give an accurate image of cigarette exports? c. The total number of cigarettes consumed in the United States in 1960 was 484 billion, very close to the number consumed in 1995, 487 billion. Does that mean smoking was as popular in 1995 as it was in 1960? Explain your answer. d. Write a paragraph summarizing what the data tell you about the consumption and exports of cigarettes since 1960, including average rates of change. 15. Examine the accompanying table on life expectancy. Average Number of Years of Life Expectancy in the United States by Race and Sex Since 1900 Life Expectancy at Birth by Year 1900 1950 2000 2010
White Males
White Females
Black Males
Black Females
46.6 66.5 74.8 76.5
48.7 72.2 80.0 81.3
32.5 58.9 68.2 70.2
33.5 62.7 74.9 77.2
Source: U.S. National Center for Health Statistics, Statistical Abstract of the United States, 2010.
a. What group had the highest life expectancy in 1900? In 2010? What group had the lowest life expectancy in 1900? In 2010? b. Which group had the largest average rate of change in life expectancy between 1900 and 2010? c. Write a short summary of the patterns in U.S. life expectancy from 1900 to 2010 using average rates of change to support your points. 16. The table below shows U.S. marital status. Marital Status of Population 15 Years Old and Older Year
Number of Unmarried Males (in thousands)
Number of Unmarried Females (in thousands)
1950 1960 1970 1980 1990 2000 2008
17,735 18,492 23,450 30,134 36,121 43,429 51,383
19,525 22,024 29,618 36,950 43,040 50,133 57,756
Source: U.S. Bureau of the Census, www.census.gov.
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a. Calculate the average rate of change in the number of unmarried males between 1950 and 2008. Interpret your results. b. Calculate the average rate of change in the number of unmarried females between 1950 and 2008. Interpret your results.
Change in the Average Rate of Change
67
c. Compare the two results. d. What does this tell you, if anything, about the percentages of unmarried males and females?
2.2 Change in the Average Rate of Change We can obtain an even better sense of patterns in the U.S. population if we look at how the average rate of change varies over time. One way to do this is to pick a fixed interval for time and then calculate the average rate of change for each successive time period. Since we have the U.S. population data in 10-year intervals, we can calculate the average rate of change for each successive decade. The third column in Table 2.3 shows the results of these calculations. Each entry represents the average population growth per year (the average annual rate of change) during the previous decade. A few of these calculations are worked out in the last column of the table. Average Annual Rates of Change of U.S. Population: 1790–2010
Year
Population (millions)
Average Annual Rate of change for Prior Decade (millions/yr)
1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
3.9 5.3 7.2 9.6 12.9 17.1 23.2 31.4 39.8 50.2 63.0 76.2 92.2 106.0 123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4 309.2 (est.)
Data not available 0.14 0.19 0.24 0.33 0.42 0.61 0.82 0.84 1.04 1.28 1.32 1.60 1.38 1.72 0.90 1.91 2.80 2.40 2.32 2.22 3.27 2.78
Sample Calculations 0.14 (5.3 3.9)/(1800 1790)
0.42 (17.1 12.9)/(1840 1830)
0.90 (132.2 123.2)/(1940 1930)
Table 2.3 Source: U.S. Bureau of the Census, Statistical Abstract of the United States: 2010.
What is happening to the average rate of change over time? Start at the top of the third column and scan down the numbers. Notice that until 1910 the average rate of change increases every year. Not only is the population growing every decade until 1910, but it is growing at an increasing rate. It’s like a car that is not only moving forward but also accelerating. A feature that was not so obvious in the original data is now evident: In the intervals 1910 to 1920, 1930 to 1940, and 1960 to
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1990 we see an increasing population but a decreasing rate of growth. It’s like a car decelerating—it is still moving forward but it is slowing down. E X A M P L E
1
S O L U T I O N
Changes in the rate of change in U.S. population a. Graph and interpret the average rate of change of the population over time. b. What does this tell you about the U.S. population? a. Figure 2.7 clearly shows how the average rate of change in population fluctuates over time. Average Annual Rates of Change in the U.S. Population by Decade
Rate of change (in millions per year) over prior decade
3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 2010 Year
Figure 2.7
The first point, corresponding to the year 1800, shows an average rate of change of 0.14 million people/year for the decade 1790 to 1800. The rate 1.72, corresponding to the year 1930, means that from 1920 to 1930 the population was increasing at a rate of 1.72 million people/year. b. The rate of growth was steadily increasing up until about 1910. Why did it change after that? A possible explanation for the slowdown in growth rate in the decade prior to 1920 might be World War I and the 1918 worldwide flu epidemic. By 1920 the flu killed nearly 20,000,000 people, including about 500,000 Americans. The steepest decline in the average rate of change is between 1930 and 1940. One obvious suspect for the big slowdown in population growth in the 1930s is the Great Depression. The average rate of change increases again between 1940 and 1960, an indication of the baby boomers born after World War II. It drops off from the 1960s through the 1980s, possibly due to the introduction of birth control. The rate increases once more in the 1990s. This latest surge in the growth rate is attributed partially to the “baby boom echo” (the result of baby boomers having children) and to a rise in birth rates and immigration. EXPLORE & EXTEND
2.2 DA T A
FEDDEBT
Changes in the Average Rate of Change in the Federal Debt In Explore & Extend 2.1 (pages 63–64), the federal debt table and graph suggest a steady increase in the debt. But the average rate of change of the federal debt (in the accompanying table and graph) is a better gauge for showing how rapidly the debt is growing or slowing down.
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69
Accumulated Gross Federal Debt
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Debt (billions of $) 260 257 274 291 322 381 542 909 1,818 3,207 4,921 5,674 7,932 12,130 (est.)
Average Rate of Change (billions of $ per year) n.a. ⫺1 4 3 6 12 32 73 182 278 343 151 452 840
Average Rate of Change of Federal Debt 900
Average rate of change (billions of $ over prior year)
Year
800 700 600 500 400 300 200 100 0 1940
1960
Source: www.census.gov.
1980
2000
Year
a. What does it mean that the average rate of change between 1945 and 1950 is negative (i.e., ⫺1)? b. After 1950, the average rate of change is positive. What does that tell you about the national debt? c. The average rate of change decreases between 1995 and 2000. Interpret this change. d. Over what interval does the average rate of change dramatically increase? Identify what was happening in the U.S. economy then. e. Create a 60-second summary about the average rate of change in the federal debt.
Algebra Aerobics 2.2
World Population
Year 1800 1850 1900 1950 2000 2050
Total Population (millions) 980 1260 1650 2520 6090 9317 (est.)
Annual Average Rate of Change (over prior 50 years) n.a.
Source: Population Division of the United Nations, www.un.org/popin.
World Population 10000 Population (in millions)
1. The accompanying table and graph show estimates for world population between 1800 and 2050. a. Fill in the third column of the table by calculating the annual average rate of change. b. Graph the annual average rate of change versus time.
8000 6000 4000 2000
1800
1850
1900
1950
2000
2050
Year
c. During what 50-year period was the average annual rate of change the largest? d. Describe in general terms what happened (and is predicted to happen) to the world population and its average rate of change between 1800 and 2050. 2. A graph illustrating a corporation’s profits indicates a positive average rate of change between 2006 and 2007, another positive rate of change between 2007
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and 2008, a zero rate of change between 2008 and 2009, and a negative rate of change between 2009 and 2010. Describe the graph and the company’s financial situation over the years 2006–2010. 3. The data table shows educational data collected on 18to 24-year-olds between 1960 and 2007 by the National Center for Educational Statistics. The table shows the number of students who graduated from high school or completed a GED (a high school equivalency exam) during the indicated year. High School Completers
Year
Number (thousands)
1960 1970 1980 1990 2000 2007
1679 2757 3089 2355 2756 2955
Average Rate of Change (thousands per year)
a. Fill in the blank cells with the appropriate average rates of change for high school completers. b. Describe the pattern in the number of high school completers between 1960 and 2007. c. In this context what does it mean when the average rate of change is positive? Give a specific example from your data. d. What does it mean when the average rate of change is negative? Give another specific example. e. What does it mean when two adjacent average rates of change are positive, but the second one is smaller than the first?
n.a.
Exercises for Section 2.2 Graphing program is optional for Exercises 4 and 11 and access to Internet recommended for Exercise 11. 1. Calculate the average rate of change between adjacent points for the following function. (The first few are done for you.)
x
ƒ (x)
0 1 2 3 4 5
0 1 8 27 64 125
Average Rate of Change n.a. 1 7
a. Is the function ƒ (x) increasing, decreasing, or constant throughout? b. Is the average rate of change increasing, decreasing, or constant throughout? 2. Calculate the average rate of change between adjacent points for the following function. The first one is done for you. x
ƒ (x)
0 1 2 3 4 5
0 1 16 81 256 625
a. Is the function ƒ (x) increasing, decreasing, or constant throughout? b. Is the average rate of change increasing, decreasing, or constant throughout? 3. The accompanying table shows the number of registered motor vehicles in the United States.
Average Rate of Change n.a. 1
Year
Registered Motor Vehicles (millions)
1970 1980 1990 2000 2010
108 156 189 218 270 (est.)
Annual Average Rate of Change (over prior decade) n.a.
a. Fill in the third column in the table. b. During which decade was the average rate of change the smallest? c. During which decade was the average rate of change the largest? d. Write a paragraph describing the change in registered motor vehicles between 1970 and 2010. 4. (Graphing program optional.) The accompanying table indicates the number of juvenile arrests (in thousands) in the United States for aggravated assault.
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2.2
Year
Juvenile Arrests (thousands)
Annual Average Rate of Change over Prior 5 Years
1985 1990 1995 2000 2005
36.8 54.5 68.5 49.8 36.9
n.a.
40
y
x
ƒ (x)
x
g(x)
0 10 20 30 40 50
5 25 45 65 85 105
0 10 20 30 40 50
270 240 210 180 150 120
a. Are the functions ƒ (x) and g(x) increasing, decreasing, or constant throughout? b. Is the average rate of change of each function increasing, decreasing, or constant throughout? 6. Calculate the average rate of change between adjacent points for each of the functions in Tables A–D and place the values in a third column in each table. (The first entry is “n.a.”) Then for each function decide which statement best describes it. Table C
x
ƒ (x)
x
h(x)
0 1 2 3 4 5
1 3 9 27 81 243
0 10 20 30 40 50
50 55 60 65 70 75
Table D
x
g(x)
x
k(x)
0 15 30 45 60 75
200 155 110 65 20 ⫺25
0 1 2 3 4 5
40 31 24 19 16 15
15
0
x 6
Graph A
y
6
y
x 0
6
Graph B
x 0
Graph C
6
8. Match each data table with its graph. Table A
5. Calculate the average rate of change between adjacent points for the following functions and place the values in a third column in each table. (The first entry is “n.a.”)
Table B
As x increases, the function increases at a constant rate. As x increases, the function increases at an increasing rate. As x increases, the function decreases at a constant rate. As x increases, the function decreases at a decreasing rate.
a. b. c. d.
7. Each of the following functions has a graph that is increasing. If you calculated the average rate of change between sequential equal-size intervals, which function can be said to have an average rate of change that is: a. Constant? b. Increasing? c. Decreasing?
a. Fill in the third column in the table by calculating the annual average rate of change. b. Graph the annual average rate of change versus time. c. During what 5-year period was the annual average rate of change the largest? d. Describe the change in juvenile arrests during these years by referring both to the number and to the annual average rate of change.
Table A
71
15
Table B
x
y
x
y
x
y
0 1 2 3 4 5 6
2 5 8 11 14 17 20
0 1 2 3 4 5 6
0 0.5 2 4.5 8 12.5 18
0 1 4 9 16 25 36
0 2 4 6 8 10 12
y
0
Table C
20
Graph D
x 40
y
0
20
Graph E
x 8
y
0
Graph F
x 6
9. Refer to the first two data tables (A and B) in Exercise 8. Insert a third column in each table and label the column “average rate of change.” (Again the first entry is “n.a.”) a. Calculate the average rate of change over adjacent data points. b. Identify whether the table represents an average rate of change that is constant, increasing, or decreasing. c. Explain how you could tell this by looking at the corresponding graph in Exercise 8. 10. Following are data on the U.S. population over the time period 1830–1930 (extracted from Table 2.1). U.S. Population Year 1830 1850 1870 1890 1910 1930
Population (in millions)
Average Rate of Change (millions/yr)
12.9 23.2
n.a.
63.0 92.2
0.83 1.16 1.55
Source: U.S. Bureau of the Census, www.census.gov.
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a. Fill in the missing parts of the table. b. Which 20-year interval experienced the largest average rate of change in population? c. Which 20-year interval experienced the smallest average rate of change in population?
Number of U.S. Newspapers
11. (Technology recommended.) The accompanying data give a picture of the two major methods of D A T A news communication in the United States. (See also Excel or graph link files NEWPRINT and ONAIRTV.) a. Use the U.S. population numbers from Table 2.1 (at the beginning of this chapter) to calculate and compare the number of copies of newspapers per person in 1920 and in 2000. b. Create a table that displays the annual average rate of change in TV stations for each decade between 1950 and 2000 and the 8-yr period between 2000 and 2008. Create a similar table that displays the annual average rate of change in newspapers published for the same period. Graph both of the results. c. If new TV stations continue to come into existence at the same rate as from 2000 to 2008, how many would there be in the year 2010? If possible, check your predictions on the Internet. Is your estimate too high or too low? d. What trends do you see in the dissemination of news as reflected in these data?
Year
Newspapers (thousands of copies printed)
Number of Newspapers Published
1920 1930 1940 1950 1960 1970 1980 1990 2000 2008
27,791 39,589 41,132 53,829 58,882 62,108 62,202 62,324 55,800 48,598
2042 1942 1878 1772 1763 1748 1745 1611 1480 1408
Source: U.S. Bureau of the Census, www.census.gov.
Number of U.S. Commercial TV Stations Year
Number of Commercial TV Stations
1950 1960 1970 1980 1990 2000 2008
98 515 677 734 1092 1248 1353
Source: U.S. Bureau of the Census, www.census.gov.
2.3 The Average Rate of Change Is a Slope On a graph, the average rate of change is the slope of the line connecting two points. The slope is an indicator of the steepness of the line.
Calculating Slopes The reading “Slopes” describes many of the practical applications of slopes, from cowboy boots to handicap ramps.
If (x1, y1) and (x2, y2) are two points, then the change in y equals y2 y1. This difference is often denoted by y, read as “delta y,” where is the Greek letter capital D (think of D as representing difference): y y2 y1. Similarly, the change in x (delta x) can be represented by x x2 x1. See Figure 2.8. Slope = y /x y (x2, y2)
y = y2 – y1
(x1, y1)
x = x2 – x1
(x2, y1)
x
Figure 2.8
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The Average Rate of Change Is a Slope
73
The average rate of change represents a slope. Given two points (x1, y1) and (x2, y2), average rate of change 5
change in y y y 2 y1 5 5 2 5 slope change in x x x2 2 x1
When calculating a slope, it doesn’t matter which point is first Given two points, (x1, y1) and (x2, y2), it doesn’t matter which one we use as the first point when we calculate the slope. In other words, we can calculate the slope between (x1, y1) and (x2, y2) as y2 2 y1 x2 2 x1
or as
y1 2 y2 x1 2 x2
The two calculations result in the same value. Note: In calculating the slope, we need to be consistent in the order in which the coordinates appear in the numerator and the denominator. If y1 is the first term in the numerator, then x1 must be the first term in the denominator. E X A M P L E
1
S O L U T I O N
Slope through two points Plot the two points (2, 6) and (7, 12) and calculate the slope of the line passing through them. Treating (2, 6) as (x1, y1) and (7, 12) as (x2, y2) (Figure 2.9), then slope 5
y2 2 y1 12 2 (26) 18 5 5 52 x2 2 x1 7 2 (22) 9 12
y (7, 12)
12 – (–6)
x –8
8
(–2, –6)
7 – (–2)
–12
Figure 2.9
We could also have used 6 and 2 as the first terms in the numerator and denominator, respectively: slope 5
y1 2 y2 26 2 12 218 5 5 52 x1 2 x2 22 2 7 29
Either way we obtain the same answer. 2
Decrease in rural population The percentage of the U.S. population living in rural areas decreased from 84.7% in 1850 to 21.0% in 2000. Plot the data, then calculate and interpret the average rate of change in the rural population over time.
S O L U T I O N
If we treat year as the input and percentage as the output, our given data can be represented by the points (1850, 84.7) and (2000, 21.0). (See Figure 2.10.)
E X A M P L E
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Percentage of the U.S. Population Living in Rural Areas 100 (1850, 84.7) 80 Percentage
74
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60 40
(2000, 21.0)
20 1800
1850
1900
1950
2000
2050
Year
Figure 2.10 Source: U.S. Bureau of the Census, www.census.gov.
change in percentage points of rural population change in time (21.0 2 84.7) percentage points 5 (2000 2 1850) years 263.7 percentage points 5 150 years < 20.42 percentage points per year
The average rate of change 5
The sign of the average rate of change is negative since the percentage of people living in rural areas was decreasing. (The negative slope of the graph in Figure 2.10 confirms this.) The value tells us that, on average, the percentage living in rural areas decreased by 0.42 percentage points (or about one-half of 1%) each year between 1850 and 2000. The change per year may seem small, but in a century and a half the rural population went from being the overwhelming majority (84.7%) to about one-fifth (21%) of the population. If the slope of the line connecting any two data points on a graph is positive, then the output increases as the input increases. If the slope is negative, the output decreases as the input increases. If the slope is zero, the line is flat and the output is constant for all input. E X A M P L E
3
Plotting civil disturbances a. Use Table 2.4 to plot the number of civil disturbances in U.S. cities (in three-month intervals) between 1967 and 1977, and then connect the points. Without doing any Civil Disturbances in U.S. Cities Year
Period
Number of Disturbances
1968
Jan.–Mar. Apr.–June July–Sept. Oct.–Dec. Jan.–Mar. Apr.–June July–Sept. Oct.–Dec. Jan.–Mar. Apr.–June July–Sept. Oct.–Dec.
6 46 25 3 5 27 19 6 26 24 20 6
1969
1970
Year
Period
Number of Disturbances
1971
Jan.–Mar. Apr.–June July–Sept. Oct.–Dec. Jan.–Mar. Apr.–June July–Sept. Oct.–Dec.
12 21 5 1 3 8 5 5
1972
Table 2.4 Source: D.S. Moore and G.P. McCabe, Introduction to the Practice of Statistics. Copyright © 1989 by W.H. Freeman and Company. Used with permission.
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calculations, indicate on the graph when the average rate of change between adjacent points is positive (⫹), negative (⫺), and zero (0). b. Describe the pattern of civil disturbances. S O L U T I O N
a. The data are plotted in Figure 2.11. Each line segment is labeled ⫹, ⫺, or 0, indicating whether the average rate of change between adjacent points is positive, negative, or zero. The largest positive average rate of change, or steepest upward slope, seems to be between the January-to-March and April-to-June in 1968. The largest negative average rate of change, or steepest downward slope, appears later in the same year (1968) between the July-to-September and October-toDecember.
– –
–
1968
1969
1970
1971
–
Oct.–Dec.
0 July–Sept.
Jan.–Mar.
+ Apr.–June
July–Sept.
Jan.–Mar.
– + Apr.–June
Oct.–Dec.
July–Sept.
Jan.–Mar.
Apr.–June
Oct.–Dec.
Apr.–June
July–Sept.
– +
Oct.–Dec.
– Jan.–Mar.
Oct.–Dec.
+ –
+ Apr.–June
–
+
+
–
July–Sept.
Jan.–Mar.
Number of disturbances
Civil Disturbances: 1968–1972 50 45 40 35 + 30 25 20 15 10 5 0
1972
Figure 2.11 Source: D.S. Moore and G.P. McCabe, Introduction to the Practice of Statistics. Copyright © 1989 by W.H. Freeman and Company. Used with permission.
b. Civil disturbances between 1968 and 1972 occurred in cycles: The largest numbers generally occurred in the spring months and the smallest in the winter months. The peaks decrease over time. What was happening in America that might correlate with the peaks? This was a tumultuous period in our history. Many previously silent factions of society were finding their voices. Recall that in April 1968 Martin Luther King was assassinated, and in March 1973 the last American troops were withdrawn from Vietnam, technically ending the Vietnam War. On April 30, 1975, frantic pro-U.S. Vietnamese tried a final escape via American helicopters on the U.S. embassy roof. The next day, ten Marines from the U.S. embassy were the last American soldiers to depart, concluding the U.S. presence in Vietnam.
Algebra Aerobics 2.3 Violent Crimes in the United States 2,500 Violent crimes (in thousands)
1. a. Plot each pair of points and then calculate the slope of the line that passes through them. i. (4, 1) and (8, 11) ii. (23, 6) and (2, 6) iii. (0, 23) and (25, 21) b. Recalculate the slopes in part (a), reversing the order of the points. Check that your answers are the same. 2. Specify the intervals on the graph of violent crimes for which the average rate of change between adjacent data points is approximately zero.
2,000 1,500 1,000 500 0 1992
1994
1996
1998
2000
Year
2002
2004
2006
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3. Specify the intervals on the graph of tornado deaths for which the average rate of change between adjacent data points appears positive, negative, or zero. Deaths Due to Tornadoes (2000–2009) 140 Number of deaths
120 100 80 60
5. Find the slope of the line through the points (2, 9) and (2 1 h, 9 1 2h) where h 2 0. 6. Consider points P1 5 (0, 0), P2 5 (1, 1), P3 5 (2, 4), and P4 5 (3, 9). a. Verify that these four points lie on the graph of y 5 x2. b. Find the slope of the line segments connecting P1 and P2, P2 and P3, and P3 and P4. c. What do these slopes suggest about the graph of the function within those intervals?
40 20 0 2000
2002
2004 2006 Year
2008
2010
4. What is the missing y-coordinate that would produce a slope of 4, if a line were drawn through the points (3, 22) and (5, y)?
7. Show that the two forms are equivalent. (Hint: multiply 21 by .) 21 y2 2 y1 y 2 y2 and 1 x2 2 x1 x1 2 x2
Exercises for Section 2.3 1. Find the slope of a straight line that goes through: a. (⫺5, ⫺6) and (2, 3) b. (⫺5, 6) and (2, ⫺3)
b. Calculate the exact slope for each line segment A, B, and C.
12
2. Find the slope of each line using the points where the graph intersects the x and y axes.
y
10
A
8 12
y
8
y
C
6 4
B
2
–6
–6
6
x 0
2
4
Graph A
Graph B
3. Find the slope of each line using the points where the line crosses the x- or y-axis. y
6
5. Given the following graph: a. Estimate the slope for each line segment A–F . b. Which line segment is the steepest? c. Which line segment has a slope of zero?
y
8
y E
x –4
6
6 –4
–8
2
0
x
x
4
x –1
1
2
3
4
–8
A
x 12
B C
D
F
–16 –6 Graph A
–10 Graph B
4. Examine the line segments A, B, and C. a. Which line segment has a slope that is positive? That is negative? That is zero?
6. Plot each pair of points and calculate the slope of the line that passes through them. a. (3, 5) and (8, 15) d. (⫺2, 6) and (2, ⫺6) b. (⫺1, 4) and (7, 0) e. (⫺4, ⫺3) and (2, ⫺3) c. (5, 9) and (⫺5, 9)
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7. The following problems represent calculations of the slopes of different lines. Solve for the variable in each equation. 150 2 75 182 2 150 a. c. 5m 54 20 2 10 28 2 x 70 2 y 620 b. d. 5 0.5 5 0.6 028 x 2 10 8. Find the slope m of the line through the points (0, b) and (x, y), then solve the equation for y. 9. Find the value of t if m is the slope of the line that passes through the given points. a. (3, t) and (2, 1), m 4 b. (5, 6) and (t, 9), m 5 23 10. a. Find the value of x so that the slope of the line through (x, 5) and (4, 2) is 31. b. Find the value of y so that the slope of the line through (1, 3) and (4, y) is 2. c. Find the value of y so that the slope of the line through (2, 3) and (5, y) is 0. d. Find the value of x so that the slope of the line through (2, 2) and (x, 10) is 2. e. Find the value of y so that the slope of the line through 1 (100, 10) and (0, y) is 2 10 . f. Find at least one set of values for x and y so that the slope of the line through (5, 8) and (x, y) is 0.
17. In the previous exercise, which pairs of points produce a line with a positive slope? 18. A study on numerous streams examined the effects of a warming climate. It found an increase in water temperature of about 0.7C for every 1C increase in air temperature. a. Find the rate of change in water temperature with respect to air temperature. What are the units? b. If the air temperature increased by 5C, by how much would you expect the stream temperature to increase? 19. Handicapped Vietnam veterans successfully lobbied for improvements in the architectural standards for wheelchair access to public buildings. a. The old standard was a 1-foot rise for every 10 horizontal feet. What would the slope be for a ramp built under this standard? b. The new standard is a 1-foot rise for every 12 horizontal feet. What would the slope of a ramp be under this standard? c. If the front door is 3 feet above the ground, how long would the handicapped ramp be using the old standard? Using the new? 20. The following graph shows the relationship between education and health in the United States. Years of Education vs. Health Issues 90
11. Points that lie on the same straight line are said to be collinear. Determine if the following points are collinear. a. (2, 3), (4, 7), and (8, 15) b. (3, 1), (2, 4), and (7, 8)
13. Graph a line through each pair of points and then calculate its slope. a. The origin and (6, 2) b. The origin and (4, 7) 14. Find some possible values of the y-coordinates for the points (3, y1) and (6, y2) such that the slope m 0. 15. Calculate the slope of the line passing through each of the following pairs of points. a. (0, !2) and ( !2, 0) b. (0, 232 ) and (232, 0) c. (0, b) and (b, 0) d. What do these pairs of points and slopes all have in common? 16. Which pairs of points produce a line with a negative slope? a. (5, 5) and (3, 3) d. (4, 3) and (12, 0) b. (2, 6) and (1, 4) e. (0, 3) and (4, 10) c. (3, 7) and (3, 7) f. (4, 2) and (6, 2)
Per c
en t a ge w ho d id n r teste ot h ave d for a flu H.I.V . shot in la Over st ye weig ar ht o r ob ese Ha ve at le a st on ea Have suffe l co red from pa in in last ye ho lic ar dri Sm nk oke pe rm on th Have a heart condition Nev e r had a ma m mo In po gram o r or (wom fair h en 4 ealth 0+) Neve
80 70 Percentage of group
12. a. Find the slope of the line through each of the following pairs of points. i. (1, 4) and (2, 4) ii. (7, 3) and (7, 3) iii. (2, 6) and (5, 6) b. Summarize your findings.
77
60 50 40 30 20
Have an ulcer
10 Ha ve diabetes 0
2
4
6
8
10
12
14
16
Years spent in school
Source: “A Surprising Secret to a Long Life, According to Studies: Stay in School,” New York Times, January 3, 2007.
a. Estimate the percentages for those with 8 years of education and for those with 16 years of education (college graduates) who: i. Are overweight or obese ii. Have at least one alcoholic drink per month iii. Smoke b. For each category in part (a), calculate the average rate of change with respect to years of education. What do they all have in common?
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c. What does the graph suggest about the link between education and health? Could there be other factors at play?
21. Read the Anthology article “Slopes” on the course website and then describe two practical applications of slopes, one of which is from your own experience.
2.4 Putting a Slant on Data Whenever anyone summarizes a set of data, choices are being made. One choice may not be more “correct” than another. But these choices can convey, either accidentally or on purpose, very different impressions.
Exploration 2.1 gives you a chance to put your own “spin” on data.
Slanting the Slope: Choosing Different End Points Within the same data set, one choice of end points may paint a rosy picture, while another choice may portray a more pessimistic outcome.
E X A M P L E
1
Poverty in the United States The data in Table 2.5 and the scatter plot in Figure 2.12 show the number of people below the poverty level in the United States from 1960 to 2008. How could we use the information to make the case that the poverty level has decreased? Has increased?
Year
Number of People in Poverty (in thousands)
1960 1970 1980 1990 2000 2008
39,851 25,420 29,272 33,585 31,581 38,110
Table 2.5 Source: U.S. Bureau of the Census, www. census.gov.
Number of people in poverty (thousands)
Putting a Slant on Poverty People in Poverty in the United States
45,000 Slope = –36,000 people/year
40,000
35,000
30,000 Slope = 334,000 people/year 25,000 20,000 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year
Figure 2.12 Number of people in poverty in the United States between 1960 and 2008.
S O L U T I O N
Optimistic case: To make an upbeat case that poverty numbers have decreased, we could choose as end points (1960, 39851) and (2008, 38110). Then change in no. of people in poverty (000s) change in years 38,110 2 39,851 5 2008 2 1960 21741 5 48 < 236 thousand people/year
average rate of change 5
So between 1960 and 2008 the number of impoverished individuals decreased on average by 36 thousand (or 36,000) each year. We can see this reflected in Figure 2.12 in the negative slope of the line connecting (1960, 39851) and (2008, 38110).
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79
Pessimistic case: To make a depressing case that poverty numbers have risen, we could choose (1970, 25420) and (2008, 38110) as end points. Then change in no. of people in poverty (000s) change in years 38,110 2 25,420 5 2008 2 1970 12,690 5 38 < 334 thousand people/year
average rate of change 5
So between 1970 and 2008 the number of impoverished individuals increased on average by 334 thousand (or 334,000) per year. This is reflected in Figure 2.12 in the positive slope of the line connecting (1970, 25420) and (2008, 38110). Both average rates of change are correct, but they give very different impressions of the changing number of people living in poverty in America.
Slanting the Data with Words and Graphs
The U.S. population increased by only 2.78 million/year between 2000 and 2010 Stretching the scale of the horizontal axis relative to the vertical axis makes the slope of the line look almost flat and hence minimizes the impression of change (Figure 2.13).
Slight Growth in the U.S. Population 350 Population (in millions)
See “4: Distortion by Clipping and Scaling” in Rates of Change to manipulate median household income.
If we wrap data in suggestive vocabulary and shape graphs to support a particular viewpoint, we can influence the interpretation of information. In Washington, D.C., this would be referred to as “putting a spin on the data.” Take a close look at the following three examples. Each contains exactly the same underlying facts: the same average rate of change calculation and a graph with a plot of the same two data points (2000, 281.4) and (2010, 309.2), representing the U.S. population (in millions) in 2000 and in 2010.
300 250 200 150 100 50 0 2000
2005
2010
Year
Figure 2.13
The U.S. population had an explosive growth of over 2.78 million/year between 2000 and 2010 Cropping the vertical axis (which now starts at 280 instead of 0) and stretching the scale of the vertical axis relative to the horizontal axis makes the slope of the line look steeper and strengthens the impression of dramatic change (Figure 2.14).
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Explosive Growth in the U.S. Population 310
Population (in millions)
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300
290
280 2000
2005 Year
2010
Figure 2.14
The U.S. population grew at a reasonable rate of 2.78 million/year between 2000 and 2010 Visually, the steepness of the line in Figure 2.15 seems to lie roughly halfway between the previous two graphs. In fact, the slope of 2.78 million/year is precisely the same for all three graphs.
Reasonable Growth in the U.S. Population 350
Population (in millions)
80
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300
250 2000
2005
2010
Year
Figure 2.15
How could you decide upon a “fair” interpretation of the data? You might try to put the data in context by asking: How does the growth between 2000 and 2010 compare with other decades in the history of the United States? How does it compare with growth in other countries at the same time? Was this rate of growth easily accommodated, or did it strain national resources and overload the infrastructure? A statistical claim is never completely free of bias. For every statistic that is quoted, others have been left out. This does not mean that you should discount all statistics. However, you will be best served by a thoughtful approach when
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interpreting the statistics you encounter daily. By getting in the habit of asking questions and then coming to your own conclusions, you will develop good sense about understanding data.
EXPLORE & EXTEND
Different Views of the Dow Jones In Exercise 2, Section 1.2, we looked at one time interval of the Dow Jones Industrial Average, the best-known index of U.S. stocks. The following charts of the Dow Jones all end in August 2009 but begin in earlier and earlier years, each chart giving drastically different impressions. Describe the overall impression of each separate graph, using flamboyant and overblown language.
March 2009–Aug. 2009 10,000 9,500 9,000 8,500 8,000 7,500 7,000
March 2009
April 2009
May 2009
June 2009
July 2009
Oct. 2007–Aug. 2009 16,000 14,000 12,000 10,000 8,000
2007
2008
2009
Jan. 1970–Aug. 2009 16,000 14,000 12,000 10,000 8,000 6,000 4,000 2,000 ’70 ’71 ’72 ’73 ’74 ’75 ’76 ’77 ’78 ’79 ’80 ’81 ’82 ’83 ’84 ’85 ’86 ’87 ’88 ’89 ’90 ’91 ’92 ’93 ’94 ’95 ’96 ’97 ’98 ’99 ’00 ’01 ’02 ’03 ’04 ’05 ’06 ’07 ’08 ’09
2.4
Source: fivecent.com
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Algebra Aerobics 2.4 1. The table shows the percentage of the U.S. population in poverty between 1960 and 2008.
3. Sketch a graph and compose a few sentences to forcefully convey the views of the following persons. a. You are an antiwar journalist reporting on American casualties during a war. In week 1 there were 17, in week 2 there were 29, and in week 3 there were 26. b. You are the president’s press secretary in charge of reporting war casualties (listed in part (a)) to the public. 4. The graph below appeared as part of an advertisement in the Boston Globe. Identify at least three strategies used to persuade you to buy gold.
% of U.S. Population in Poverty
Gold Explodes Experts Predict $1,500.00 an Ounce $380
Source: U.S. Bureau of the Census, www.census.gov.
$360
Up $80 in 11 Months
$350 $340 $330 $320 $310
Feb '03
Jan '03
Dec '02
Nov '02
Oct '02
Sept '02
$300 Apr '02
Use the data for 2008 and for a different year from the table to make the case that poverty: a. Has declined dramatically. b. Has remained relatively stable. c. Has increased substantially. 2. Assume you are the financial officer of a corporation whose stock earnings were $1.02 per share in 2008, and $1.12 per share in 2009, and $1.08 per share in 2010. How could you make a case for: a. Dramatic growth? b. Dramatic decline?
Gold prices/ounce
$370
Aug '02
22.2 12.6 13.0 13.5 11.3 12.6 12.3 12.5 13.2
July '02
1960 1970 1980 1990 2000 2005 2006 2007 2008
June '02
%
May '02
Year
Exercises for Section 2.4 Graphing program is optional for Exercise 10. Course software is required for Exercise 12.
Assuming the end date is at the beginning of May, estimate an earlier beginning date (different in each part (a)–(c)) to make the case that: a. Mortgage rates have dramatically declined. b. Mortgage rates have “exploded.” c. Mortgage rates have stayed the same.
1. Examine the following graph of 30-year fixed mortgage rates.
2. Compare the accompanying graphs. a. Which line appears to have the steeper slope? b. Use the intercepts to calculate the slope. Which graph actually does have the steeper slope?
30-Year Fixed Mortgage Rates: February to May 2010 5.28 5.24
Rate
5.20
National, 30 yr fixed mtg
y 24
4
5.12
2
5.08
x
x
5.04 5.00
y
6
5.16
–8 Feb ’10
Source: bankrate.com.
Mar
Apr
–6
–4
–2
–2
May
Graph A
2
–8
–4
4 –12
Graph B
8
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3. Compare the accompanying graphs. a. Which line appears to have the steeper slope? b. Use the intercepts to calculate the slope. Which graph has the more negative (and hence steeper) slope? 24 18 12 6
y
5 4 3 2 1
x –2
–1 –6
1
–1
2
Graph A
y
–1
1
x 2
Graph B
4. The following graphs show the same function graphed on different scales. a. In which graph does Q appear to be growing at a faster rate? b. In which graph does Q appear to be growing at a near zero rate? c. Explain why the graphs give different impressions.
100
Q
10
Q t 50
–50
–100
–10
Graph A
83
6. Generate two graphs and on each draw a line through the points (0, 3) and (4, 6), choosing x and y scales such that: a. The first line appears to have a slope of almost zero. b. The second line appears to have a very large positive slope. 7. a. Generate a graph of a line through the points (0, 2) and (5, 2). b. On a new grid, choose different scales so that the line through the same points appears to have a large positive slope. c. What have you discovered? 8. What are three adjectives (like “explosive”) that would imply rapid growth? 9. What are three adjectives (like “severe”) that would imply rapid decline? 10. (Graphing program optional.) Examine the data given on women in the U.S. military forces.
Women in Uniform: Female Active-Duty Personnel
t 5
–5
Putting a Slant on Data
Graph B
5. The following graph shows the National Assessment of Educational Progress (NAEP) math scores for California students, grades 4 and 8, as well as national math scores.
Year
Total
Army
Navy
1970 1980 1990 2000 2005 2008
41,479 171,418 227,018 204,498 189,465 200,337
16,724 69,338 83,621 72,021 71,400 73,902
8,683 34,980 59,907 53,920 54,800 50,008
Marine Corps 2,418 6,706 9,356 9,742 8,498 12,290
Air Force 13,654 60,394 74,134 68,815 54,767 64,137
Source: Department of Defense.
California Math Achievement 290 270 250 230 210 2000
2003
2005
2007
Grade 4 (CA)
Grade 8 (CA)
National Grade 4
National Grade 8
Source: National Assessment of Educational Progress (NAEP).
a. Summarize the message this graph conveys concerning achievement in math scores. b. What strategy is used to make the increase in math scores seem modest? How could you make the scores seem even more modest? c. What strategy could you use to make the increase more dramatic?
a. Make the case with graphs and numbers that women are a growing presence in the U.S. military. b. Make the case with graphs and numbers that women are a declining presence in the U.S. military. c. Write a paragraph that gives a balanced picture of the changing presence of women in the military using appropriate statistics to make your points. What additional data would be helpful? 11. On the following page there is a graph on AIDS in America similar to one that appeared in Chapter 1. The left-hand axis shows both the number of new AIDS cases and deaths from AIDS. In this new version, the right-hand axis shows the number of people living with AIDS. a. Find something encouraging to say about these data by using numerical evidence, including estimated average rates of change. b. Find numerical support for something discouraging to say about the data. c. How might we explain the enormous increase in new AIDS cases reported from 1985 to a peak in 1992–1993, and the drop-off the following year?
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600,000 500,000 400,000
60,000
300,000 200,000
30,000
100,000 0 1985
Number living with AIDS
New AIDS and deaths from AIDS
AIDS in America 90,000
14. The accompanying graph shows the number of Nobel Prizes awarded in science for various countries between 1901 and 1974. It contains accurate information but gives the impression that the number of prize winners declined drastically in the 1970s, which was not the case. What flaw in the construction of the graph leads to this impression?
Nobel Prizes Awarded in Science, for Selected Countries, 1901–1974
0 1990
1995 Year
New AIDS Cases
2000
2005
30
Deaths from AIDS
Living with AIDS
Source: Centers for Disease Control.
12. (Course software required.) Open “L6: Changing Axis Scales” in Linear Functions. Generate a line in the upper left-hand box. The same line will appear graphed in the three other boxes but with the axes scaled differently. Describe how the axes are rescaled in order to create such different impressions. 13. The accompanying graphs show the same data on the median income of a household headed by a black person as a percentage of the median income of a household headed by a white person. Describe the impression each graph gives and how that was achieved. (Data from the U.S. Bureau of the Census.)
Number of Nobel Prizes
84
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25 United States 20 15
United Kingdom
Germany 10
U.S.S.R
5
France
0 1901– 1910
1911– 1920
1921– 1930
1931– 1940
1941– 1950
1961– 1971– 1970 1974
1951– 1960
Source: E. R. Tufte, The Visual Display of Quantitative Information (Cheshire, Conn.: Graphics Press, 1983).
15. The following graph shows the trend in National Assessment of Educational Progress (NAEP) math scores of 17-year-old students by gender from 1973 to 2008.
Median Black Income as a Percentage of Median White Income
Trend in NAEP Mathematics Average Scores for 17-Year-Old Students, by Gender 340
64%
330 63%
320
Percentage
309
310
62%
300 61%
304
301
301 297
290
296
305 306
299
309 309 310
303 305 304
305
310 308 307 305
307
309 303
304
280
60%
0 59% 2000
’73 2002
2004 Year
’78
’82
’86
’90’92’94’96 ’99
’04
’08
Year
2006
Graph A
= Male
Median Black Income as a Percentage of Median White Income
= Female
Source: nationsreportcard.gov.
100%
Percent
80%
a. Summarize the message this graph conveys concerning achievement in math scores by gender. b. What strategy could be used to make the gap between genders seem greater?
60% 40% 20% 0% 2000 2001 2002
2003 2004 2005 2006 Year Graph B
2007
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Linear Functions: When Rates of Change Are Constant
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2.5 Linear Functions: When Rates of Change Are Constant In many of our examples so far, the average rate of change has varied depending on the choice of end points. Now we will examine the special case when the average rate of change remains constant.
What If the U.S. Population Had Grown at a Constant Rate? A Hypothetical Example In Section 2.2 we calculated the average rate of change in the population between 1790 and 1800 as 0.14 million people/year. We saw that the average rate of change was different for different decades. What if the average rate of change had remained constant? What if in every decade after 1790 the U.S. population had continued to grow at the same rate of 0.14 million people/year? That would mean that starting with a population estimated at 3.9 million in 1790, the population would have grown by 0.14 million each year. The slopes of all the little line segments connecting adjacent population data points would be identical, namely 0.14 million people/year. The graph would be a straight line, indicating a constant average rate of change. On the graph of actual U.S. population data, the slopes of the line segments connecting adjacent points are increasing, so the graph curves upward. Figure 2.16 compares the actual and hypothetical results.
U.S. Population: A Hypothetical Example 350
Population (in millions)
300
250
Actual growth
200
150
100 Hypothetical growth at a constant rate
50
0 1790
1830
1870
1910 Year
1950
1990 2010
Figure 2.16
Any function that has the same average rate of change on every interval has a graph that is a straight line. The function is called linear. This hypothetical example represents a linear function. When the average rate of change is constant, we can drop the word “average” and just say “rate of change.” A linear function has a constant rate of change. Its graph is a straight line.
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The Relationship between Velocity and Distance Open up the course software “C3: Average Velocity and Distance” in Rates of Change. a. Press and drag several velocity values. Describe the corresponding effect on the distance graph. What is the relationship between the two graphs? b. Set all the velocity values to the same value (of your choice). What is the effect on the total distance graph? c. If you set the velocities all at another fixed value, what would the distance graph look like now? d. What could you conclude from parts (b) and (c)?
Real Examples of a Constant Rate of Change Linear functions are frequently used to model real life situations. E X A M P L E
1
Median weight for female infants According to the standardized growth and development charts used by many American pediatricians, the median weight for girls during their first six months of life increases at an almost constant rate. Starting at 7.0 pounds at birth, female median weight increases by approximately 1.5 pounds per month. If we assume that the median weight for females, W, is increasing at a constant rate of 1.5 pounds per month, then W is a linear function of age in months, A. a. Generate a table that gives the median weight for females, W, for the first six months of life and create a graph of W as a function of A. b. Find an equation for W as a function of A. What is an appropriate domain for this function? An appropriate range? c. Express the equation for part (b) using only units of measure. a. For female infants at birth (A 0 months), the median weight is 7.0 lb (W 7.0 lb). The rate of change, 1.5 pounds/month, means that as age increases by 1 month, weight increases by 1.5 pounds. See Table 2.6. The dotted line in Figure 2.17 shows the trend in the data. Graph of Girls' Weight
Median Weight for Girls A, Age (months) 0 1 2 3 4 5 6
W
W, Weight (lb) 7.0 8.5 10.0 11.5 13.0 14.5 16.0
Table 2.6 Source: Data derived from the Ross Growth and Development Program, Ross Laboratories, Columbus, OH.
20 Weight (in pounds)
S O L U T I O N
15
10
5
0
2 4 Age (in months)
6
A
Figure 2.17 Median weight for girls.
b. To find a linear equation for W (median weight in pounds) as a function of A (age in months), we can study the table of values in Table 2.6. W 5 initial weight 1 weight gained 5 initial weight 1 rate of growth ? number of months 5 7.0 lb 1 1.5 lb/month ? number of months
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The equation would be W 5 7.0 1 1.5A An appropriate domain would be 0 # A # 6, and range 7.0 # W # 16.0. c. Since our equation represents quantities in the real world, each term in the equation has units attached to it. The median weight W is in pounds (lb), rate of change is in lb/month, and age, A, is in months. If we display the equation W 5 7.0 1 1.5A showing only the units, we have lb 5 lb 1 a
lb bmonth month
The rules for canceling units are the same as the rules for canceling numbers in fractions. So, lb 5 lb 1 a
lb bmonth month
lb 5 lb 1 lb This equation makes sense in terms of the original problem since pounds (lb) added to pounds (lb) should give us pounds (lb). 2
Median weight for baby boys vs. baby girls a. If the median birth weight for baby boys is the same as for baby girls, but boys put on weight at a faster rate, which numbers in the previous weight model would change? Which would stay the same? b. What would be different about the two graphs?
S O L U T I O N
a. The initial birth weight for males would be 7 lb (the same as for females), but the rate of growth (or slope) for males would be 1.5 lb/month (greater than that for females.) b. Both graphs would have the same vertical intercept, but the male graph would be steeper.
E X A M P L E
3
Depreciation of computer cost You spend $1200 on a computer and for tax purposes choose to depreciate it (or assume it decreases in value) to $0 at a constant rate over a 5-year period. a. Calculate the rate of change of the assumed value of the equipment over 5 years. What are the units? b. Create a table and graph showing the value of the equipment over 5 years. c. Create a function for the value of the computer over time in years. Why is this a linear function? d. What is an appropriate domain for this function? What is the range?
S O L U T I O N
a. After 5 years, your computer is worth $0. If V(t) is the value of your computer in dollars as a function of t, the number of years you own the computer, then the rate of change of V(t) from t 0 to t 5 is
E X A M P L E
rate of change 5
change in value V 5 change in time t
52
$1200 5 2$240/year 5 years
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Thus, the worth of your computer drops at a constant rate of $240 per year. The rate of change in V(t) is negative because the worth of the computer decreases over time. The units for the rate of change are dollars per year. b. Table 2.7 and Figure 2.18 show the depreciated value of the computer.
Value of Computer Depreciated over 5 Years Numbers of Years
Value of Computer ($)
0 1 2 3 4 5
1200 960 720 480 240 0
Table 2.7
$1400 Value of computer
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$1200 $1000 $800 $600 $400 $200 $0
0
1
2 3 Number of years
4
5
Figure 2.18 Value of computer over 5 years.
c. To find a linear function, V(t), think about how we found the table of values. value of computer 5 initial value 1 (rate of decline) ? (number of years) V(t) 5 $1200 1 (2$240/year) ? t V(t) 5 1200 2 240t This equation describes V(t) as a function of t because for every value of t, there is one and only one value of V(t). It is a linear function because the rate of change is constant. d. The domain is 0 ⱕ t ⱕ 5 and the range is 0 ⱕ V(t) ⱕ 1200.
The General Equation for a Linear Function The equations in Examples 1 and 3 can be rewritten in terms of the output (dependent variable) and the input (independent variable). weight 5 initial weight 1 rate of growth ? number of months value of computer 5 initial value 1 rate of decline ? number of years output 5 initial value 1 rate of change ? input 3 3 5 5 x y m b So, the general linear equation can be written in the form y 5 b 1 mx where we use the traditional mathematical choices of y for the output (dependent variable) and x for the input (independent variable). We let m stand for the rate of change; so ⌬ output change in y ⌬y 5 5 ⌬ input change in x ⌬x 5 slope of the graph of the line
m5
In mathematical models of the form y ⫽ b ⫹ mx, we often call b the initial or base value. Why? Many models exclude negative numbers (like the prior examples on
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infant weight or computer depreciation), so the smallest value in the domain is often 0. When x 0, then
y x
y5b1m?0
y x
y5b x
The point (0, b) satisfies the equation and is the vertical intercept. Since the coordinate b tells us where the line crosses the y-axis, we often just refer to b as the vertical or y-intercept (see Figure 2.19).
A Linear Function A function y ƒ (x) is called linear if it can be represented by an equation of the form
Figure 2.19 Graph of y b mx.
y 5 b 1 mx Its graph is a straight line where m is the slope, the rate of change of y with respect to x. So if (x1, y1) and (x2, y2 ) are any two distinct points on the line, then the slope 5
y2 2 y1 y 5 5 m 5 average rate of change x2 2 x1 x
b is the vertical or y-intercept and is the value of y when x 0. If the function is a mathematical model, b typically represents an initial or starting value.
The equation y b mx could, of course, be written in the equivalent form y mx b. This may be the traditional format that you first learned. E X A M P L E
4
S O L U T I O N
E X A M P L E
5
Identifying the vertical intercept and slope For each of the following equations, identify the value of b and the value of m. a. y 5 24 1 3.25x c. g(x) 4x 3.25 b. f(x) 5 3.25x 2 4 d. y 5 3.25 2 4x a. b. c. d.
b 4 and m 3.25 b 4 and m 3.25 b 3.25 and m 4 b 3.25 and m 4
Comparing legal costs In the following equations, L(h) represents the legal fees (in dollars) charged by four different law firms and h represents the number of hours of legal advice. L1 (h) 5 500 1 200h
L3 (h) 5 800 1 350h
L2 (h) 5 1000 1 150h
L4 (h) 5 500h
a. Which initial fee is the highest? b. Which rate per hour is the highest? c. If you need 5 hours of legal advice, which legal fee will be the highest? S O L U T I O N
a. L2 has the highest initial fee of $1000. b. L4 has the highest rate of $500 per hour.
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c. Evaluate each equation for h 5 hours. L1 (5) 5 500 1 200(5) 5 $1500 L 2 (5) 5 1000 1 150(5) 5 $1750
L3 (5) 5 800 1 350(5) 5 $2550 L 4 (5) 5 500(5) 5 $2500
For 5 hours of legal advice, L3 has the highest legal fee.
Algebra Aerobics 2.5 1. From Figure 2.17, estimate the weight W of a baby girl who is 4.5 months old. Then use the equation W 5 7.0 1 1.5A to calculate the corresponding value for W. How close is your estimate? 2. From the same graph, estimate the age of a baby girl who weighs 11 pounds. Then use the equation to calculate the value for A. 3. a. If C 5 15P 1 10 describes the relationship between the number of persons (P) in a dining party and the total cost in dollars (C) of their meals, what is the unit of measure for 15? For 10? b. The equation W 5 7.0 1 1.5A (modeling weight, W, as a function of age, A) expressed in units of measure only is lb 5 lb 1 a
lb b months month
Express C 5 15P 1 10 from part (a) in units of measure only. 4. (True story.) A teenager travels to Alaska with his parents and wins $1200 in a rubber ducky race. (The race releases 5000 yellow rubber ducks marked with successive integers from one bridge over a river and collects them at the next bridge.) Upon returning home he opens up a “Rubber Ducky Savings Account,” deposits his winnings, and continues to deposit $50 each month. If D 5 amount in the account and M 5 months since the creation of the account, then D 5 1200 1 50 M describes the amount in the account after M months. a. What are the units for 1200? For 50? b. Express the equation using only units of measure. 5. Assume S 5 0.8Y 1 19 describes the projected relationship between S, sales of a company (in millions of dollars), for Y years from today. a. What are the units of 0.8 and what does it represent? b. What are the units of 19 and what does it represent?
c. What would be the projected company sales in three years? d. Express the equation using only units of measure. 6. Assume C 5 0.45N represents the total cost C (in dollars) of operating a car for N miles. a. What does 0.45 represent and what are its units? b. Find the total cost to operate a car that has been driven 25,000 miles. c. Express the equation using only units of measure. 7. The relationship between the balance B (in dollars) left on a mortgage loan (including interest) and N, the number of monthly payments, is given by B 5 302,400 2 840N. a. What is the monthly mortgage payment? b. What does 302,400 represent? c. What is the balance on the mortgage after 10 years? 20 years? 30 years? (Hint: Remember there are 12 months in a year.) 8. Identify the slope, m, and the vertical intercept, b, of the line for each equation. a. y 5 5x 1 3 e. ƒ (x) 5 7.0 2 x b. y 5 5 1 3x f. h(x) 5 211x 1 10 c. y 5 5x g. y 5 1 2 23 x d. y 5 3 h. 2y 1 6 5 10x 9. If ƒ a. b. c.
(x) 5 50 2 25x: Why does ƒ (x) describe a linear function? Evaluate ƒ (0) and ƒ (2). Use your answers in part (b) to verify that the slope is 225. 10. Identify the functions that are linear. For each linear function, identify the slope and the vertical intercept. a. ƒ (x) 5 3x 1 5 c. ƒ(x) 5 3x2 1 2 b. ƒ (x) 5 x d. ƒ(x) 5 4 2 23 x 11. Write an equation for the line in the form y 5 b 1 mx for the indicated values. a. m 5 3 and b 5 4 b. m 5 21 and passes through the origin c. m 5 0 and b 5 23
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12. Write the equation of the graph of each of the four lines below in y 5 b 1 mx form. Use the y-intercept and the slope indicated on each graph. 8
y
2
y
m⫽4
4
m⫽–2 –1
–2
–1
–2
1
2
1
y
4
x 2
x x
Graph A
–2
Graph B
m⫽1
m⫽0
x –3
y
–1 Graph C
2
4
–2
–1
1
2
–3 Graph D
Four linear graphs
Exercises for Section 2.5 You might wish to hone your algebraic mechanical skills with three programs in the course software; Linear Functions: “L1: Finding m & b”, “L3: Finding a Line through 2 Points” and “L4: Finding 2 Points on a Line.” They offer practice in predicting values for m and b, generating linear equations, and finding corresponding solutions. 1. Consider the equation A 5 300 1 50n. a. Find the value of A for n 5 0, 1, 20. b. Express your answers to part (a) as points with coordinates (n, A). 2. Consider the equation B 5 450 1 40n. a. Find the value of B for n 5 0, 1, 20. b. Express your answers to part (a) as points with coordinates (n, B). 3. Determine if any of the following points satisfy one or both of the equations in Exercises 1 and 2. a. (300, 0) b. (10, 850) c. (15, 1050) 4. Suppose during a 5-year period the profit P( Y ) (in billions of dollars) for a large corporation was given by P(Y ) 5 7 1 2Y, where Y represents the year. a. Fill in the chart. Y P(Y )
0
1
2
3
4
b. What are the units of P( Y )? c. What does the 2 in the equation represent, and what are its units? d. What was the initial profit? 5. Consider the equation D 5 3.40 1 0.11n a. Find the values of D for n 5 0, 1, 2, 3, 4. b. If D represents the average consumer debt, in thousands of dollars, over n years, what does 0.11 represent? What are its units? c. What does 3.40 represent? What are its units? 6. Suppose the equations E(n) 5 5000 1 1000n and G(n) 5 12,000 1 800n give the total cost of operating an electrical (E) versus a gas (G) heating/cooling system in a home for n years.
a. Find the cost of heating a home using electricity for 10 years. b. Find the cost of heating a home using gas for 10 years. c. Find the initial (or installation) cost for each system. d. Determine how many years it will take before $40,000 has been spent in heating/cooling a home that uses: i. Electricity ii. Gas 7. If the equation E(n) 5 5000 1 1000n gives the total cost of heating/cooling a home after n years, rewrite the equation using only units of measure. 8. Over a 5-month period at Acadia National Park in Maine, the mean night temperature increased on average 5 degrees Fahrenheit per month. If the initial temperature is 25 degrees Fahrenheit, create a function for the night temperature N(t ) for month t, where 0 # t # 4. 9. A residential customer in the Midwest purchases gas from a utility company that charges according to the formula C(g) 5 11 1 10.50(g), where C(g) is the cost, in dollars, for g thousand cubic feet of gas. a. Find C(0), C(5), and C(10). b. What is the cost if the customer uses no gas? c. What is the rate per thousand cubic feet charged for using the gas? d. How much would it cost if the customer uses 96 thousand cubic feet of gas (the amount an average Midwest household consumes during the winter months)? 10. Create the formula for converting degrees centigrade, C, to degrees Fahrenheit, F, if for every increase of 5 degrees centigrade the Fahrenheit temperature increases by 9 degrees, with an initial point of (C, F ) 5 (0, 32). 11. Determine the vertical intercept and the rate of change for each of these formulas: a. P 5 4s c. C 5 2pr b. C 5 pd d. C 5 59 F 2 17.78 12. A hiker can walk 2 miles in 45 minutes. a. What is his average speed in miles per hour? b. What formula can be used to find the distance traveled, d, in miles in t hours?
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13. The cost C(x) of producing x items is determined by adding fixed costs to the product of the cost per item and the number of items produced, x. Below are three possible cost functions C(x), measured in dollars. Match each description in parts (e) 2 (g) with the most likely cost function and explain your choices. a. C(x) 5 125,000 1 75x b. C(x) 5 400,000 1 0.30x c. C(x) 5 250,000 1 800x e. The cost of producing a computer f. The cost of producing a college algebra text g. The cost of producing a CD 14. A new $25,000 car depreciates in value by $5000 per year. Construct a linear function for the value V of the car after t years. 15. The state of Pennsylvania has a 6% sales tax on taxable items. (Note: Clothes, food, and certain pharmaceuticals are not taxed in Pennsylvania.) a. Create a formula for the total cost (in dollars) of an item C(p) with a price tag p. (Be sure to include the sales tax.) b. Find C(9.50), C(115.25), and C(1899). What are the units? 16. a. If S(x) 5 20,000 1 1000x describes the annual salary in dollars for a person who has worked for x years for the Acme Corporation, what is the unit of measure for 20,000? For 1000? b. Rewrite S(x) as an equation using only units of measure. c. Evaluate S(x) for x values of 0, 5, and 10 years. d. How many years will it take for a person to earn an annual salary of $43,000? 17. The following represent linear equations written using only units of measure. In each case supply the missing unit. a. inches 5 inches 1 (inches/hour) ? (?) b. miles 5 miles 1 (?) ? (gallons) c. calories 5 calories 1 (?) ? (grams of fat) 18. Identify the slopes and the vertical intercepts of the lines with the given equations. a. y 5 3 1 5x d. Q 5 35t 2 10 b. ƒ (t) 5 2t e. ƒ (E) 5 10,000 1 3000E c. y 5 4
19. For each of the following, find the slope and the vertical intercept, then sketch the graph. (Hint: Find two points on the line.) a. y 5 0.4x 2 20 b. P 5 4000 2 200C 20. Construct an equation and sketch the graph of its line with the given slope, m, and vertical intercept, b. (Hint: Find two points on the line.) a. m 5 2, b 5 23 c. m 5 0, b 5 50 b. m 5 234, b 5 1 In Exercises 21 to 23 find an equation, generate a small table of solutions, and sketch the graph of the line with the indicated attributes. 21. A line that has a vertical intercept of 22 and a slope of 3. 22. A line that crosses the vertical axis at 3.0 and has a rate of change of 22.5. 23. A line that has a vertical intercept of 1.5 and a slope of 0. 24. Estimate b (the y-intercept) and m (the slope) for each of the accompanying graphs. Then, for each graph, write the corresponding linear function. 8
y
8
x
–8
8
x
–8
–8
8
–8
Graph A
Graph C
8
y
8
x
–8
8
y
x
–8
–8 Graph B
y
8
–8 Graph D
2.6 Visualizing Linear Functions The values for b and m in the general form of the linear equation, y b mx, tell us about the graph of the function. The value for b tells us where to anchor the line on the y-axis. The value for m tells us whether the line climbs or falls and how steep it is. EXPLORE & EXTEND
2.6
Making Predictions About the Effects of m and b Each chapter in which a new function is introduced has a “Visualizing” section accompanied by course software called “sliders.” For linear functions, open up the software “L1: m and b Sliders” to explore the effect of m and b on the graph of y mx b. (We recommend that you do this with a partner.)
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a. Fix a value for b. Then construct and compare four graphs (of different colors) using the “m” slider. If you fix another value for b and construct four new graphs, do your previous conclusions still hold? b. Fix a value for m. Then once again construct and compare four graphs (of different colors), now using the “b” slider. Pick a new m and repeat the process. What are your conclusions? c. Write a 60-second summary on the effect of m and b on the graph of y ⫽ mx ⫹ b.
The Effect of b In the equation y ⫽ b ⫹ mx, the value b is the vertical intercept, so it anchors the line at the point (0, b) (see Figure 2.20).
8y y = 3 + 0.5x
(b = 3)
y = 0.5x
(b = 0)
y = –2 + 0.5x
(b = –2)
(0, 3) (0, 0) –8
x 8
(0, –2)
–8
Figure 2.20 The effect of b, the vertical intercept.
E X A M P L E
1
Differing graphs Explain how the graph of y ⫽ 4 ⫹ 5x differs from the graphs of the following functions: a. y ⫽ 8 ⫹ 5x b. y ⫽ 2 ⫹ 5x c. y ⫽ 5x ⫹ 4
S O L U T I O N
Although all of the graphs are straight lines with a slope of 5, they each have a different vertical intercept. The graph of y ⫽ 4 ⫹ 5x has a vertical intercept at 4. a. The graph of y ⫽ 8 ⫹ 5x intersects the y-axis at 8, four units above the graph of y ⫽ 4 ⫹ 5x. b. The graph of y ⫽ 2 ⫹ 5x intersects the y-axis at 2, two units below the graph of y ⫽ 4 ⫹ 5x. c. Since y ⫽ 5x ⫹ 4 and y ⫽ 4 ⫹ 5x are equivalent equations, they have the same graph.
The Effect of m The sign of m The sign of m in the equation y ⫽ b ⫹ mx determines whether the line climbs (slopes up) or falls (slopes down) as we move from left to right on the graph. If m is positive, the line climbs from left to right (as x increases, y increases). If m is negative, the line falls from left to right (as x increases, y decreases) (see Figure 2.21).
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x y ⫽ mx (m ⬍ 0) negative slope
Figure 2.21 The effect of the sign of m.
E X A M P L E
2
Matching graphs and equations Pair each graph in Figure 2.22 with a matching equation.
12
y A
ƒ (x) 5 9 2 0.4x g(x) 5 24 1 x h(x) 5 9 2 0.2x i(x) 5 24 1 0.25x j(x) 5 9 1 0.125x k(x) 5 4 2 0.25x
B C D E x 8 F
–8
–12
Figure 2.22 Graphs of multiple linear functions.
S O L U T I O N
A is the graph of j(x). B is the graph of h(x). C is the graph of ƒ (x). D is the graph of g(x). E is the graph of k(x). F is the graph of i(x).
The magnitude of m The magnitude (absolute) value of m determines the steepness of the line. Recall that the absolute value of m is the value of m stripped of its sign; for example, k ⫺3 k ⫽ 3.
The magnitude (absolute value) of any number x is written as k x k and strips x of its sign. So k x k is always ⱖ 0. For example, k ⫺7 k ⫽ k 7 k ⫽ 7.
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The greater the magnitude k m k , the steeper the slope of the line. This makes sense since m is the slope or rate of change of y with respect to x. E X A M P L E
3
Comparing the steepness of slopes a. Which function in Figure 2.23 (with positive slopes) has the steepest slope?
16
y
y ⫽ 7⫹5x y ⫽ 7⫹3x y ⫽ 7⫹x
16
y
y ⫽ 7⫺x x
x –8
8
–16
Figure 2.23 Graphs with positive values for m.
–8
8
–16
y ⫽ 7⫺3x y ⫽ 7⫺5x
Figure 2.24 Graphs with negative values for m.
b. Which function in Figure 2.24 (with negative slopes) has the steepest slope? c. Considering both Figures 2.23 and 2.24 which of all the lines is steepest? S O L U T I O N
a. The steepness of the lines increases as the magnitude of m (⫽ k m k ) increases. So in Figure 2.23 the slope (m ⫽ 5) of y ⫽ 7 ⫹ 5x is steeper than both the slope (m ⫽ 3) of y ⫽ 7 ⫹ 3x and the slope (m ⫽ 1) of y ⫽ 7 ⫹ x. b. In Figure 2.24, the slope (m ⫽ ⫺5) of y ⫽ 7 ⫺ 5x is steeper than both the slope (m ⫽ ⫺3) of y ⫽ 7 ⫺ 3x and the slope (m ⫽ ⫺1) of y ⫽ 7 ⫺ x, since k ⫺5 k ⫽ 5 ⬎ |⫺3| ⫽ 3 ⬎ k ⫺1k ⫽ 1. c. In Figure 2.24, the slopes (m) are all negative, but the steepness ( k m k ) depends upon the absolute value of the slope. So the lines y ⫽ 7 ⫺ 5x and y ⫽ 7 ⫹ 5x have the same steepness of 5 since k ⫺5 k ⫽ k 5 k ⫽ 5.
E X A M P L E
4
Finding the steepest slope from equations Without graphing the following functions, how can you tell which graph will have the steepest slope? a. ƒ (x) ⫽ 5 ⫺ 2x b. g(x) ⫽ 5 ⫹ 4x c. h(x) ⫽ 3 ⫺ 6x
S O L U T I O N
The graph of the function h will be steeper than the graphs of the functions f and g since the magnitude of m is greater for h than for f or g. The greater the magnitude of m, the steeper the graph of the line. For ƒ (x), the magnitude of the slope is k ⫺2 k ⫽ 2. For g(x), the magnitude of the slope is k 4 k ⫽ 4. For h(x), the magnitude of the slope is k ⫺6 k ⫽ 6.
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E X A M P L E
5
Finding the steepest slope from graphs Which of the graphs in Figure 2.25 has the steeper slope? 50
y
25
50
y
25
A
B
x –10
–5
5
10
x –50
–25
25
–25
–25
–50
–50
Graph A
50
Graph B
Figure 2.25 Comparing slopes. S O L U T I O N
Remember from Section 2.4 that the steepness of a linear graph is not related to its visual impression, but to the numerical magnitude of the slope. The scales of the horizontal axes are different for the two graphs in Figure 2.25, so the impression of relative steepness is deceiving. Line A passes through (0, 0) and (10, 25), so its slope is (25 ⫺ 0)/(10 ⫺ 0) ⫽ 2.5. Line B passes through (0, 0) and (50, 50) so its slope is (50 ⫺ 0)/(50 ⫺ 0) ⫽ 1. So line A has a steeper slope than line B.
Domain and Range The general linear function y ⫽ b ⫹ mx has a defined domain (for the input) and range (for the output). Barring any constraints on the input or output then: Domain: The domain is all real numbers; that is, all x in the interval (⫺⬁, ⫹⬁). Range: The range is all real numbers; that is, all y in the interval (⫺⬁, ⫹⬁), except for the horizontal line y ⫽ b (where m ⫽ 0), where the range is the single number b. Specific examples however may place restrictions on the domain and range, such as in Section 2.5, Example 1 on the median weight of female infants and Example 3 on the depreciation of computer costs. E X A M P L E
6
Domain and Range Identify the domain and range for: a. y ⫽ 2.5 ⫹ 4x
S O L U T I O N
b. y ⫽ 2.5 ⫺ 4x
c. y ⫽ 2.5
a. Domain: all x in the interval (⫺⬁, ⫹⬁). Range: all y in the interval (⫺⬁, ⫹⬁). b. Domain: all x in the interval (⫺⬁, ⫹⬁). Range: all y in the interval (⫺⬁, ⫹⬁). c. Domain: all x in the interval (⫺⬁, ⫹⬁). Range: y ⫽ 2.5 The Graph of a Linear Function The graph of the linear function y ⫽ b ⫹ mx is a straight line. The y-intercept, b, tells us where the line crosses the y-axis. The slope, m, tells us how fast the line is climbing or falling. The larger the magnitude (or absolute value) of m, the steeper the graph. If the slope, m, is positive, the line climbs from left to right. If m is negative, the line falls from left to right.
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Algebra Aerobics 2.6 1. Place these numbers in order from smallest to largest. 2.
3.
4.
5.
6.
k 212 k , k 27 k , k 23 k , k 21 k , 0, 4, 9 Without graphing the function, explain how the graph y 5 6x 2 2 differs from the graph of a. y 5 6x c. y 5 22 1 3x b. y 5 2 1 6x d. y 5 22 2 2x Without graphing, order the graphs of the functions from least steep to steepest. a. y 5 100 2 2x c. y 5 23x 2 5 b. y 5 1 2 x d. y 5 3 2 5x On an x-y coordinate system, draw a line with a positive slope and label it ƒ (x). a. Draw a line g(x) with the same slope but a y-intercept three units above ƒ (x). b. Draw a line h(x) with the same slope but a y-intercept four units below ƒ (x). c. Draw a line k(x) with the same steepness as ƒ (x) but with a negative slope. Which function has the steepest slope? Create a table of values for each function and graph the function to show that this is true. a. ƒ (x) 5 3x 2 5 b. g(x) 5 7 2 8x Create three functions with a y-intercept of 4 and three different negative slopes. Indicate which function has the steepest slope.
7. Match the four graphs below with the given functions. a. ƒ (x) 5 2 1 3x b. g(x) 5 22 2 3x c. h(x) 5 12 x 2 2 d. k(x) 5 2 2 3x 10
y
10
–10
10
x
–10
10
x
–10
–10
Graph A 10
y
Graph C y
y 10
–10
10
x
–10
10
–10
x
–10
Graph B
Graph D
Exercises for Section 2.6 1. Assuming m is the slope, identify the graph(s) where: a. m 5 3 b. k m k 5 3 c. m 5 0 d. m 5 23 y
6
6
2. Which line(s), if any, have a slope m such that: a. k m k 5 2? b. m 5 12 ?
y 10
y
A –6
–6
6
B
x
x
C
9 5
–6
–6 Graph C
Graph A 6
y 6
y
–5
x –6
6
–6
6
–6
–6
Graph B
Graph D
x
0
5
x
3. For each set of conditions, construct a linear equation and draw its graph. a. A slope of zero and a y-intercept of 23 b. A positive slope and a vertical intercept of 23 c. A slope of 23 and a vertical intercept that is positive
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4. Construct a linear equation for each of the following conditions. a. A negative slope and a positive y-intercept b. A positive slope and a vertical intercept of 210.3 c. A constant rate of change of $1300/year 5. Match the graph with the correct equation. x a. y 5 x b. y 5 2x c. y 5 d. y 5 4x 2 15
y
x
x
–10
3
10
–6 –15 Graph A 3
Graph C
y
15
y
x –3
x
3
6
–6
–3 –15 Graph D
Graph B
6. Match the function with its graph. (Note: There is one graph that has no match.) a. f(x) 5 24 1 3x b. g(x) 5 23x 1 4 c. h(x) 5 4 1 3x 6
5
–5
5
x
–5
5
x
–5
–4 Graph A
8. Which equation has the steepest slope? a. y 5 2 2 7x b. y 5 2x 1 7 c. y 5 22 1 7x 9. Given the function Q(t) 5 13 2 5t, construct a related function whose graph: a. Lies five units above the graph of Q(t) b. Lies three units below the graph of Q(t) c. Has the same vertical intercept as Q(t) d. Has the same slope as Q(t) e. Has the same steepness as Q(t) but the slope is positive
y 6
–3
7. In each part construct three different linear equations that all have: a. The same slope b. The same vertical intercept
10. Given the equation C(n) 5 30 1 15n, construct a related equation whose graph: a. Is steeper b. Is flatter (less steep) c. Has the same steepness, but the slope is negative 11. On the same grid, graph (and label with the correct equation) three lines that go through the point (0, 2) and have the following slopes: a. m 5 12 b. m 5 2 c. m 5 56 12. On the same grid, graph (and label with the correct equation) three lines that go through the point (0, 2) and have the following slopes: a. m 5 212 b. m 5 22 c. m 5 256 13. Given the following graphs of three straight lines: a. Which has the steepest slope? b. Which has the flattest slope? c. If the slopes of the lines A, B, and C are m1, m2, and m3, respectively, list them in increasing numerical order. d. In this example, why does the line with the steepest slope have the smallest numerical value? y
Graph C
4
A
7
B
C
–5
5
x
x –5
–6 Graph B
5 –3 Graph D
x
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2.7 Constructing Graphs and Equations of Linear Functions The program “L4: Finding 2 Points on a Line” in Linear Functions creates a linear function and asks you to identify two points that lie on its graph.
E X A M P L E
1
There are many different strategies for finding the equations and graphs of linear functions.
Finding the Graph Given the equation A forestry study measured the diameter of the trunk of a red oak tree over 5 years. The scientists created a linear model D( Y) ⫽ 1 ⫹ 0.13Y, where D( Y) ⫽ diameter in inches and Y ⫽ number of years from the beginning of the study. a. What do the numbers 1 and 0.13 represent in this context? b. Sketch a graph of the function model.
S O L U T I O N
a. The number 1 represents a starting diameter of 1 inch. The number 0.13 represents the annual growth rate of the oak’s diameter (change in diameter/change in time), 0.13 inches per year. b. The linear function D(Y) ⫽ 1 ⫹ 0.13Y tells us that 1 is the vertical intercept, so the point (0, 1) lies on the graph. The graph represents solutions to the equation. So to find a second point, we can evaluate D(Y ) for any other value of Y. If we set Y ⫽ 1, then D( Y) ⫽ 1 ⫹ (0.13 ⭈ 1) ⫽ 1 ⫹ 0.13 ⫽ 1.13. So (1, 1.13) is another point on the line. Since two points determine a line, we can sketch our line through (0, 1) and (1, 1.13) (see Figure 2.26). 1.8 D(y)
Diameter (inches)
1.6 1.4
(1, 1.13)
1.2 1.0 0.8 0.6 0.4 0.2 0
Y 0
1
2 3 Year of study
4
5
Figure 2.26 The diameter of a red oak over time.
2
Given the vertical intercept and the slope The top speed a snowplow can travel on dry pavement is 40 miles per hour, which decreases by 0.8 miles per hour with each inch of snow on the highway. a. Construct an equation describing the relationship between snowplow speed and snow depth. b. Determine a reasonable domain and corresponding range, and then graph the function.
S O L U T I O N
a. If we think of the snow depth, D, as determining the snowplow speed, P, then we need an equation of the form P 5 b 1 mD. If there is no snow, then the snowplow can travel at its maximum speed of 40 mph; that is, when D 5 0, then P 5 40. So the point (0, 40) lies on the line, making the vertical intercept b 5 40. The rate of change, m, (change in snowplow speed)/(change in snow depth) 5 20.8 mph per inch of snow. So the desired equation is
E X A M P L E
P 5 40 2 0.8D
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b. Consider the snowplow as only going forward (i.e., not backing up). Then the snowplow speed does not go below 0 mph. So if we let P 5 0 and solve for D, we have 0 5 40 2 0.8D 0.8D 5 40 D 5 50 So when the snow depth reaches 50 inches, the plow is no longer able to move. A reasonable domain then would be 0 ⱕ D ⱕ 50. The corresponding range would be 0 ⱕ P ⱕ 40. (See Figure 2.27.)
Snowplow speed (mph)
100
11/27/10
50
P
40 30 20 10 0
10 20 30 40 50 Snow depth (inches)
D 60
Figure 2.27 Snowplow speed versus snow depth.
3
Given a point off the y-axis and the slope Given a point (2, 3) and a slope of m ⫽ ⫺5/4, describe at least two ways you could find a second point to plot a line with these characteristics without constructing the equation.
S O L U T I O N
Plot the point (2, 3). a. If we write the slope as (⫺5)/4, then a change of 4 in x corresponds to a change of ⫺5 in y. So starting at (2, 3), moving horizontally four units to the right (adding 4 to the x-coordinate), and then moving vertically five units down (subtracting 5 from the y-coordinate) gives us a second point on the line at (2 ⫹ 4, 3 ⫺ 5) ⫽ (6, ⫺2). Now we can plot our second point (6, ⫺2) and draw the line through it and our original point, (2, 3) (see Figure 2.28).
E X A M P L E
y
6
4 (2, 3) –5
x –2
8 (6, –2) –4
Figure 2.28 Graph of the line through (2, 3) with slope ⫺5/4.
b. If we are modeling real data, we are more likely to convert the slope to decimal form. In this case m ⫽ (⫺5)/4 ⫽ ⫺1.25. We can generate a new point on the line by starting at (2, 3) and moving 1 unit to the right and down (since m is negative) ⫺1.25 units to get the point (2 ⫹ 1, 3 ⫺ 1.25) ⫽ (3, 1.75).
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101
Given a general description A recent study reporting on the number of smokers showed: a. b. c. d.
A linear increase in Georgia A linear decrease in Utah A concave down decrease in Hawaii A concave up increase in Oklahoma
Generate four rough sketches that could represent these situations. See Figure 2.29.
Hawaii
Year
Oklahoma No. of smokers
Utah No. of smokers
No. of smokers
Georgia
No. of smokers
S O L U T I O N
Year
Year
Year
Figure 2.29 The change in the number of smokers over time in four states.
Finding the Equation The program “L.3: Finding a Line Through 2 Points” in Linear Functions will give you practice in this skill.
E X A M P L E
5
To determine the equation of any particular linear function y ⫽ b ⫹ mx, we only need to find the specific values for the slope, m, and the vertical intercept, b. Given a graph Find the equation of the linear function graphed in Figure 2.30.
y 4
x 4
–4
–4
Figure 2.30 Graph of a linear function.
S O L U T I O N
We can use any two points on the graph to calculate m, the slope. If, for example, we take (⫺3, 0) and (3, ⫺2), then slope 5
change in y 22 2 0 22 21 5 5 5 change in x 3 2 (23) 6 3
The y-intercept is ⫺1, so b ⫽ ⫺1. Hence the equation is y 5 21 2 13 x.
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E X A M P L E
6
S O L U T I O N
Given two points Pediatric growth charts suggest a linear relationship between age (in years) and median height (in inches) for children between 2 and 12 years. Two-year-olds have a median height of 35 inches, and 12-year-olds have a median height of 60 inches. a. Generate the average rate of change of height with respect to age. (Be sure to include units.) Interpret your result in context. b. Generate an equation to describe height as a function of age. What is an appropriate domain? Range? c. What would this model predict as the median height of 8-year-olds? a. Average rate of change 5
change in height 60 2 35 25 5 5 5 2.5 inches/year change in age 12 2 2 10
The chart suggests that, on average, children between the ages of 2 and 12 grow 2.5 inches each year. b. If we think of height, H (in inches), depending on age, A (in years), then we want an equation of the form H ⫽ b ⫹ m ⭈ A. From part (a) we know m ⫽ 2.5, so our equation is H ⫽ b ⫹ 2.5A. To find b, we can substitute the values for any known point into the equation. When A ⫽ 2, then H ⫽ 35. Substituting in, we get H 5 b 1 2.5A 35 5 b 1 (2.5 ? 2) 35 5 b 1 5 30 5 b So the final form of our equation is H 5 30 1 2.5A where the domain is 2 ⱕ A ⱕ 12 and the range is 35 ⱕ H ⱕ 60. c. When A 5 8 years, our model predicts that the median height is H 5 30 1 (2.5 ? 8) 5 30 1 20 5 50 inches. E X A M P L E
7
Given a table a. Determine if the data in Table 2.8 represent a linear relationship between values of blood alcohol concentration and number of drinks consumed for a 160-pound person. (One drink is defined as 5 oz of wine, 1.25 oz of 80-proof liquor, or 12 oz of beer.) D, Number of Drinks
A, Blood Alcohol Concentration
2 4 6 10
0.047 0.094 0.141 0.235
Table 2.8
Note that federal law requires states to have 0.08 as the legal BAC limit for driving drunk. b. If the relationship is linear, determine the corresponding equation. S O L U T I O N
a. We can generate a third column in the table that represents the average rate of change between consecutive points (see Table 2.9). Since the average rate of change of A with respect to D remains constant at 0.0235, these data represent a linear relationship.
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b. The rate of change is the slope, so the corresponding linear equation will be of the form A 5 b 1 0.0235D D
A
Average Rate of Change
2
0.047
4
0.094
6
0.141
10
0.235
n.a. 0.047 0.094 2 0.047 5 5 0.0235 422 2 0.047 0.141 2 0.094 5 5 0.0235 624 2 0.235 2 0.141 0.094 5 5 0.0235 10 2 6 4
(1)
Table 2.9
To find b, we can substitute any of the original paired values for D and A, for example, (4, 0.094), into Equation (1) to get 0.094 5 b 1 (0.0235 ? 4) 0.094 5 b 1 0.094 0.094 2 0.094 5 b 05b So the final equation is or just
A 5 0 1 0.0235D A 5 0.0235D
So when D, the number of drinks, is 0, A, the blood alcohol concentration, is 0, which makes sense.
The slope m between an arbitrary point (x, y) and a specific point (x1, y1) is m5
y 2 y1 x 2 x1
If we solve this equation for y, we get m (x ⫺ x1) ⫽ y ⫺ y1 or y 5 m(x 2 x1 ) 1 y1 This is called a linear function in point-slope form. E X A M P L E
8
S O L U T I O N
Using the point-slope formula to find the equation a. If (2, 3) is a point on a line with slope 5, use the point-slope formula to find the equation of the line. b. Confirm that the equation is equivalent to an equation in the form y ⫽ mx ⫹ b. a. Given the point (2, 3) on a line with slope 5, then y23 x22 5(x 2 2) 5 y 2 3 y 5 5(x 2 2) 1 3. 55
So in point slope form b. So in the form y ⫽ mx ⫹ b
y ⫽ 5(x ⫺ 2) ⫹ 3 ⫽ 5x ⫺ 10 ⫹ 3 y ⫽ 5x ⫺ 7
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Algebra Aerobics 2.7 For Problems 1 and 2, find an equation, make an appropriate table, and sketch the graph of: 1. A line with slope 1.2 and vertical intercept 24. 2. A line with slope 2400 and vertical intercept 300 (be sure to think about scales on both axes). 3. Write an equation for the line graphed below. 5
y
b. Beginning with the vertical intercept, if we move one unit horizontally to the left, then we need to move up ___ units vertically to stay on the line and arrive at point (__, __). 7. For both Graph A and Graph B, identify two points on each line, determine the slope, then write an equation for the line. 10
y
5
x –5
Q
x
5 –10
10
–5
5
t
–5 –10
Graph of a linear function.
4. a. Find an equation to represent the current salary after x years of employment if the starting salary of an employee is $35,000 with annual increases of $3000. b. Create a small table of values and sketch a graph. 5. Complete this statement regarding the graph of y 5 6.2 1 3x: Beginning with any point on the graph of the line, we could find another point by moving up ___ units for each unit that we move horizontally to the right. 6. Given the equation y 5 8 2 4x, complete the following statements. a. Beginning with the vertical intercept, if we move one unit horizontally to the right, then we need to move down ___ units vertically to stay on the line and arrive at point (__, __).
–5
Graph A
Graph B
8. a. Write the equations of three lines each with vertical intercept of 6. b. Write the equations of three lines with slope ⫺3. 9. Use the point-slope strategy to find linear equations through a. The point (⫺2, 1) with a slope of 4 b. The point (3, 5) with a slope of ⫺2/3 c. The point (⫺1, 3) with a slope of ⫺10 d. The point (1.2, 4.5) with a slope of 2 10. Write an equation for the line: a. Through (2, 5) and (4, 11) b. Through (⫺3, 2) and (6, 1) c. Through (4, ⫺1) and (⫺2, ⫺7)
Exercises for Section 2.7 Graphing program is optional in Exercises 8 and 10.
a.
1. Write an equation for the line through (22, 3) that has slope: a. 5 b. 234 c. 0 2. Write an equation for the line through (0, 50) that has slope: a. 220 b. 5.1 c. 0 3. Calculate the slope and write an equation for the linear function represented by each of the given tables. a. b. x y A W 2 4
7.6 5.1
5 7
12 16
4. Determine which of the following tables represents a linear function. If it is linear, write the equation for the linear function.
b.
x
y
0 1 2 3 4
3 8 13 18 23
q
R
0 1 2 3 4
0.0 2.5 5.0 7.5 10.0
c.
d.
x
g(x)
0 1 2 3 4
0 1 4 9 16
t
r
10 20 30 40 50
5.00 2.50 1.67 1.25 1.00
e.
f.
x
h(x)
20 40 60 80 100
20 260 2140 2220 2300
p
T
5 10 15 20 25
0.25 0.50 0.75 1.00 1.25
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5. Plot each pair of points, then determine the equation of the line that goes through the points. a. (2, 3), (4, 0) c. (2, 0), (0, 2) b. (22, 3), (2, 1) d. (4, 2), (25, 2) 6. Find the equation for each of the lines A–C on the accompanying graph. 4
y
A B 5 –5
x
C
–6
7. Put the following equations in y ⫽ b ⫹ mx form, then identify the slope and the vertical intercept. a. 2x 2 3y 5 6 d. 2y 2 3x 5 0 b. 3x 1 2y 5 6 e. 6y 2 9x 5 0 c. 13 x 1 12 y 5 6 f. 12 x 2 23 y 5 216 8. (Graphing program optional.) Solve each equation for y in terms of x, then identify the slope and the y-intercept. Graph each line by hand. Verify your answers with a graphing utility if available. a. 24y 2 x 2 8 5 0 c. 24x 2 3y 5 9 1 1 b. 2 x 2 4 y 5 3 d. 6x 2 5y 5 15 9. Complete the table for each of the linear functions, and then sketch a graph of each function. Make sure to choose an appropriate scale and label the axes. a. x ƒ (x) 5 0.10x 1 10 2100 0 100 b.
x
h(x) 5 50x ⫹ 100
20.5 0 0.5 10. (Graphing program optional.) The equation K 5 4F 2 160 models the relationship between F, the temperature in degrees Fahrenheit, and K, the number of chirps per minute for the snow tree cricket. a. Assuming F is the independent variable and K is the dependent variable, identify the slope and vertical intercept in the given equation. b. Identify the units for K, 4, F, and 2160. c. What is a reasonable domain for this model? d. Generate a small table of points that satisfy the equation. Be sure to choose realistic values for F from the domain of your model.
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e. Calculate the slope directly from two data points. Is this value what you expected? Why? f. Graph the equation, indicating the domain. 11. Find an equation to represent the cost of attending college classes if application and registration fees are $150 and classes cost $250 per credit. 12. a. Write an equation that describes the total cost to produce x items if the startup cost is $200,000 and the production cost per item is $15. b. Why is the total average cost per item less if the item is produced in large quantities? 13. Your bank charges you a $2.50 monthly maintenance fee on your checking account and an additional $0.10 for each check you cash. Write an equation to describe your monthly checking account costs. 14. If a town starts with a population of 63,500 that declines by 700 people each year, construct an equation to model its population size over time. How long would it take for the population to drop to 53,000? 15. A teacher’s union has negotiated a uniform salary increase for each year of service up to 20 years. If a teacher started at $26,000 and 4 years later had a salary of $32,000: a. What was the annual increase? b. What function would describe the teacher’s salary over time? c. What would be the domain for the function? 16. Your favorite aunt put money in a savings account for you. The account earns simple interest; that is, it increases by a fixed amount each year. After 2 years your account has $8250 in it and after 5 years it has $9375. a. Construct an equation to model the amount of money in your account. b. How much did your aunt put in initially? c. How much will your account have after 10 years? 17. You read in the newspaper that the river is polluted with 285 parts per million (ppm) of a toxic substance, and local officials estimate they can reduce the pollution by 15 ppm each year. a. Derive an equation that represents the amount of pollution, P, as a function of time, t. b. The article states the river will not be safe for swimming until pollution is reduced to 40 ppm. If the cleanup proceeds as estimated, in how many years will it be safe to swim in the river? 18. The women’s recommended weight formula from Harvard Pilgrim Healthcare says, “Give yourself 100 lb for the first 5 ft plus 5 lb for every inch over 5 ft tall.” a. Find a mathematical model for this relationship. Be sure you clearly identify your variables. b. Specify a reasonable domain and range for the function and then graph the function. c. Use your model to calculate the recommended weight for a woman 5 feet, 4 inches tall; and for one 5 feet, 8 inches tall.
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19. In 1977 a math professor bought her condominium in Cambridge, Massachusetts, for $70,000. The value of the condo has risen steadily so that in 2007 real estate agents tell her the condo is now worth $850,000. a. Find a formula in point-slope form, to represent these facts about the value of the condo V(t), as a function of time, t, in years. b. If she retires in 2010, what does your formula predict her condo will be worth then? 20. The y-axis, the x-axis, the line x 5 6, and the line y 5 12 determine the four sides of a 6-by-12 rectangle in the first quadrant (where x . 0 and y . 0) of the xy plane. Imagine that this rectangle is a pool table. There are pockets at the four corners and at the points (0, 6) and (6, 6) in the middle of each of the longer sides. When a ball bounces off one of the sides of the table, it obeys the “pool rule”: The slope of the path after the bounce is the negative of the slope before the bounce. (Hint: It helps to sketch the pool table on a piece of graph paper first.) a. Your pool ball is at (3, 8). You hit it toward the y-axis, along the line with slope 2. i. Where does it hit the y-axis? ii. If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? iii. Show that the ball ultimately returns to (3, 8). Would it do this if the slope had been different from 2? What is special about the slope 2 for this table? b. A ball at (3, 8) is hit toward the y-axis and bounces off it at Q0, 16 3 R. Does it end up in one of the pockets? If so, what are the coordinates of that pocket? c. Your pool ball is at (2, 9). You want to shoot it into the pocket at (6, 0). Unfortunately, there is another ball at (4, 4.5) that may be in the way. i. Can you shoot directly into the pocket at (6, 0)? ii. You want to get around the other ball by bouncing yours off the y-axis. If you hit the y-axis at (0, 7), do you end up in the pocket? Where do you hit the line x 5 6? iii. If bouncing off the y-axis at (0, 7) didn’t work, perhaps there is some point (0, b) on the y-axis from which the ball would bounce into the pocket at (6, 0). Try to find that point. 21. Find the equation of the line shown on the accompanying graph. Use this equation to create two new graphs, taking care to label the scales on your new axes. For one of your graphs, choose scales that make the line appear steeper than in the original graph. For your second graph, choose scales that make the line appear less steep than in the original graph. 4
y
x
–4
4
–4
22. You are living in Paris but will be traveling in London. The exchange rate that a French bank gave in January 2010 was 1 euro (standard European currency) for 0.87 pound (currency in Great Britain). The bank charges 3 euros for any transaction. a. Write a general equation for changing euros into pounds (including transaction fees). b. Graph pounds as a function of euros. c. Would it make any sense to exchange 10 euros? 23. According to Moody’s www.econ.com. in the fourth quarter of 2006, 6.5% of homeowners had negative equity (that is, the market value of the home falls below the outstanding debt to pay off the mortgage). The percentage increased at a roughly constant rate until the fourth quarter of 2008, gaining about 2.5 percentage points per quarter. a. Construct an equation describing the relationship of the percent of homeowners having negative equity vs. the number of quarters since the fourth quarter of 2006. b. Determine a reasonable domain and then graph the function. c. Given this model, what percentage of homeowners had negative equity in the fourth quarter of 2008? 24. a. Create a third column in Tables A and B, and insert values for the average rate of change. (The first entry will be “n.a.”.) t
d
t
d
0 1 2 3 4 5
400 370 340 310 280 250
0 1 2 3 4 5
1.2 2.1 3.2 4.1 5.2 6.1
Table A
Table B
b. In either table, is d a linear function of t? If so, construct a linear equation relating d and t for that table. 25. Adding minerals or organic compounds to water lowers its freezing point. Antifreeze for car radiators contains glycol (an organic compound) for this purpose. The accompanying table shows the effect of salinity (dissolved salts) on the freezing point of water. Salinity is measured in the number of grams of salts dissolved in 1000 grams of water. So our units for salinity are in parts per thousand, abbreviated ppt. Is the relationship between the freezing point and salinity linear? If so, construct an equation that models the relationship. If not, explain why. Relationship between Salinity and Freezing Point Salinity (ppt) 0 5 10 15 20 25
Freezing Point (8C) 0.00 20.27 20.54 20.81 21.08 21.35
Source: Data adapted from P.R. Pinel, Oceanography: An Introduction to the Planet Oceanus (St. Paul, MN: West, 1992), p. 522.
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2.8
26. The accompanying data show rounded mean values for blood alcohol concentration (BAC) for people of different weights, according to how many drinks (5 oz wine, 1.25 oz 80-proof liquor, or 12 oz beer) they have consumed.
100 lb
140 lb
180 lb
2 4 6 8 10
0.075 0.150 0.225 0.300 0.375
0.054 0.107 0.161 0.214 0.268
0.042 0.083 0.125 0.167 0.208
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b. Examine the data on BAC for a 140-pound person. Are the data linear? If they’re not precisely linear, what might be a reasonable estimate for the average rate of change of blood alcohol concentration, A, with respect to number of drinks, D? Find a formula to estimate blood alcohol concentration, A, as a function of number of drinks, D, for a 140-pound person. Can you make any general conclusions about BAC as a function of number of drinks for all of the weight categories? c. Examine the data on BAC for people who consume two drinks. Are the data linear? If so, find a formula to express blood alcohol concentration, A, as a function of weight, W, for people who consume two drinks. Can you make any general conclusions about BAC as a function of weight for any particular number of drinks?
Blood Alcohol Concentration for Selected Weights Number of Drinks
Special Cases
a. Examine the data on BAC for a 100-pound person. Are the data linear? If so, find a formula to express blood alcohol concentration, A, as a function of the number of drinks, D, for a 100-pound person.
2.8 Special Cases Direct Proportionality The simplest relationship between two variables is when one variable is equal to a constant multiple of the other. For instance, in the previous section (Example 7) we saw that A 0.0235D; blood alcohol concentration A equals a constant, 0.0235, times D, the number of drinks. We say that A is directly proportional to D.
How to recognize direct proportionality Linear functions of the form (m 2 0)
y 5 mx
describe a relationship where y is directly proportional to x. If two variables are directly proportional to each other, the graph will be a straight line that passes through the point (0, 0), the origin. Figure 2.31 shows the graphs of two relationships in which y is directly proportional to x, namely y 2x and y x.
y –x
4
y y 2x
x –4
4
(0, 0) origin
