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THIRD EDITION

College Algebra
GRAPHS AND MODELS
Raymond A. Barnett
Merritt College

Michael R. Ziegler
Marquette University

Karl E. Byleen
Marquette University

Dave Sobecki
Miami University Hamilton

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COLLEGE ALGEBRA: GRAPHS AND MODELS, THIRD EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the
Americas, New York, NY 10020. Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Previous editions © 2005, 2000. No part of this publication may be reproduced or distributed in any form or
by any means, or stored in a database or retrieval system, without the prior written consent of The McGrawHill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission,
or broadcast for distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the

United States.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 9 8
ISBN 978–0–07–305195–6
MHID 0–07–305195–0
ISBN 978–0–07–334187–3 (Annotated Instructor’s Edition)
MHID 0–07–334187–8
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Library of Congress Cataloging-in-Publication Data
College algebra : graphs and models. –– 3rd ed. / Raymond A. Barnett ... [et al.].
p. cm.––(Barnett, Ziegler, and Byleen's precalculus series)

Includes index.
ISBN 978–0–07–305195–6 — ISBN 0–07–305195–0 (hard copy : alk. paper)
1. Algebra––Textbooks. 2. Algebra––Graphic methods––Textbooks. I. Barnett, Raymond A.
QA152.3.B37 2009
512.9––dc22
2007042212

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About the Authors

Raymond A. Barnett, a native of and educated in California, received his B.A. in
mathematical statistics from the University of California at Berkeley and his M.A. in
mathematics from the University of Southern California. He has been a member of the
Merritt College Mathematics Department and was chairman of the department for four
years. Associated with four different publishers, Raymond Barnett has authored or coauthored 18 textbooks in mathematics, most of which are still in use. In addition to
international English editions, a number of the books have been translated into Spanish. Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University; Charles Burke, City College of San Francisco; John Fujii,
Merritt College; and Karl Byleen, Marquette University.
Michael R. Ziegler received his B.S. from Shippensburg State College and his M.S.
and Ph.D. from the University of Delaware. After completing postdoctoral work at the
University of Kentucky, he was appointed to the faculty of Marquette University where
he currently holds the rank of Professor in the Department of Mathematics, Statistics,
and Computer Science. Dr. Ziegler has published more than a dozen research articles
in complex analysis and has co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl Byleen.
Karl E. Byleen received his B.S., M.A., and Ph.D. degrees in mathematics from the

University of Nebraska. He is currently an Associate Professor in the Department of
Mathematics, Statistics, and Computer Science of Marquette University. He has published a dozen research articles on the algebraic theory of semigroups and co-authored
more than a dozen undergraduate mathematics textbooks with Raymond Barnett and
Michael Ziegler.
Dave Sobecki earned a B.A. in math education from Bowling Green State University,
then went on to earn an M.A. and a Ph.D. in mathematics from Bowling Green. He is
an Associate Professor in the Department of Mathematics and Statistics at Miami University in Hamilton, Ohio. He has written or co-authored five journal articles, eleven
books and five interactive CD-ROMs. Dave lives in Fairfield, Ohio, with his wife (Cat)
and dogs (Large Coney and Macleod). His passions include Ohio State football, Cleveland
Indians baseball, heavy metal music, travel, and home improvement projects.

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Barnett, Ziegler, Byleen and Sobecki’s Precalculus Series
College Algebra, Eighth Edition
This book is the same as Precalculus without the three chapters on trigonometry.
ISBN 0-07-286738-8, ISBN 978-0-07-286738-1
Precalculus, Sixth Edition
This book is the same as College Algebra with three chapters of trigonometry added.
The trigonometry functions are introduced by a unit circle approach.
ISBN 0-07-286739-6, ISBN 978-0-07-286739-8
College Algebra with Trigonometry, Eighth Edition
This book differs from Precalculus in that College Algebra with Trigonometry uses
right triangle trigonometry to introduce the trigonometric functions.
ISBN 0-07-331264-9, ISBN 978-0-07-331264-4
College Algebra: Graphs and Models, Third Edition
This book is the same as Precalculus: Graphs and Models without the three chapters

on trigonometry. This text assumes the use of a graphing calculator.
ISBN 0-07-305195-0, ISBN 978-0-07-305195-6
Precalculus: Graphs and Models, Third Edition
This book is the same as College Algebra: Graphs and Models with three additional
chapters on trigonometry. The trigonometric functions are introduced by a unit circle
approach. This text assumes the use of a graphing calculator.
ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3
College Algebra with Trigonometry: Graphs and Models
This book is the same as Precalculus: Graphs and Models except that the trigonometric functions are introduced by right triangle trigonometry. This text assumes the
use of a graphing calculator.
ISBN 0-07-291699-0, ISBN 978-0-07-291699-7

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Contents
Preface vii
Application Index xvi

CHAPTER
1-1
1-2
1-3
1-4
1-5
1-6

1


Functions, Graphs,
and Models 1
Using Graphing Calculators 2
Functions 21
Functions: Graphs and Properties 46
Functions: Graphs and Transformations 67
Operations on Functions; Composition 84
Inverse Functions 99
Chapter 1 Review 119
Chapter 1 Group Activity: Mathematical Modeling:
Choosing a Cell Phone Provider 126

CHAPTER
2-1
2-2
2-3
2-4
2-5
2-6
2-7

CHAPTER

2

Modeling with Linear and
Quadratic Functions 127
Linear Functions 128
Linear Equations and Models 151

Quadratic Functions 172
Complex Numbers 190
Quadratic Equations and Models 206
Additional Equation-Solving Techniques 226
Solving Inequalities 241
Chapter 2 Review 258
Chapter 2 Group Activity: Mathematical
Modeling in Population Studies 265

Cumulative Review Exercises
Chapters 1–2 267

4-1
4-2
4-3
4-4
4-5

3-1
3-2
3-3
3-4
3-5
3-6

3

Exponential and Logarithmic
Functions 381
Exponential Functions 382

Exponential Models 399
Logarithmic Functions 416
Logarithmic Models 430
Exponential and Logarithmic Equations 440
Chapter 4 Review 462
Chapter 4 Group Activity: Comparing Regression
Models 456

Cumulative Review Exercises
Chapters 3–4 457

CHAPTER
5-1
5-2
5-3
5-4

Polynomial and Rational
Functions 273
Polynomial Functions and Models 274
Polynomial Division 291
Real Zeros and Polynomial Inequalities 303
Complex Zeros and Rational Zeros of
Polynomials 320
Rational Functions and Inequalities 336
Variation and Modeling 361
Chapter 3 Review 371
Chapter 3 Group Activity: Interpolating
Polynomials 378


6-2
6-3
6-4
6-5
6-6
6-7

5

Modeling with Systems of
Equations and Inequalities 463
Systems of Linear Equations in Two
Variables 464
Systems of Linear Equations in Three
Variables 482
Systems of Linear Inequalities 495
Linear Programming 510
Chapter 5 Review 522
Chapter 5 Group Activity: Heat Conduction 526

CHAPTER
6-1

CHAPTER

4

6

Matrices and Determinants 527

Systems of Linear Equations: Gauss–Jordan
Elimination 528
Matrix Operations 547
Inverse of a Square Matrix 565
Matrix Equations and Systems of Linear
Equations 578
Determinants 588
Properties of Determinants 597
Determinants and Cramer’s Rule 604
Chapter 6 Review 610
Chapter 6 Group Activity: Using Matrices to
Find Cost, Revenue, and Profit 616

Cumulative Review Exercises
Chapters 5–6 618

V

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CHAPTER
7-1
7-2
7-3
7-4
7-5
7-6


7

Sequences, Induction, and
Probability 621
Sequences and Series 622
Mathematical Induction 633
Arithmetic and Geometric Sequences 644
The Multiplication Principle, Permutations, and
Combinations 659
Sample Spaces and Probability 676
The Binomial Formula 696
Chapter 7 Review 704
Chapter 7 Group Activity: Sequences Specified
by Recursion Formulas 709

APPENDIX

APPENDIX
CHAPTER
8-1
8-2
8-3
8-4

8

Additional Topics in Analytic
Geometry 711
Conic Sections; Parabola 712

Ellipse 723
Hyperbola 735
Systems of Nonlinear Equations 750
Chapter 8 Review 761
Chapter 8 Group Activity: Focal Chords 765

Cumulative Review Exercises
Chapters 7–8 766

A

Basic Algebra Review A-1
Algebra and Real Numbers A-2
Exponents A-13
Radicals A-27
Polynomials: Basic Operations A-36
Polynomials: Factoring A-47
Rational Expressions: Basic Operations A-58
Appendix A Review *
Appendix A Group Activity: Rational
and Irrational Numbers *
*Available online at www.mhhe.com/barnett
A-1
A-2
A-3
A-4
A-5
A-6

B


Review of Equations and
Graphing A-69
B-1 Linear Equations and Inequalities A-70
B-2 Cartesian Coordinate System A-82
B-3 Basic Formulas in Analytic Geometry A-91

C

APPENDIX
Special Topics A-105
C-1 Significant Digits A-106
C-2 Partial Fractions A-109
C-3 Descartes’ Rule of Signs A-118
C-4 Parametric Equations *
*Available online at www.mhhe.com/barnett
APPENDIX

D

Geometric Formulas A-123

Student Answers SA-1

VI

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Preface
Enhancing a Tradition of Success
We take great satisfaction from the fact that more than 100,000 students have learned college algebra from a Barnett Series textbook. Ray Barnett is one of the masters of college
textbook writing. His central approach is proven and remains effective for today’s students.
The third edition of College Algebra: Graphs and Models has benefited greatly
from the numerous contributions of new coauthor Dave Sobecki of Miami University
Hamilton. Dave brings a fresh approach to the material and many good suggestions
for improving student accessibility. Every aspect of the revision focuses on making
the text more relevant to students, while retaining the precise presentation of the mathematics for which the Barnett name is renowned.
Specifically we concentrated on the areas of writing, worked examples, exercises,
technology, and design. Based on numerous reviews, advice from expert consultants,
and direct correspondence with many users of previous editions, we feel that this edition is more relevant than ever before. We hope you will agree.
Writing Without sacrificing breadth or depth or coverage, we have rewritten explanations to make them clearer and more direct. As in previous editions, the text emphasizes computational skills, real-world data analysis and modeling, and problem solving rather than theory.
Examples In the new edition, even more solved examples in the book provide graphical solutions side-by-side with algebraic solutions. By seeing the same answer result
from their symbol manipulations and from graphical approaches, students gain insight
into the power of algebra and make important conceptual and visual connections.
Likewise, we added expanded color annotations to many examples, explaining the
solution steps in words. Each example is then followed by a similar matched problem
for the student to solve. Answers to the matched problems are located at the end of
each section for easy reference. This active involvement in learning while reading helps
students develop a more thorough understanding of concepts and processes.
Exercises With an eye to improving student performance and to make the book
more useful for instructors, we have extensively revised the exercise sets. We added
hundreds of new writing questions as well as exercises at the easy to moderate level
and expanded the variety of problem types to ensure a gradual increase in difficulty
level throughout each exercise set.
Technology Although technology is employed throughout, we strive to balance
algebraic skill development with the use of technology as an aid to learning and problem solving. We assume that students using the book will have access to one of the
various graphing calculators or computer programs that are available to perform the

following operations:
• Simultaneously display multiple graphs in a user-selected viewing window
• Explore graphs using trace and zoom

VII

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VIII

Preface

• Approximate roots and intersection points
• Approximate maxima and minima
• Plot data sets and find associated regression equations
• Perform basic matrix operations, including row reduction and inversion
Most popular graphing calculators perform all of these operations. The majority of the
graphing calculator images in this book are “screen shots” from a Texas Instruments
TI-84 Plus graphing calculator. Students not using that TI calculator should be able to
produce similar results on any calculator or software meeting the requirements listed.
The proper use of such calculators is covered in Section 1-1.

A Central Theme
In the Barnett series, the function concept serves as a unifying theme. A brief look
at the table of contents reveals this emphasis. A major objective of this book is the
development of a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to
proceed through this book with greater confidence and understanding.


Design: A New Book with a New Look
The third edition of College Algebra: Graphs and Models presents the subject in the
precise and straightforward way that users have come to rely on in the Barnett textbooks, now updated for students in the twenty-first century. The changes to the text
are manifested visually in a new design. We think the pages of this edition offer a
more contemporary and inviting visual backdrop for the mathematics. We are confident that the design and other changes to the text described here will help to improve
your students’ enjoyment of and success in the course.

Features
New to the Third Edition
An extensive reworking of the narrative throughout the chapters has made the language less formal and more engaging for students.
A new full-color design gives the book a more contemporary feel and will appeal
to students who are accustomed to high production values in books, magazines, and
nonprint media.
Even more examples now feature side-by-side solutions integrating algebraic and
graphical solution methods. This format encourages students to investigate mathematical principles and processes graphically and numerically, as well as algebraically.
The increased use of annotated algebraic steps, in small colored type, to walk
students through each critical step in the problem-solving process helps students follow the authors’ reasoning and improve their own problem-solving strategies.
An Annotated Instructor’s Edition is now available and contains answers to
exercises in the text, including answers to section, chapter review, and cumulative
review exercises. These answers are printed in a second color, adjacent to corresponding exercises, for ease of use by the instructor.
More balanced exercise sets give instructors maximum flexibility in assigning
homework. We added exercises at the easy to moderate level and expanded the vari-

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Preface


IX

ety of problem types to ensure a gradual increase in difficulty level throughout each
exercise set. The division of exercise sets into A (routine, easy mechanics), B (more
difficult mechanics), and C (difficult mechanics and some theory) is no longer explicit
in the student edition of the text: the letter designations appear only in the Annotated
Instructor’s Edition. This change was made in order to avoid fueling students’ anxiety about challenging exercises. As in previous editions, students at all levels can be
challenged by the exercises in this text.
Hundreds of new writing questions encourage students to think about the important concepts of the section before solving computational problems. Problem numbers
that appear in blue indicate problems that require students to apply their reasoning
and writing skills to the solution of the problem.

Features Retained
Examples and matched problems introduce concepts and demonstrate problem-solving
techniques using side-by-side algebraic and graphical solution methods. Each carefully solved example is followed by a similar Matched Problem for the student to work
through while reading the material. Answers to the matched problems are located at
the end of each section, for easy reference. This active involvement in the learning
process helps students develop a more thorough understanding of algebraic and graphical concepts and processes.
Graphing calculator technology is integrated throughout the text for visualization, investigation, and verification. Graphing calculator screens displayed in the text
are actual output. Although technology is employed throughout, the authors strive to
balance algebraic skill development with the use of technology as an aid to learning
and problem solving.
Annotated steps of examples and developments are found throughout the text to
help students through the critical stages of problem-solving. Think Boxes (color
dashed boxes) are used to enclose steps that, with some experience, many students
will be able to perform mentally.
Applications throughout the third edition give the student substantial experience
in modeling and solving real-world problems, fulfilling a primary objective of the text.
Over 500 application exercises help convince even the most skeptical student that

mathematics is relevant to everyday life. Chapter Openers are written to highlight
interesting applications and an Applications Index is included to help locate applications from particular fields.
Explore-discuss boxes are interspersed throughout each section. They foster conceptual understanding by asking students to think about a relationship or process
before a result is stated. Verbalization of mathematical concepts, processes, and results
is strongly encouraged in these investigations and activities.
Group activities at the end of each chapter involve multiple concepts discussed
in the chapter. These activities strongly encourage the verbalization of mathematical
concepts, results, and processes. All of these special activities are highlighted to
emphasize their importance.
Foundations for calculus icons are used to mark concepts that are especially pertinent to a student’s future study of calculus.
Interpretation of graphs icons are used to mark exercises that ask students to
make determinations about equations or functions based on graphs.

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X

Preface

Key Content Changes
Chapter 1, Functions, Graphs, and Models Functions are now introduced in
terms of relations, providing more flexibility in discussing correspondences between
sets of objects.
Chapter 2, Modeling with Linear and Quadratic Functions Quadratic inequalities are now covered using a more general test point method, so that all nonlinear
inequalities are solved using the same method.
Chapter 3, Polynomial and Rational Functions Section 3-1 (Polynomial Functions and Models) has been split into two sections, so Section 3-2 is now entirely
devoted to division of polynomials, including remainder and factor theorems crucial

to the study of zeros of polynomials later in the chapter. The chapter also features a
new section (3-6) on direct, inverse, joint, and combined variation. This new section
supports the book’s emphasis on mathematical modeling.
Chapter 5, Modeling with Systems of Equations and Inequalities A new Section 5-2 on Systems of Linear Equations in Three Variables was added, while Section
5-1 on Systems of Linear Equations in Two Variables was reorganized.
Chapter 6 Matrices and Determinants
ear systems is now found in Section 6-1.

The material on matrix solutions to lin-

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Preface

XI

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Video Lectures on Digital Video Disk (DVD) In the videos, J. D. Herdlick of
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closed-captioned for the hearing impaired, subtitled in Spanish, and meet the Americans

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XII

Preface

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Instructor’s Solutions Manual Prepared by Dave Sobecki, and available on MathZone, the Instructor’s Solutions Manual provides comprehensive, worked-out solutions
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Preface

XIII

Acknowledgments
In addition to the authors, many others are involved in
the successful publication of a book. We wish to thank

personally the following people who reviewed the third
edition manuscript and offered invaluable advice for
improvements:
Laurie Boudreaux, Nicholls State University
Emma Borynski, Durham Technical Community College
Barbara Burke, Hawaii Pacific University
Sarah Cook, Washburn University
Donna Densmore, Bossier Parish Community College
Alvio Dominguez, Miami-Dade College (Wolfson Campus)
Joseph R. Ediger, Portland State University
Angela Everett, Chattanooga State Technical Community
College
Mike Everett, Santa Ana College
Toni Fountain, Chattanooga State Technical Community
College
Scott Garten, Northwest Missouri State University
Perry Gillespie, Fayetteville State University
Dana Goodwin, University of Central Arkansas
Judy Hayes, Lake-Sumter Community College
James Hilsenbeck, University of Texas–Brownsville
Lynda Hollingsworth, Northwest Missouri State University
Michelle Hollis, Bowling Green Community College of
WKU
Linda Horner, Columbia State Community College
Tracey Hoy, College of Lake County
Byron D. Hunter, College of Lake County
Michelle Jackson, Bowling Green Community College of
WKU
Tony Lerma, University of Texas–Brownsville
Austin Lovenstein, Pulaski Technical College

Jay A. Malmstrom, Oklahoma City Community College
Lois Martin, Massasoit Community College
Mikal McDowell, Cedar Valley College
Rudy Meangru, LaGuardia Community College
Dennis Monbrod, South Suburban College
Sanjay Mundkar, Kennesaw State University
Elaine A. Nye, Alfred State College
Dale Oliver, Humboldt State University
Jorge A. Perez, LaGuardia Community College
Susan Pfeifer, Butler Community College
Dennis Reissig, Suffolk County Community College

Brunilda Santiago, Indian River Community College
Nicole Sifford, Rivers Community College
James Smith, Columbia State Community College
Joyce Smith, Chattanooga State Technical Community
College
Shawn Smith, Nicholls State University
Margaret Stevenson, Massasoit Community College
Pam Stogsdill, Bossier Parish Community College
John Verzani, College of Staten Island
Deanna Voehl, Indian River Community College
Ianna West, Nicholls State University
Fred Worth, Henderson State University
We also wish to thank
Hossein Hamedani for providing a careful and thorough
check of all the mathematical calculations in the book
(a tedious but extremely important job).
Dave Sobecki for developing the supplemental manuals
that are so important to the success of a text.

Jeanne Wallace for accurately and efficiently producing
most of the manuals that supplement the text.
Mitchel Levy for scrutinizing our exercises in the manuscript and making recommendations that helped us to
build balanced exercise sets.
Tony Palermino for providing excellent guidance in
making the writing more direct and accessible to students.
Pat Steele for carefully editing and correcting the manuscript.
Jay Miller for his careful technical proofread of the first
pages.
Katie White for organizing the revision process, determining the objectives for the new edition, and supervising the preparation of the manuscript.
Sheila Frank for guiding the book smoothly through all
publication details.
All the people at McGraw-Hill who contributed their
efforts to the production of this book, especially Dawn
Bercier.
Producing this new edition with the help of all these
extremely competent people has been a most satisfying
experience.

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EXAMPLE

4

Side-by-Side Solutions: Algebraic
and Graphical


Using Exponential Function Properties
Find all solutions to 4xϪ3 ϭ 8.

Many solved examples in the book provide
graphical solutions side-by-side with algebraic
solutions. By seeing the same answer result
from their symbol manipulations and from
graphical approaches, students gain insight
into the power of algebra and make important
conceptual and visual connections.

SOLUTIONS

Algebraic Solution
Notice that the two bases, 4 and 8, can both be written
as a power of 2. This will enable us to use Property 2
to equate exponents.
4xϪ3 ϭ 8
(22)xϪ3 ϭ 23
22xϪ6 ϭ 23
2x Ϫ 6 ϭ 3
2x ϭ 9

xϭ

Graphical Solution
Graph y1 ϭ 4xϪ3 and y2 ϭ 8. Use the INTERSECT
command to obtain x ϭ 4.5 (Fig. 10).
10


Express 4 and 8 as powers of 2.
(ax)y ‫ ؍‬axy

Ϫ10

10

Property 2
Add 6 to both sides.
Ϫ10

Divide both sides by 2.

9
2

Z Figure 10

CHECK
✓

4(9/2)Ϫ3 ϭ 43/2 ϭ ( 14)3 ϭ 23 ϭ 8

Examples and Matched Problems
Integrated throughout the text, completely
worked examples and practice problems are
used to introduce concepts and demonstrate
problem-solving techniques—algebraic, graphical, and numerical. Each Example is followed
by a similar Matched Problem for the student

to work through while reading the material.
Answers to the matched problems are located
at the end of each section, for easy reference.
This active involvement in the learning
process helps students develop a thorough
understanding of algebraic concepts and
processes.

EXAMPLE

5

Table 3 Home Ownership
Rates
Year

Home Ownership
Rate (%)

1940

43.6

1950

55.0

1960

61.9


1970

62.9

1980

64.4

1990

64.2

2000

67.4

Home Ownership Rates
The U.S. Census Bureau published the data in Table 3 on home ownership rates.
(A) Let x represent time in years with x ϭ 0 representing 1900, and let y represent
the corresponding home ownership rate. Use regression analysis on a graphing
calculator to find a logarithmic function of the form y ϭ a ϩ b ln x that models
the data. (Round the constants a and b to three significant digits.)
(B) Use your logarithmic function to predict the home ownership rate in 2010.
SOLUTIONS

(A) Figure 1 shows the details of constructing the model on a graphing calculator.
(B) The year 2010 corresponds to x ϭ 110. Evaluating y1 ϭ Ϫ36.7 ϩ 23.0 ln x at
x ϭ 110 predicts a home ownership rate of 71.4% in 2010.


100

0

120

0

(a) Data

(b) Regression equation

(c) Regression equation entered
in equation editor

(d) Graph of data and regression
equation

Z Figure 1

MATCHED PROBLEM

5

Refer to Example 5. The home ownership rate in 1995 was 64.7%.
(A) Find a logarithmic regression equation for the expanded data set.
(B) Predict the home ownership rate in 2010.

ANSWERS


TO MATCHED PROBLEMS

1. 95.05 decibels
2. 7.80
3. 2.67
5. (A) Ϫ31.5 ϩ 21.7 ln x
(B) 70.5%

XIV

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Balanced Exercise Sets
Exercises

In Problems 1–8, determine the validity of each statement. If a
statement is false, explain why.
1. If x2 ϭ 5, then x ϭ Ϯ15

2. 125 ϭ Ϯ5

3. (x ϩ 5)2 ϭ x2 ϩ 25

4. (2x Ϫ 1)2 ϭ 4x2 Ϫ 1


5. ( 1x Ϫ 1 ϩ 1)2 ϭ x

6. ( 1x Ϫ 1)2 ϩ 1 ϭ x

7. If x3 ϭ 2, then x ϭ 8

8. If x1/3 ϭ 2, then x ϭ 8

In Problems 9–14, transform each equation of quadratic type
into a quadratic equation in u and state the substitution used in
the transformation. If the equation is not an equation of
quadratic type, say so.
3
6
4
9. 2xϪ6 Ϫ 4xϪ3 ϭ 0
10. Ϫ ϩ 2 ϭ 0
x
7
x
11. 3x3 Ϫ 4x ϩ 9 ϭ 0

12. 7xϪ1 ϩ 3xϪ1/2 ϩ 2 ϭ 0

10
7
4
ϩ 2Ϫ 4ϭ0
9
x

x

14. 3x3/2 Ϫ 5x1/2 ϩ 12 ϭ 0

13.

37. 2x2/3 ϩ 3x1/3 Ϫ 2 ϭ 0

39. (m2 Ϫ m)2 Ϫ 4(m2 Ϫ m) ϭ 12
40. (x2 ϩ 2x)2 Ϫ (x2 ϩ 2x) ϭ 6
41. 1u Ϫ 2 ϭ 2 ϩ 12u ϩ 3

16. Would raising both sides of an equation to the third power
ever introduce extraneous solutions? Why or why not?
17. Write an example of a false statement that becomes true
when you square both sides. What would every possible example have in common?
18. How can you recognize when an equation is of quadratic type?
In Problems 19–32, solve algebraically and confirm graphically, if possible.

42. 13t ϩ 4 ϩ 1t ϭ Ϫ3

43. 13y Ϫ 2 ϭ 3 Ϫ 13y ϩ 1
44. 12x Ϫ 1 Ϫ 1x Ϫ 4 ϭ 2
45. 17x Ϫ 2 Ϫ 1x ϩ 1 ϭ 13
46. 13x ϩ 6 Ϫ 1x ϩ 4 ϭ 12
47. 24x2 ϩ 12x ϩ 1 Ϫ 6x ϭ 9
48. 6x Ϫ 24x2 Ϫ 20x ϩ 17 ϭ 15
49. 3nϪ2 Ϫ 11nϪ1 Ϫ 20 ϭ 0

50. 6xϪ2 Ϫ 5xϪ1 Ϫ 6 ϭ 0


51. 9yϪ4 Ϫ 10yϪ2 ϩ 1 ϭ 0
53. y

15. Explain why squaring both sides of an equation sometimes
introduces extraneous solutions.

1/2

Ϫ 3y

1/4

52. 4xϪ4 Ϫ 17xϪ2 ϩ 4 ϭ 0
54. 4xϪ1 Ϫ 9xϪ1/2 ϩ 2 ϭ 0

ϩ2ϭ0

55. (m Ϫ 5) ϩ 36 ϭ 13(m Ϫ 5)
4

2

56. (x Ϫ 3)4 ϩ 3(x Ϫ 3)2 ϭ 4
57. Explain why the following “solution” is incorrect:
1x ϩ 3 ϩ 5 ϭ 12
x ϩ 3 ϩ 25 ϭ 144
x ϭ 116
58. Explain why the following “solution” is incorrect.
2x2 Ϫ 16 ϭ 2x ϩ 3

x Ϫ 4 ϭ 2x ϩ 3

19. 14x Ϫ 7 ϭ 5

20. 14 Ϫ x ϭ 4

21. 15x ϩ 6 ϩ 6 ϭ 0

22. 110x ϩ 1 ϩ 8 ϭ 0

3
23. 1x ϩ 5 ϭ 3

4
24. 1x Ϫ 3 ϭ 2

In Problems 59–62, solve algebraically and confirm graphically, if possible.

25. 1x ϩ 5 ϩ 7 ϭ 0

26. 3 ϩ 12x Ϫ 1 ϭ 0

59. 15 Ϫ 2x Ϫ 1x ϩ 6 ϭ 1x ϩ 3

27. y4 Ϫ 2y2 Ϫ 8 ϭ 0

28. x4 Ϫ 7x2 Ϫ 18 ϭ 0

60. 12x ϩ 3 Ϫ 1x Ϫ 2 ϭ 1x ϩ 1


29. 3x ϭ 2x Ϫ 2

30. x ϭ 25x ϩ 9

61. 2 ϩ 3yϪ4 ϭ 6yϪ2

31. 2x2 Ϫ 5x ϭ 1x Ϫ 8

32. 12x ϩ 3 ϭ 2x2 Ϫ 12

In Problems 63–66, solve two ways: by isolating the radical and
squaring, and by substitution. Confirm graphically, if possible.

2

2

In Problems 33–56, solve algebraically and confirm graphically,
if possible.
33. 15n ϩ 9 ϭ n Ϫ 1

34. m Ϫ 13 ϭ 1m ϩ 7

35. 13x ϩ 4 ϭ 2 ϩ 1x

36. 13w Ϫ 2 Ϫ 1w ϭ 2

College Algebra: Graphs and Models, third
edition, contains more than 4,000 problems.
Each Exercise set is designed so that an average or below-average student will experience

success and a very capable student will be
challenged. Exercise sets are found at the end
of each section in the text. The Annotated
Instructor’s Edition features A (routine, easy
mechanics), B (more difficult mechanics), and
C (difficult mechanics and some theory) designations to denote these levels and help
instructors in the assignment building process.
Problem numbers that appear in blue indicate
problems that require students to apply their
reasoning and writing skills to the solution of
the problem.

38. x2/3 Ϫ 3x1/3 Ϫ 10 ϭ 0

Ϫ7 ϭ x

62. 4mϪ2 ϭ 2 ϩ mϪ4

63. m Ϫ 7 1m ϩ 12 ϭ 0

64. y Ϫ 6 ϩ 1y ϭ 0

65. t Ϫ 11 1t ϩ 18 ϭ 0

66. x ϭ 15 Ϫ 21x

APPLICATIONS

Applications


101. CONSTRUCTION A gardener has a 30 foot by 20 foot rectangular plot of ground. She wants to build a brick walkway of uniform
width on the border of the plot (see the figure). If the gardener wants
to have 400 square feet of ground left for planting, how wide (to two
decimal places) should she build the walkway?

cottage for a resort area. A cross-section of the cottage is an
isosceles triangle with a base of 5 meters and an altitude of 4 meters. The front wall of the cottage must accommodate a sliding
door positioned as shown in the figure.

DOOR DETAIL
Page 1 of 4

x

20 feet

w

4 meters
30 feet

h

102. CONSTRUCTION Refer to Problem 101. The gardener buys
enough bricks to build 160 square feet of walkway. Is this sufficient
to build the walkway determined in Problem 101? If not, how wide
(to two decimal places) can she build the walkway with these bricks?

5 meters


103. CONSTRUCTION A 1,200 square foot rectangular garden is enclosed with 150 feet of fencing. Find the dimensions of the garden to (A) Express the area A(w) of the door as a function of the width
the nearest tenth of a foot.
w and state the domain of this function. [See the hint for Prob104. CONSTRUCTION The intramural fields at a small college will lem 105.]
cover a total area of 140,000 square feet, and the administration has (B) A provision of the building code requires that doorways
budgeted for 1,600 feet of fence to enclose the rectangular field. Find must have an area of at least 4.2 square meters. Find the width
of the doorways that satisfy this provision.
the dimensions of the field.
(C) A second provision of the building code requires all door105. ARCHITECTURE A developer wants to erect a rectangular build- ways to be at least 2 meters high. Discuss the effect of this reing on a triangular-shaped piece of property that is 200 feet wide and quirement on the answer to part B.
400 feet long (see the figure).
107. TRANSPORTATION A delivery truck leaves a warehouse
and travels north to factory A. From factory A the truck travels
east to factory B and then returns directly to the warehouse (see
the figure on the next page). The driver recorded the truck’s
Property
A
odometer reading at the warehouse at both the beginning and
Property Line
l
the end of the trip and also at factory B, but forgot to record it at
factory A (see the table on the next page). The driver does recall
Proposed
that it was farther from the warehouse to factory A than it was
w
Building
from factory A to factory B. Because delivery charges are based
200 feet

One of the primary objectives of this book is
to give the student substantial experience in
modeling and showing real-world problems.

More than 500 application exercises help convince even the most skeptical student that
mathematics is relevant to everyday life. The
most difficult application problems are marked
with two stars (* *), the moderately difficult
application problems with one star (*), and
easier application problems are not marked.
An Application Index is included immediately
preceding Chapter 1 to locate particular applications.

(A) Express the area A(w) of the footprint of the building as a
function of the width w and state the domain of this function.
[Hint: Use Euclid’s theorem* to find a relationship between the
length l and width w.]
(B) Building codes require that this building have a footprint of
at least 15,000 square feet. What are the widths of the building
that will satisfy the building codes?
(C) Can the developer construct a building with a footprint of
100. NAVIGATION A speedboat takes 1 hour longer to go 24 miles up 25,000 square feet? What is the maximum area of the footprint
a river than to return. If the boat cruises at 10 miles per hour in still of a building constructed in this manner?
water, what is the rate of the current?
106. ARCHITECTURE An architect is designing a small A-frame
99. AIR SEARCH A search plane takes off from an airport at 6:00 A.M.
and travels due north at 200 miles per hour. A second plane takes off at
6:30 A.M. and travels due east at 170 miles per hour. The planes carry
radios with a maximum range of 500 miles. When (to the nearest
minute) will these planes no longer be able to communicate with each
other?

REBEKAH DRIVE


2-6

FIRST STREET
400 feet

*Euclid’s theorem: If two triangles are similar, their corresponding
sides are proportional:
c

a
b

aЈ

cЈ
bЈ

a
b
c
ϭ
ϭ
a¿
b¿
c¿

XV

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APPLICATION INDEX
Adiabatic process, 658
Advertising, 377, 451, 457
Aeronautical engineering, 734
Aeronautics, 169, 170
Agriculture, 439, 456, 509, 517–518, 547
AIDS cases, 411
Airfreight, 564
Air search, 223
Airspeed, 150, 476–477, 480
Air temperature, 44, 150, 257, 658
Alcohol consumption, 224
Altitude and oxygen percentage, 125
Animal nutrition, 525, 547
Anthropology, 370
Approximation, 255
Architecture, 223, 264, 376, 734, 749,
768, A–109
Astronomy, 438, 439, 460, 658, 722
Atmospheric pressure, 443, 659
Automobile rental, 66
Average cost, 360
Averaging tests, 620
Bacterial growth, 401, 402, 413, 658
Biology, 370, A–26
Boat speed, 477
Boiling point of water, 44

Boy-girl composition of families, 684
Braking distance and speed, 370
Break-even analysis, 251–252, 255–257,
263, 265, 269, 480
Breaking distance, A–26
Bungee jumper, 632
Business, 150, 263, 270, 480, 525, 657,
A–95
Carbon-14 dating, 405–406, 450–451,
455
Card hands, 672–673
Car rental, 45, 59
Cell division, 658
Cell phone charges, 125, 126
Cell phone subscribers, 490, 492
Celsius/Fahrenheit, 256
Chemistry, 170, 438–439, 480, 546
Cigarette consumption, 224–225
Circuit analysis, 587
Code-word counting, 662–663

Coin problem, A–46
Coin toss probabilities, 682–683, 689–691
College tuition, 170, 171
Combined area, 183
Combined outcomes, 660–661
Combined variation, 367–368
Committee selection, 687
Communications, 764
Competitive rowing, 169

Completion of years of college, 319
Compound interest, 391–395, 442, 450,
460, 461, 709
Computer design, 413–414
Computer-generated tests, 662
Computer science, 60–61, 65, 125, 270,
A–26
Construction, 65, 82, 183, 189, 223, 224,
240, 263, 269, 270, 290, 314–315,
319, 335, 360, 377, 438, 760, A–58,
A–73, A–109
Consumer debt, 38–40
Continuous compound interest, 393–395
Cost analysis, 150, 156–157, 169, 263,
563
Credit union debt, 40
Cryptography, 574–575, 577–578, 616
Data analysis, 38–41, 125–126, 490–492,
A–91–92, A–96
Delivery charges, 66
Demand, 94, 111, 270
Demographics, 150
Depreciation, 150, 263, 269, 415
Depth of a well, 234–236, 240
Design, 215–216, 236–237, 236–238,
240, 263, 264, 734, 760
Diamond prices, 161–164
Diet, 475–476, 522, 525, 588, 615, 620
Distance-rate-time, 157–159
Distance-rate-time problems, 158–159

Divorce, 291
Dominance relation, 564–565
Drawing cards, 686–687
Drug use, 264
Earthquake intensity, 433–435, 448
Earthquakes, 170, 438, 451, 455, 461
Earth science, 170, 256, 414, 482, A–26

XVI

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Ecology, 439
Economics, 94–95, 657, 709, 767, A–26,
A–73
Economy stimulation, 654–655
Efficiency, 460
Electrical circuit, A–73
Electricity, 370
Empirical probabilities for an insurance
company, 682–683
Employee training, 359, 415
Engineering, 189, 370, 371, 658, 722,
734, 764, 768, A–109
Environmental science, 98
Epidemics, 408–409
Estimating weight, 285
Evaporation, 83, 99
Explosive energy, 438
Fabrication, 335

Falling objects, 44, 184, 189, 262, 363,
364, 481, 482, 632, 658
Finance, 398, 399, 481, 494, 620, 658
Fish weight, 285
Fixed costs, 251, 253
Flight ground speed, 170
Flight navigation, 150
Fluid flow, 83, 98–99
Food chain, 658
Force of stretched spring, 150, 362
Gaming, 413
Gas mileage, 188–189
Genealogy, 658
Genetics, 370
Geology, 149
Geometry, 240, 262, 318, 371, 377, 460,
494, 525, 546, 587, 659, 760, 767,
A–46
Global warming, 149
Half-life, 403, 404
Health care, 290–291
Heat conduction, 526
Height of bungee jumper, 124
History of technology, 414
Home ownership, 319, 437
Horsepower and speed, 370
Hydroelectric power consumption, 286


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APPLICATION INDEX

Illumination, 370
Immigration, 149, 450
Income analysis, 45, 263
Income tax, 66
Infectious diseases, 410
Insecticides, 414
Installation charges, 45
Insurance company probabilities, 692–693
Interest, 391
Internet growth, 262
Internet hosts, 492
Inventory value, 563–564
Inverse variation, 365
Investment allocation, 583–584
Investment analysis, 393
Investment comparison, 393
Joint variation, 366–367
Labor costs, 558–560, 563, 615–616
Labor-hours, 513
Learning curve, 407–408
Learning theory, 360
Life expectancy, 495
Loan repayment, 709
Logistic growth in an epidemic,
408–409
Manufacturing, 15–17, 20, 65, 124–125,
240, 290, 318, 377, 494, 620

Marine biology, 414, 451, 455
Market analysis, 696, 709
Market research, 98, 124
Markup policy, 150, 563
Marriage, 291
Maximizing revenue, 56–57, 189–190
Maximum area, 183–184, 189, 263, 270
Medical research, 450
Medicare, 455
Medicinal lithotripsy, 730–731
Medicine, 125, 150, 263, 402, 415, 455
Meteorology, 19, 98, 149, 150, 170, 262
Mixture problems, 159–160
Mixtures, 159–160, 170
Money growth, 398, 399, 455
Motion of object, 45
Motion picture industry, 45, 188
Music, 365, 370, 371, 658
Naval architecture, 734
Navigation, 150, 169–170, 223, 745–746
Net cash flow, 6–7
Newton’s law of cooling, 414, 415, 451
Nuclear power, 416, 750
Numbers, 760
Nutrition, 481, 509–510, 525–526, 547,
564

Officer selection, 668–669
Olympic games, 125, 169, 171
Optimal speed, 219–220, 220, 225, 265,

271
Ozone levels, A–91 – A–92
Packaging, 335, A–47
Parabolic reflector, 719–720
Pendulum, A–35
Photic zone, 414, 451
Photography, 370, 415, 451, 659
Physics, 44, 45, 124, 149, 150, 370, 376,
460, 658, A–35, A–95, A–96
Physiology, 360
Plant nutrition, 509, 522
Political science, 264, 768
Politics, 149, 564
Pollution, 521–522
Population growth, 266, 400–401, 413,
414, 450, 455, 460, 495, 657–658
Position of moving object, 124
Present value, 398, 399, 455
Price and demand, 20, 94, 111, 118, 124,
189, 190, 251–253, 270, 478
Price and revenue, 21
Price and supply, 118, 478,
A–96
Pricing, 263, 270
Prize money, 650–651
Production costs, 82
Production scheduling, 481, 488–490,
494, 504–506, 510–513, 546–547,
555, 587
Profit, 19, 64–65, 94–95, 188, 251–253,

256, 290, 318, 376
Profit analysis, 263, 460
Profit and loss analysis, 269
Projectile motion, 184–185, 189, 250,
256, A–132 – 134
Propagation of a rumor, 409
Psychology, 360, 371, 510, 521
Purchasing, 521, 542–543, 620
Puzzle, 546, 615, 658–659
Quality control, 709
Radioactive decay, 403–404
Radioactive tracers, 414
Rate of change, 142–144, 144
Rate-time, 480
Relativistic mass, A–35
Rental charges, 57–58, 156, 157, 219, 220
Replacement time, 360
Research and development analysis, 46
Resource allocation, 494–495, 509, 521,
525, 587, 615
Retention, 360

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XVII

Revenue, 64, 111–112, 118, 125, 188,
189, 190, 290
Revenue analysis, 45, 610
Richter scale, 433, 434, 435

Rocket flight, 435–436
Rolling two dice, 679–680, 686
Safety research, 83
Salary increment, 632
Sales analysis, 45
Sales commissions, 66, 169, 552–553
Selecting officers, 668–669
Selecting subcommittees, 671–672
Serial number counting, 673
Service charges, 66
Shipping, 270, 319, 460
Signal light, 722
Simple interest, 376
Sociology, 510, 522
Solid waste disposal, 216–218
Sound detection, 170
Sound intensity, 431–433, 438, 455, 461
Space vehicles, 438
Sports, 125, 169, A–108
Sports medicine, 263
Stock prices, 67, 149
Stopping distance, 225, 271
Storage, 335
Subcommittee selection, 671–672
Supply and demand, 164–167, 171,
264–265, 477–478, 481
Telephone charges, 66, 125, 126
Temperature, 19, 149, 150, 170, A–96
Timber harvesting, 82–83
Time and speed, 370

Time measurement with atomic clock, A–23
Time spent studying, 262
Tire mileage, 66
Transportation, 65, 223–224, 521, 709, 760
Underwater pressure, 144
Variable costs, 251, 253
Vibration of air in pipe, 365, 371
Volume of cylindrical shell, A–44
Weather balloon, 257
Weight and speed, 370
Weight estimates, 285
Well depth, 234–235
Wildlife management, 415, 455
Women in the workforce, 377
Work, 376
World population, 450
Zeno’s paradox, 659


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CHAPTER

1


Functions, Graphs,
and Models
C
THE function concept is one of the most important ideas in
mathematics. To study math beyond the elementary level, you
absolutely need to have a solid understanding of functions and
their graphs. In this chapter, you’ll learn the fundamentals of
what functions are all about, and how to use them. In subsequent chapters, this will pay off as you study particular types
of functions in depth. In the first section of this chapter, we discuss the techniques involved in using an electronic graphing
device like a graphing calculator. In the remaining sections, we
introduce the concept of functions and discuss general properties of functions and their graphs. Everything you learn in
this chapter will increase your chance of success in this
course, and in almost any other course you may take that involves mathematics.

OUTLINE
1-1

Using Graphing Calculators

1-2

Functions

1-3

Functions: Graphs and
Properties

1-4


Functions: Graphs and
Transformations

1-5

Operations on Functions;
Composition

1-6

Inverse Functions
Chapter 1 Review
Chapter 1 Group Activity:
Mathematical Modeling:
Choosing a Cell Phone
Provider

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2

CHAPTER 1

1-1

FUNCTIONS, GRAPHS, AND MODELS


Using Graphing Calculators
Z Using Graphing Calculators
Z Understanding Screen Coordinates
Z Using the Trace, Zoom, and Intersect Commands
Z Mathematical Modeling

The use of technology to aid in drawing and analyzing graphs is revolutionizing mathematics education and is the primary motivation for this book. Your ability to interpret
mathematical concepts and to discover patterns of behavior will be greatly increased
as you become proficient with an electronic graphing device. In this section we introduce some of the basic features of electronic graphing devices. Additional features will
be introduced as the need arises. If you have already used an electronic graphing device
in a previous course, you can use this section to quickly review basic concepts. If you
need to refresh your memory about a particular feature, consult the Technology Index
at the end of this book to locate the textbook discussion of that particular feature.

Z Using Graphing Calculators
We will begin with the use of electronic graphing devices to graph equations. We will
refer to any electronic device capable of displaying graphs as a graphing utility. The
two most common graphing utilities are handheld graphing calculators and computers with appropriate software. It’s essential that you have such a device handy as you
proceed through this book.
Since many different brands and models exist, we will discuss graphing calculators only in general terms. Refer to your manual for specific details relative to your
own graphing calculator.*
An image on the screen of a graphing calculator is made up of darkened rectangles
called pixels (Fig. 1). The pixel rectangles are the same size, and don’t change in size during any application. Graphing calculators use pixel-by-pixel plotting to produce graphs.
Z Figure 1 Pixel-by-pixel plotting
on a graphing calculator.

(a) Image on
a graphing
calculator.


(b) Magnification to show pixels.
*Manuals for most brands of graphing calculators are readily available on the Internet.

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S E C T I O N 1–1

(a) Standard window variable values
10

Ϫ10

10

Ϫ10

(b) Standard viewing window

Z Figure 2 A standard viewing
window and its dimensions.

Using Graphing Calculators

3

The accuracy of the graph depends on the resolution of the graphing calculator.
Most graphing calculators have screen resolutions of between 50 and 75 pixels per

inch, which results in fairly rough but very useful graphs. Some computer systems
can print very high quality graphs with resolutions greater than 1,000 pixels per inch.
Most graphing calculator screens are rectangular. The graphing screen on a graphing calculator represents a portion of the plane in the rectangular coordinate system.
But this representation is an approximation, because pixels are not really points, as is
clearly shown in Figure 1. Points are geometric objects without dimensions (you can
think of them as “infinitely small”), whereas a pixel has dimensions. The coordinates
of a pixel are usually taken at the center of the pixel and represent all the infinitely
many geometric points within the pixel. Fortunately, this does not cause much of a
problem, as we will see.
The portion of a rectangular coordinate system displayed on the graphing screen
is called a viewing window and is determined by assigning values to six window variables: the lower limit, upper limit, and scale for the x axis and the lower limit, upper
limit, and scale for the y axis. Figure 2(a) illustrates the names and values of standard
window variables, and Figure 2(b) shows the resulting standard viewing window.
The names Xmin, Xmax, Xscl, Ymin, Ymax, and Yscl will be used for the six
window variables. Xscl and Yscl determine the distance between tick marks on the x
and y axes, respectively. Xres is a seventh window variable on some graphing calculators that controls the screen resolution; we will always leave this variable set to the
default value 1. The window variables may be displayed slightly differently by your
graphing calculator. In this book, when a viewing window of a graphing calculator is
pictured in a figure, the values of Xmin, Xmax, Ymin, and Ymax are indicated by
labels to make the graph easier to read [see Fig. 2(b)]. These labels are always centered on the sides of the viewing window, regardless of the location of the axes.
REMARK: We think it’s important that actual output from existing graphing calculators
be used in this book. The majority of the graphing calculator images in this book are
screen dumps from a Texas Instruments TI-84 graphing calculator. Occasionally we
use screen dumps from a TI-86 graphing calculator, which has a wider screen. You
may not always be able to produce an exact replica of a figure on your graphing calculator, but the differences will be relatively minor.
We now turn to the use of a graphing calculator to graph equations that can be
written in the form

y ϭ (some expression in x)


(1)

Graphing an equation of the type shown in equation (1) using a graphing calculator
is a simple three-step process:
Z GRAPHING EQUATIONS USING A GRAPHING CALCULATOR
Step 1. Enter the equation.
Step 2. Enter values for the window variables. (A good rule of thumb for
choosing Xscl and Yscl, unless there are reasons to the contrary, is
to choose each about one-tenth the width of the corresponding variable range.)
Step 3. Press the GRAPH command.

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4

CHAPTER 1

FUNCTIONS, GRAPHS, AND MODELS

The following example illustrates this procedure for graphing the equation
y ϭ x2 Ϫ 4. (See Example 1 of Appendix B, Section B-2 for a hand-drawn sketch of
this equation.)

EXAMPLE

1


Graphing an Equation with a Graphing Calculator
Use a graphing calculator to graph y ϭ x2 Ϫ 4 for Ϫ5 Յ x Յ 5 and Ϫ5 Յ y Յ 15.
SOLUTION

Press the Yϭ key to display the equation editor and enter the equation [Fig. 3(a)]. Press
WINDOW to display the window variables and enter the given values for these variables
[Fig. 3(b)]. Press GRAPH to obtain the graph in Figure 3(c). (The form of the screens in
Figure 3 may differ slightly, depending on the graphing calculator used.)
15

5

Ϫ5

(a) Enter equation.

(b) Enter window variables.

Ϫ5

(c) Press the graph command.

Z Figure 3 Graphing is a three-step process.

MATCHED PROBLEM

1*

Use a graphing calculator to graph y ϭ 8 Ϫ x2 for Ϫ5 Յ x Յ 5 and Ϫ10 Յ y Յ 10.


For Example 1, we displayed a screen shot for each step in the
graphing procedure. Generally, we will show only the final results, as illustrated in
Figure 3(c).
Often, it is helpful to think about an appropriate viewing window before starting to graph an equation. This can help save time, as well as increase your odds of
seeing the whole graph.
REMARK:

*Answers to matched problems in a given section are found near the end of the section, before the
exercise set.

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S E C T I O N 1–1

EXAMPLE

Using Graphing Calculators

5

Finding an Appropriate Viewing Window

2

Find an appropriate viewing window in which to graph the equation y ϭ 1x Ϫ 13
with a graphing calculator.
SOLUTIONS


Algebraic Solution
We begin by thinking about reasonable x values for this
equation. Since y values are determined by an expression under a root, only x values that result in x Ϫ 13
being nonnegative will have an associated y value. So we
write and solve the inequality

Graphical Solution
We first enter the equation y1 ϭ 1x Ϫ 13 in a graphing calculator (Fig. 4).

x Ϫ 13 Ն 0
x Ն 13
This tells us that there will be points on the graph only
for x values 13 or greater.
Next, we make a table of values for selected x values to see what y values are appropriate. Note that we
chose x values that make it easy to compute y.
x

13

14

17

22

29

38


y

0

1

2

3

4

5

Z Figure 4

We then make a table of values for selected x values
after trying a variety of choices for x (Fig. 5).*

To clearly display all of these points and leave some space
around the edges, we choose Xmin ϭ 10, Xmax ϭ 40,
Ymin ϭ Ϫ1, and Ymax ϭ 10.
Z Figure 5

We find that there are no points on the graph for x values less than 13, and there appear to be points for all x
values greater than or equal to 13.
To clearly display all of these points and leave
some space around the edges, we choose Xmin ϭ 10,
Xmax ϭ 40, Ymin ϭ Ϫ1, and Ymax ϭ 10.


MATCHED PROBLEM

2

Find an appropriate viewing window in which to graph the equation
y ϭ 2 ϩ 1x ϩ 15 with a graphing calculator.

*Many graphing calculators can construct a table of values like the one in Figure 5 using the TBLSET
and TABLE commands.

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6

CHAPTER 1

FUNCTIONS, GRAPHS, AND MODELS

The next example illustrates how a graphing calculator can be used as an aid to
sketching the graph of an equation by hand. The example illustrates the use of algebraic, numeric, and graphic approaches; an understanding of all three approaches will
be a big help in problem solving.

EXAMPLE

3

Using a Graphing Calculator as an Aid

to Hand Graphing—Net Cash Flow
The net cash flow y in millions of dollars of a small high-tech company from
1998–2006 is given approximately by the equation
y ϭ 0.4x3 Ϫ 2x ϩ 1

Ϫ4 Յ x Յ 4

(2)

where x represents the number of years before or after 2002, when the board of directors appointed a new CEO.
(A) Construct a table of values for equation (2) for each year starting with 1998 and
ending with 2006. Compute y to one decimal place.
(B) Obtain a graph of equation (2) in the viewing window of your graphing calculator. Plot the table values from part A by hand on graph paper, then join these
points with a smooth curve using the graph in the viewing window as an aid.
(a)

SOLUTIONS

(b)

Z Figure 6

(A) After entering the given equation as y1, we can find the value of y for a given
value of x by storing the value of x in the variable X and simply displaying y1,
as shown in Figure 6(a). To speed up this process, we can compute an entire
table of values directly, as shown in Figure 6(b). We organize these results in
Table 1.
Recall that x represents years before or after 2002, and y represents cash flow
in millions of dollars.
Table 1 Net Cash Flow

Year

Z Figure 7

1998

1999

2000

2001

2002

2003

2004

2005

2006

x

Ϫ4

Ϫ3

Ϫ2


Ϫ1

0

1

2

3

4

y (million $)

Ϫ16.6

Ϫ3.8

1.8

2.6

1

Ϫ0.6

0.2

5.8


18.6

(B) To create a graph of equation (2) in the viewing window of a graphing calculator, we select values for the viewing window variables that cover a little more
than the values shown in Table 1, as shown in Figure 7. We add a grid to the
viewing window to obtain the graphing calculator graph shown in Figure 8(a).
(On many graphing calculators, this option is on the FORMAT screen.) The corresponding hand sketch is shown in Figure 8(b).

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