Stewart J. College Algebra: Concepts and Contexts

5.3 Maxima and Minima: Getting Information from a Model 439

Finding Maximum and Minimum Values • Modeling with Quadratic Functions

5.4 Quadratic Equations: Getting Information from a Model 448

Solving Quadratic Equations: Factoring • Solving Quadratic Equations: The Quadratic

Formula • The Discriminant • Modeling with Quadratic Functions

5.5 Fitting Quadratic Curves to Data 461

Modeling Data with Quadratic Functions

CHAPTER 5 Review 466

CHAPTER 5 Test 472

■ EXPLORATIONS

1 Transformation Stories 473

2 Toricelli’s Law 476

3 Quadratic Patterns 478

chapter 6 Power, Polynomial, and Rational

Functions 483

6.1 Working with Functions: Algebraic Operations 484

Adding and Subtracting Functions • Multiplying and Dividing Functions

6.2 Power Functions: Positive Powers 493

Power Functions with Positive Integer Powers • Direct Proportionality • Fractional

Positive Powers • Modeling with Power Functions

6.3 Polynomial Functions: Combining Power Functions 504

Polynomial Functions • Graphing Polynomial Functions by Factoring • End Behavior

and the Leading Term • Modeling with Polynomial Functions

6.4 Fitting Power and Polynomial Curves to Data 516

Fitting Power Curves to Data • A Linear, Power, or Exponential Model? • Fitting

Polynomial Curves to Data

6.5 Power Functions: Negative Powers 527

The Reciprocal Function • Inverse Proportionality • Inverse Square Laws

6.6 Rational Functions 536

Graphing Quotients of Linear Functions • Graphing Rational Functions

CHAPTER 6 Review 546

CHAPTER 6 Test 553

■ EXPLORATIONS

1 Only in the Movies? 554

2 Proportionality: Shape and Size 557

3 Managing Traffic 560

4 Alcohol and the Surge Function 563

viii CONTENTS

chapter 7 Systems of Equations and Data

in Categories 567

7.1 Systems of Linear Equations in Two Variables 568

Systems of Equations and Their Solutions • The Substitution Method • The Elimination

Method • Graphical Interpretation: The Number of Solutions • Applications: How Much

Gold Is in the Crown?

7.2 Systems of Linear Equations in Several Variables 580

Solving a Linear System • Inconsistent and Dependent Systems • Modeling with Linear

Systems

7.3 Using Matrices to Solve Systems of Linear Equations 590

Matrices • The Augmented Matrix of a Linear System • Elementary Row Operations •

Row-Echelon Form • Reduced Row-Echelon Form • Inconsistent and Dependent

Systems

7.4 Matrices and Data in Categories 602

Organizing Categorical Data in a Matrix • Adding Matrices • Scalar Multiplication of

Matrices • Multiplying a Matrix Times a Column Matrix

7.5 Matrix Operations: Getting Information from Data 611

Addition, Subtraction, and Scalar Multiplication • Matrix Multiplication • Getting

Information from Categorical Data

7.6 Matrix Equations: Solving a Linear System 619

The Inverse of a Matrix • Matrix Equations • Modeling with Matrix Equations

CHAPTER 7 Review 627

CHAPTER 7 Test 634

■ EXPLORATIONS

1 Collecting Categorical Data 635

2 Will the Species Survive? 637

■ Algebra Toolkit A: Working with Numbers T1

A.1 Numbers and Their Properties T1

A.2 The Number Line and Intervals T7

A.3 Integer Exponents T14

A.4 Radicals and Rational Exponents T20

■ Algebra Toolkit B: Working with Expressions T25

B.1 Combining Algebraic Expressions T25

B.2 Factoring Algebraic Expressions T33

B.3 Rational Expressions T39

CONTENTS ix

■ Algebra Toolkit C: Working with Equations T47

C.1 Solving Basic Equations T47

C.2 Solving Quadratic Equations T56

C.3 Solving Inequalities T62

■ Algebra Toolkit D: Working with Graphs T67

D.1 The Coordinate Plane T67

D.2 Graphs of Two-Variable Equations T71

D.3 Using a Graphing Calculator T80

D.4 Solving Equations and Inequalities Graphically T85

ANSWERS A1

INDEX I1

x CONTENTS

In recent years many mathematicians have recognized the need to revamp the tradi-

tional college algebra course to better serve today’s students. A National Science

Foundation–funded conference, “Rethinking the Courses below Calculus,” held in

Washington, D.C., in October 2001, brought together some of the leading re-

searchers studying this issue.* The conference revealed broad agreement that the

topics presented in the course and, even more importantly, how those topics are pre-

sented are the main issues that have led to disappointing success rates among college

algebra students. Some of the major themes to emerge from this conference included

the need to spend less time on algebraic manipulation and more time on exploring

concepts; the need to reduce the number of topics but study the topics covered in

greater depth; the need to give greater priority to data analysis as a foundation for

mathematical modeling; the need to emphasize the verbal, numerical, graphical, and

symbolic representations of mathematical concepts; and the need to connect the

mathematics to real-life situations drawn from the students’ majors. Indeed, college

algebra students are a diverse group with a broad variety of majors ranging from the

arts and humanities to the managerial, social, and life sciences, as well as the phys-

ical sciences and engineering. For each of these students a conceptual understanding

of algebra and its practical uses is of immense importance for appreciating quantita-

tive relationships and formulas in their other courses, as well as in their everyday

experiences.

We think that each of the themes to come out of the 2001 conference represents

a major step forward in improving the effectiveness of the college algebra course.

This textbook is intended to provide the tools instructors and their students need to

implement the themes that fit their requirements.

This textbook is nontraditional in the sense that the main ideas of college alge-

bra are front and center, without a lot of preliminaries. For example, the first chap-

ter begins with real-world data and how a simple equation can sometimes help us

describe the data—the main concept here being the remarkable effectiveness of

equations in allowing us to interpolate and extend data far beyond the original mea-

sured quantities. This rather profound idea is easily and naturally introduced with-

out the need for a preliminary treatise on real numbers and equations (the traditional

approach). These latter ideas are introduced only as the need for them arises: As

more complex and subtle relationships in the real world are discovered, more prop-

erties of numbers and more technical skill with manipulating mathematical symbols

are required. But throughout the textbook the main concepts of college algebra and

the real-world contexts in which they occur are always paramount in the exposition.

Naturally, there are many valid paths to the teaching of the concepts of college

algebra, and each instructor brings unique strengths and imagination to the class-

room. But any successful approach must meet students where they are and then

PREFACE

*Hastings, Nancy B., et al., ed., A Fresh Start for Collegiate Mathematics: Rethinking the Courses

below Calculus, Mathematical Association of America, Washington, D.C., 2006.

guide them to a place where they can appreciate some of the interesting uses and

techniques of algebraic reasoning. We believe that real-world data are useful in cap-

turing student interest in mathematics and in helping to decipher the essential con-

nection between numbers and real-world events. Data also help to emphasize that

mathematics is a human activity that requires interpretation to have effective mean-

ing and use. But we also take care that the message of college algebra not be drowned

in a sea of data and subsidiary information. Occasionally, the clarity of a well-chosen

idealized example can home in more sharply on a particular concept. We also know

that no real understanding of college algebra concepts is possible without some tech-

nical ability in manipulating mathematical symbols—indeed, conceptual under-

standing and technical skill go hand in hand, each reinforcing the other. We have

encapsulated the essential tools of algebra in concise Algebra Toolkits at the end of

the book; these toolkits give students an opportunity to review and hone basic skills

by focusing on the concepts needed to effectively apply these skills. At crucial junc-

tures in each chapter students can gauge their need to study a particular toolkit by

completing an Algebra Checkpoint. Of course, we have included Skills exercises in

each section, which are devoted to practicing algebraic skills relevant to that section.

But perhaps students get the deepest understanding from nuts-and-bolts experimen-

tation and the subsequent discovery of a concept, individually or in groups. For this

reason we have concluded each chapter with special sections called Explorations, in

which students are guided to discover a basic principle or concept on their own. The

explorations and all the other elements of this textbook are provided as tools to be

used by instructors and their students to navigate their own paths toward conceptual

understanding of college algebra.

Content

The chapters in this book are organized around major conceptual themes. The over-

arching theme is that of functions and their power in modeling real-world phenom-

ena. (In this book a model always has an explicit purpose: It is used to get

information about the thing being modeled.) This theme is kept in the forefront in

the text by introducing the key properties of functions only where they are first

needed in the exposition. For example, composition and inverse functions are intro-

duced in the chapters on exponential and logarithmic functions, where they help to

explain the fundamental relationship between these functions, whereas transforma-

tions of functions are introduced in the chapter on quadratic functions, where they

help to explain how the graph of a quadratic function is obtained. To draw attention

to the function theme in each chapter, the title of the sections that specifically intro-

duce a new feature of functions is prefaced by the phrase “Working with functions.”

In general, throughout the text, specific topics are presented only as they are needed

and not as early preliminaries.

xii PREFACE

PROLOGUE The book begins with a prologue entitled Algebra and Alcohol, which introduces the

themes of data, functions, and modeling. The intention of the prologue is to engage

students’ attention from the outset with a real-world problem of some interest and

importance: How can we predict the effects of different levels of drinking? After giv-

ing some background to the problem in the prologue, we return to it throughout the

book, showing how we can answer more questions about the problem as we learn

more algebra in successive chapters.

CHAPTER 1 Data, Functions, and Models This chapter begins with real-life data and their

graphical representation. This sets the stage for simple linear equations that model

data. We next identify those relations that are functions and how they arise in real-

world contexts. We pay special attention to the interplay between numerical, graph-

ical, symbolic, and verbal representations of functions. In particular, the graph of a

function is identified as a rich source of valuable information about the behavior of

a function. Functions naturally lead to formulas, the concluding topic of this chap-

ter. (We include this topic because students will encounter formulas that they must

use and understand in their other courses.)

CHAPTER 2 Linear Functions and Models This chapter begins with the concept of the av-

erage rate of change of a function, which leads to the natural concept of constant rate

of change. The rest of the chapter focuses on the concept of linearity and its various

implications. Although basic linear equations are first introduced in Chapter 1, here

we discuss linear functions and their graphs in more detail, including the ideas of

slope and rate of change. Real-world applications of linearity lead to the question of

how trends in real-world data can be approximated by fitting lines to data.

CHAPTER 3 Exponential Functions and Models This chapter begins with an extended ex-

ample on population growth. This sets the stage for exponential functions, their rates

of growth, and their uses in modeling many real-world phenomena.

CHAPTER 4 Logarithmic Functions and Exponential Models This chapter introduces

logarithmic functions and logarithmic scales. Logarithmic equations are presented

as tools for getting information from exponential models. The concepts of function

composition and inverse functions are introduced here, where they serve to put the

relationship between exponential and logarithmic functions into sharp focus.

CHAPTER 5 Quadratic Functions and Models The function concept introduced in this

chapter is that of transformations of graphs, a process needed in obtaining the graph

of a general quadratic function as a transformation of the standard parabola. Graphs

of quadratic functions naturally lead to the concept of maximum and minimum val-

ues and to the solution of quadratic equations.

CHAPTER 6 Power, Polynomial, and Rational Functions This chapter is about power

functions (positive and negative powers) and their graphs. The function concept in-

troduced in this chapter is that of algebraic operations on functions. In this setting,

polynomial functions are simply sums of power functions. Rational functions are in-

troduced as shifts and combinations of the reciprocal function.

CHAPTER 7 Systems of Equations and Data in Categories In this chapter we return to the

theme of linearity by introducing systems of linear equations. The graphical repre-

sentation of a system gives a clear visual image of the meaning of a system and its

solutions. Matrices provide us with a new view of data: A matrix allows us to cate-

gorize data in well-defined rows and columns. We introduce the basic matrix opera-

tions as powerful tools for extracting information from such data, including

predicting data trends. Finally, by expressing a system of equations as a matrix, we

can use these matrix operations to solve the system. In this chapter the graphing cal-

culator is used extensively for computations involving matrices.

PREFACE xiii

TOOLKIT A Working with Numbers This toolkit is about the real number system, the prop-

erties of exponents and radicals, and the number line.

TOOLKIT B Working with Expressions This toolkit is about algebraic expressions, includ-

ing the basic properties of expanding, factoring, and adding rational expressions.

TOOLKIT C Working with Equations This toolkit is about solving linear, quadratic, and

power equations, as well as solving linear and quadratic inequalities.

TOOLKIT D Working with Graphs This toolkit is about the coordinate plane and graphs of

equations, including graphical methods for solving equations and inequalities.

xiv PREFACE

Teaching with the Help of This Book

We are keenly aware that good teaching comes in many forms and that there are

many different approaches to teaching the concepts and skills of college algebra. The

organization of the topics in this book accommodates different teaching styles. For

example, if the topics are taught in the order in which they appear in the book, then

exponential functions (Chapter 3) immediately follow linear functions (Chapter 2),

contrasting the dramatic difference in the rates of growth of these functions. Alter-

natively, the chapter on quadratic functions (Chapter 5) can be taught immediately

following the chapter on linear functions (Chapter 2), emphasizing the kinship of

these two classes of functions. In any case, we trust that this book can serve as the

foundation for a thoroughly modern college algebra course.

Exercise Sets—Concepts, Skills, Contexts Each exercise set is arranged into

Concepts, Skills, and Contexts exercises. The Concept exercises include Funda-

mentals exercises, which require students to use the language of algebra to state es-

sential facts about the topics of the section, and Think About It exercises, which are

designed to challenge students’ understanding of a concept and can serve as a basis

for class discussion. The Skills exercises emphasize the basic algebra techniques

used in the section; the Contexts exercises show how algebra is used in real-world

situations. There are sufficient exercises to give the instructor a wide choice of exer-

cises to assign.

Chapter Reviews and Chapter Tests—Connecting the Concepts Each

chapter ends with an extensive Review section beginning with a Concept Check, in

which the main ideas of the chapter are succinctly summarized. Several of the review

exercises are designated Connecting the Concepts. Each of these exercises involves

many of the ideas of the chapter in a single problem, highlighting the connections

between the various concepts. The Review ends with a Chapter Test, in which stu-

dents can gauge their mastery of the concepts and skills of the chapter.

Algebra Toolkits and Algebra Checkpoints The Algebra Toolkits present a

comprehensive review of basic algebra skills. The appropriate toolkit can be taught

whenever the need arises. The toolkits may be assigned to students to read on their

own and do the exercises (students may also do the exercises online with Enhanced

WebAssign). Several sections in the text contain Algebra Checkpoints, which con-

sist of questions designed to gauge students’ mastery of the algebra skills needed for

that section. Each checkpoint is linked to an Algebra Toolkit that explains the rele-

vant topic.

Explorations Each chapter contains several Explorations designed to guide stu-

dents to discover an algebra concept. These can be used as in-class group activities

and can be assigned at any time during the teaching of a chapter; some of the explo-

rations can serve as an introduction to the ideas of a chapter (the Instructor’s Guide

gives additional suggestions on using the explorations).

Instructor’s Guide The Instructor’s Guide, written by Professor Lynelle Weldon

(Andrews University), contains a wealth of suggestions on how to teach each sec-

tion, including key points to stress, questions to ask students, homework exercises to

assign, and many imaginative classroom activities that are sure to interest students

and bring key concepts to life.

Study Guide A Study Guide written by Professor Florence Newberger (Califor-

nia State University, Long Beach) is available to students. This unique study guide

literally guides students through the text, explaining how to read and understand the

examples and, in general, teaches students how to read mathematics. The guide pro-

vides step-by-step solutions to many of the exercises in the text that are linked to the

examples (the pencil icon in the text identifies these exercises).

Enhanced WebAssign (EWA) This is a web-based homework system that allows

instructors to assign, collect, grade, and record homework assignments online. EWA

allows for several options to help students learn, including links to relevant sections

in the text, worked-out solutions, and video instruction for most exercises. The ex-

ercises available in EWA are listed in the Instructor’s Guide.

Acknowledgments

First and foremost, we thank the instructors at Mercer County Community College

who urged us to write this book and who met with us to share their thoughts about

the need for change in the college algebra course: Don Reichman, Mary Hayes, Paul

Renato Toppo, Daniel Rose, and Daniel Guttierez.

We thank the following reviewers for their thoughtful and constructive comments:

Ahmad Kamalvand, Huston-Tillotson University; Alison Becker-Moses, Mercer

County Community College; April Strom, Scottsdale Community College; Derron

Rafiq Coles, Oregon State University; Diana M. Zych, Erie Community

College–North Campus; Ingrid Peterson, University of Kansas; James Gray,

Tacoma Community College; Janet Wyatt, Metropolitan Community

College–Longview; Judy Smalling, St. Petersburg College; Lee A. Seltzer, Jr.,

Florida Community College at Jacksonville; Lynelle Weldon, Andrews University;

Marlene Kusteski, Virginia Commonwealth University; Miguel Montanez, Miami

Dade Wolfson; Rhonda Nordstrom Hull, Clackamas Community College; Rich

West, Francis Marion University; Sandra Poinsett, College of Southern Maryland;

Semra Kilic-Bahi, Colby Sawyer College; Sergio Loch, Grand View University;

Stephen J. Nicoloff, Paradise Valley Community College; Susan Howell, University

of Southern Mississippi; Wendiann Sethi, Seton Hall University.

We are grateful to our colleagues who continually share with us their insights into

teaching mathematics. We especially thank Lynelle Weldon for writing the Instruc-

tor’s Guide and Florence Newberger for writing the Study Guide that accompanies

this book. We thank Blaise DeSesa at Penn State Abington for reading the entire

PREFACE xv

manuscript and doing a masterful job of checking the correctness of the examples and

answers to exercises. We thank Jean-Marie Magnier at Springfield Technical Com-

munity College for producing the complete and accurate solutions manual and Aaron

Watson for reading the manuscript and checking the answers to the exercises. We

thank Dr. Louis Liu for suggesting the topic of the prologue and supplying the alco-

hol study data. (Several years ago, Louis Liu was a student in one of James Stewart’s

calculus classes; he is now a medical doctor and professor of gastroenterology at the

University of Toronto.) We thank Derron Coles and his students at Oregon State Uni-

versity for class testing the manuscript and supplying us with significant suggestions

and comments. We thank Professor Rick LeBorne and his students at Tennessee Tech

for performing the experiment on Torricelli’s Law and supplying the photograph on

page 476.

We thank Martha Emry, our production service and art editor, for her ability to

solve all production problems and Barbara Willette, our copy editor, for her attention

to every detail in the manuscript. We thank Jade Myers and his staff at Matrix for

their attractive and accurate graphs and Network Graphics for bringing many of our

illustrations to life. We thank our cover designer Larry Didona for the elegant and

appropriate cover.

At Brooks/Cole we especially thank Stacy Green, developmental editor, and

Jennifer Risden, content project manager, for guiding and facilitating every aspect of

the production of this book. Of the many Brooks/Cole staff involved in this project

we particularly thank the following: Cynthia Ashton, assistant editor; Guanglei

Zhang, editorial assistant; Lynh Pham, associate media editor; Vernon Boes, art di-

rector; Rita Lombard, developmental editor for market strategies; and our marketing

team led by Myriah Fitzgibbon, marketing manager. They have all done an out-

standing job.

Numerous other people were involved in the production of this book, including

permissions editors, photo researchers, text designers, typesetters, compositors,

proofreaders, printers, and many more. We thank them all.

Above all, we thank our editor Gary Whalen. His vast editorial experience, his

extensive knowledge of current issues in the teaching of mathematics, and especially

his deep interest in mathematics textbooks have been invaluable resources in the

writing of this book.

xvi PREFACE

Student Ancillaries

Student Solutions Manual (0-495-38790-8)

Jean Marie Magnier—Springfield Technical Community College

The student solutions manual provides worked-out solutions to all of the odd-

numbered problems in the text. It also offers hints and additional problems for

practice, similar to those in the text.

Study Guide (0-495-38791-6)

Florence Newberger—California State University, Long Beach

The study guide reinforces student understanding with detailed explanations,

worked-out examples, and practice problems. It lists key ideas to master and builds

problem-solving skills. There is a section in the study guide corresponding to each

section in the text.

Instructor Ancillaries

Instructor’s Edition (0-495-55395-6)

This instructor’s version of the complete student text has the answer to every

exercise included in the answer section.

Complete Solutions Manual (0-495-38792-4)

Jean Marie Magnier—Springfield Technical Community College

The complete solutions manual contains solutions to all exercises from the text,

including Chapter Review Exercises, Chapter Tests, and Cumulative Review

Exercises.

PowerLecture with ExamView (0-495-38796-7)

The CD-ROM provides the instructor with dynamic media tools for teaching

college algebra. PowerPoint

lecture slides and art slides of the figures from the

text, together with electronic files for the test bank and solutions manual, are

available. The algorithmic ExamView

, an easy-to-use assessment system, allows

you to create, deliver, and customize tests (both print and online) in minutes.

Enhance how your students interact with you, your lecture, and each other.

Instructor’s Guide (0-495-38795-9)

Lynelle Weldon—Andrews University

The instructor’s guide contains points to stress, suggested time to allot, text

discussion topics, core materials for lecture, workshop/discussion suggestions,

group work exercises in a form suitable for handout, and suggested homework

problems.

ANCILLARIES

xvii

Stewart J. College Algebra: Concepts and Contexts

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