College algebra

■

Linear Functions

A linear function is a function of the form f 1x2 = b + mx.
■

The graph of f is a line with slope m and y-intercept b.
y

y

b
b
x

x
Ï=b
■

Ï=b+mx

Exponential Functions

An exponential function is a function of the form f 1x2 = Ca x.
■

The graph of f has one of the shapes shown.

■

If a 7 1, then a is called the growth factor and r = a - 1 is called the growth rate.

■

If a 6 1, then a is called the decay factor and r = a - 1 is called the decay rate.
y

y

1

1

0
Ï=a˛, a>1

■

0

x

x

Ï=a˛, 0
Logarithmic Functions

A logarithmic function with base a 7 1 is a function of the form f 1x2 = loga x. By definition,

loga x = y

if and only if

■

The graph of f has the general shape shown below.

■

Basic properties:

loga1 = 0,

ay = x

loga a x = x, a logax = x

loga a = 1,
y

0

1

Ï=loga x, a>1

x

■

Quadratic Functions

A quadratic function is a function of the form f 1x2 = ax 2 + bx + c.
■

The graph of f has the shape of a parabola.

■

The maximum or minimum value of f occurs at x = -

■

The function f can be expressed in the standard form f 1x2 = a1x - h2 2 + k.

■

The vertex of the graph of f is at the point (h, k).
y

b
.
2a

y

(h, k)

0

x
(h, k)
0
y=a(x-h)™+k
a<0, h>0, k>0

■

x
y=a(x-h)™+k
a>0, h>0, k>0

Power Functions

A power function is a function of the form f 1x 2 = Cx p.
■

Graphs of some power functions are shown.

Positive powers
y

y

y

x

x

x

x

Ï=x£

Ï=≈

y

Ï=x¢

Ï=x∞

Fractional powers
y

y

y

x

x

£x
Ï=œ
∑

Ï=œ∑
x

y

x

x

∞x
Ï= œ∑

¢x
Ï= œ
∑

Negative powers
y

y

y

x

Ï= x1

y

x

1

Ï=≈

x

1

Ï=x£

x

1

Ï= x¢


College Algebra
CONCEPTS AND CONTEXTS


ABOUT THE AUTHORS
JAMES STEWART received his MS from Stanford University and his PhD from
the University of Toronto. He did research at the University of London and was
influenced by the famous mathematician George Polya at Stanford University.
Stewart is Professor Emeritus at McMaster University and is currently Professor
of Mathematics at the University of Toronto. His research field is harmonic
analysis and the connections between mathematics and music. James Stewart is

the author of a bestselling calculus textbook series published by Brooks/Cole,
Cengage Learning, including Calculus, Calculus: Early Transcendentals, and
Calculus: Concepts and Contexts; a series of precalculus texts; and a series of
high-school mathematics textbooks.

LOTHAR REDLIN grew up on Vancouver Island, received a Bachelor of Science
degree from the University of Victoria, and received a PhD from McMaster
University in 1978. He subsequently did research and taught at the University of
Washington, the University of Waterloo, and California State University, Long
Beach. He is currently Professor of Mathematics at The Pennsylvania State
University, Abington Campus. His research field is topology.

SALEEM WATSON received his Bachelor of Science degree from Andrews
University in Michigan. He did graduate studies at Dalhousie University and
McMaster University, where he received his PhD in 1978. He subsequently did
research at the Mathematics Institute of the University of Warsaw in Poland. He
also taught at The Pennsylvania State University. He is currently Professor of
Mathematics at California State University, Long Beach. His research field is
functional analysis.

PHYLLIS PANMAN received a Bachelor of Music degree in violin performance
in 1987 and a PhD in mathematics in 1996 from the University of Missouri at
Columbia. Her research area is harmonic analysis. As a graduate student she
taught college algebra and calculus courses at the University of Missouri. She
continues to teach and tutor students in mathematics at all levels, including
conducting mathematics enrichment courses for middle school students.
Stewart, Redlin, and Watson have also published Precalculus: Mathematics for
Calculus, Algebra and Trigonometry, and Trigonometry.

About the Cover

Each of the images on the cover appears somewhere within the pages of the book itself—in real-world examples, exercises, or
explorations. The many and varied applications of algebra that we study in this book highlight the importance of algebra in
understanding the world around us, and many of these applications take us to places where we never thought mathematics
would go. The global montage on the cover is intended to echo this universal reach of the applications of algebra.


College Algebra
CONCEPTS AND CONTEXTS

James Stewart
McMaster University
and
University of Toronto

Lothar Redlin
The Pennsylvania State University

Saleem Watson
California State University, Long Beach

Phyllis Panman

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States


College Algebra: Concepts and Contexts
James Stewart, Lothar Redlin, Saleem Watson,
Phyllis Panman
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CONTENTS
PROLOGUE: Algebra and Alcohol P1

1 Data, Functions, and Models

chapter

1.1

1

Making Sense of Data 2
Analyzing One-Variable Data • Analyzing Two-Variable Data

1.2

Visualizing Relationships in Data 12
Relations: Input and Output • Graphing Two-Variable Data in a Coordinate Plane •
Reading a Graph

1.3

Equations: Describing Relationships in Data 25
Making a Linear Model from Data • Getting Information from a Linear Model

1.4

Functions: Describing Change

35

Definition of Function • Which Two-Variable Data Represent Functions? • Which
Equations Represent Functions? • Which Graphs Represent Functions? • Four Ways
to Represent a Function

1.5

Function Notation: The Concept of Function as a Rule 52
Function Notation • Evaluating Functions—Net Change • The Domain of a Function •
Piecewise Defined Functions

1.6

Working with Functions: Graphs and Graphing Calculators 64
Graphing a Function from a Verbal Description • Graphs of Basic Functions • Graphing
with a Graphing Calculator • Graphing Piecewise Defined Functions

1.7

Working with Functions: Getting Information from the Graph 74
Reading the Graph of a Function • Domain and Range from a Graph • Increasing and
Decreasing Functions • Local Maximum and Minimum Values

1.8

Working with Functions: Modeling Real-World Relationships

88

Modeling with Functions • Getting Information from the Graph of a Model

1.9

Making and Using Formulas

101

What Is a Formula? • Finding Formulas • Variables with Subscripts • Reading and
Using Formulas

■
1
2
3

chapter

CHAPTER 1 Review 113
CHAPTER 1 Test 126
EXPLORATIONS
Bias in Presenting Data 128
Collecting and Analyzing Data 134
Every Graph Tells a Story 138

2 Linear Functions and Models

2.1

Working with Functions: Average Rate of Change

141
142

Average Rate of Change of a Function • Average Speed of a Moving Object • Functions
Defined by Algebraic Expressions

v


vi

CONTENTS

2.2

Linear Functions: Constant Rate of Change 153
Linear Functions • Linear Functions and Rate of Change • Linear Functions and Slope •
Using Slope and Rate of Change

2.3

Equations of Lines: Making Linear Models 165
Slope-Intercept Form • Point-Slope Form • Horizontal and Vertical Lines • When Is the
Graph of an Equation a Line?

2.4

Varying the Coefficients: Direct Proportionality 177
Varying the Constant Coefficient: Parallel Lines • Varying the Coefficient of x:
Perpendicular Lines • Modeling Direct Proportionality

2.5

Linear Regression: Fitting Lines to Data 189
The Line That Best Fits the Data • Using the Line of Best Fit for Prediction • How Good
Is the Fit? The Correlation Coefficient

2.6

Linear Equations: Getting Information from a Model 201
Getting Information from a Linear Model • Models That Lead to Linear Equations

2.7

Linear Equations: Where Lines Meet 210
Where Lines Meet • Modeling Supply and Demand

■
1
2
3
4
5

chapter

CHAPTER 2 Review 219

CHAPTER 2 Test 228
EXPLORATIONS
When Rates of Change Change 229
Linear Patterns 233
Bridge Science 237
Correlation and Causation 239
Fair Division of Assets 242

3 Exponential Functions and Models

3.1

247

Exponential Growth and Decay 248
An Example of Exponential Growth • Modeling Exponential Growth: The Growth
Factor • Modeling Exponential Growth: The Growth Rate • Modeling
Exponential Decay

3.2

Exponential Models: Comparing Rates 261
Changing the Time Period • Growth of an Investment: Compound Interest

3.3

Comparing Linear and Exponential Growth 272
Average Rate of Change and Percentage Rate of Change • Comparing Linear and
Exponential Growth • Logistic Growth: Growth with Limited Resources

3.4

Graphs of Exponential Functions 286
Graphs of Exponential Functions • The Effect of Varying a or C • Finding an Exponential
Function from a Graph

3.5

Fitting Exponential Curves to Data 295
Finding Exponential Models for Data • Is an Exponential Model Appropriate? •
Modeling Logistic Growth

CHAPTER 3 Review 303
CHAPTER 3 Test 311


vii

CONTENTS

■
1
2
3
4

chapter

EXPLORATIONS
Extreme Numbers: Scientific Notation 312

So You Want to Be a Millionaire? 315
Exponential Patterns 316
Modeling Radioactivity with Coins and Dice 320

4 Logarithmic Functions and
Exponential Models

4.1

Logarithmic Functions

323

324

Logarithms Base 10 • Logarithms Base a • Basic Properties of Logarithms • Logarithmic
Functions and Their Graphs

4.2

Laws of Logarithms

334

Laws of Logarithms • Expanding and Combining Logarithmic Expressions • Change of
Base Formula

4.3

Logarithmic Scales

342

Logarithmic Scales • The pH Scale • The Decibel Scale • The Richter Scale

4.4

The Natural Exponential and Logarithmic Functions 350
What Is the Number e? • The Natural Exponential and Logarithmic Functions •
Continuously Compounded Interest • Instantaneous Rates of Growth or Decay •
Expressing Exponential Models in Terms of e

4.5

Exponential Equations: Getting Information from a Model 364
Solving Exponential and Logarithmic Equations • Getting Information from
Exponential Models: Population and Investment • Getting Information from
Exponential Models: Newton’s Law of Cooling • Finding the Age of Ancient
Objects: Radiocarbon Dating

4.6

Working with Functions: Composition and Inverse

377

Functions of Functions • Reversing the Rule of a Function • Which Functions Have
Inverses? • Exponential and Logarithmic Functions as Inverse Functions

■

1
2
3
4

chapter

CHAPTER 4 Review 393
CHAPTER 4 Test 400
EXPLORATIONS
Super Origami 401
Orders of Magnitude 402
Semi-Log Graphs 406
The Even-Tempered Clavier 409

5 Quadratic Functions and Models

5.1

Working with Functions: Shifting and Stretching

413

414

Shifting Graphs Up and Down • Shifting Graphs Left and Right • Stretching and
Shrinking Graphs Vertically • Reflecting Graphs

5.2

Quadratic Functions and Their Graphs 428
The Squaring Function • Quadratic Functions in General Form • Quadratic Functions in
Standard Form • Graphing Using the Standard Form


viii

CONTENTS

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

■
1

2
3

chapter

CHAPTER 5 Review 466
CHAPTER 5 Test 472
EXPLORATIONS
Transformation Stories 473
Toricelli’s Law 476
Quadratic Patterns 478

6 Power, Polynomial, and Rational
Functions

6.1

483

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

■
1
2
3
4

CHAPTER 6 Review 546
CHAPTER 6 Test 553
EXPLORATIONS
Only in the Movies? 554

Proportionality: Shape and Size 557
Managing Traffic 560
Alcohol and the Surge Function 563


ix

CONTENTS

chapter

7 Systems of Equations and Data
in Categories

7.1

567

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

■
1
2

CHAPTER 7 Review 627
CHAPTER 7 Test 634
EXPLORATIONS
Collecting Categorical Data 635
Will the Species Survive? 637

■ Algebra Toolkit A: Working with Numbers
A.1
A.2
A.3
A.4

Numbers and Their Properties T1
The Number Line and Intervals T7
Integer Exponents T14
Radicals and Rational Exponents T20

■ Algebra Toolkit B: Working with Expressions
B.1
B.2
B.3

T1

Combining Algebraic Expressions T25
Factoring Algebraic Expressions T33
Rational Expressions T39

T25

x

CONTENTS

■ Algebra Toolkit C: Working with Equations
C.1
C.2
C.3

Solving Basic Equations T47
Solving Quadratic Equations T56
Solving Inequalities T62

■ Algebra Toolkit D: Working with Graphs
D.1
D.2
D.3
D.4

T47

The Coordinate Plane T67
Graphs of Two-Variable Equations T71
Using a Graphing Calculator T80
Solving Equations and Inequalities Graphically T85

ANSWERS
INDEX

I1

A1

T67


PREFACE

In recent years many mathematicians have recognized the need to revamp the traditional 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 researchers studying this issue.* The conference revealed broad agreement that the
topics presented in the course and, even more importantly, how those topics are presented 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 physical sciences and engineering. For each of these students a conceptual understanding
of algebra and its practical uses is of immense importance for appreciating quantitative 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 algebra are front and center, without a lot of preliminaries. For example, the first chapter 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 measured quantities. This rather profound idea is easily and naturally introduced without 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 properties 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 classroom. But any successful approach must meet students where they are and then
*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.

xi


xii

PREFACE

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 capturing student interest in mathematics and in helping to decipher the essential connection between numbers and real-world events. Data also help to emphasize that
mathematics is a human activity that requires interpretation to have effective meaning 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 technical ability in manipulating mathematical symbols—indeed, conceptual understanding 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 junctures 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 experimentation 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 overarching theme is that of functions and their power in modeling real-world phenomena. (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 introduced in the chapters on exponential and logarithmic functions, where they help to
explain the fundamental relationship between these functions, whereas transformations 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 introduce 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.
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 giving 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.


PREFACE

xiii

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 realworld contexts. We pay special attention to the interplay between numerical, graphical, 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 chapter. (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 average 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 example 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 values 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 introduced in this chapter is that of algebraic operations on functions. In this setting,
polynomial functions are simply sums of power functions. Rational functions are introduced 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 representation 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 categorize data in well-defined rows and columns. We introduce the basic matrix operations 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 calculator is used extensively for computations involving matrices.


xiv

PREFACE

TOOLKIT A

Working with Numbers This toolkit is about the real number system, the properties of exponents and radicals, and the number line.

TOOLKIT B

Working with Expressions This toolkit is about algebraic expressions, including 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.

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. Alternatively, 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 Fundamentals exercises, which require students to use the language of algebra to state essential 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 exercises 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 students 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 consist 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 relevant topic.


PREFACE

xv

Explorations Each chapter contains several Explorations designed to guide students 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 explorations 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 section, 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 (California 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 provides 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 exercises 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 Instructor’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


xvi

PREFACE

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 Community 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 alcohol 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 University 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 director; Rita Lombard, developmental editor for market strategies; and our marketing
team led by Myriah Fitzgibbon, marketing manager. They have all done an outstanding 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.

ANCILLARIES

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 oddnumbered 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.

xvii


xviii

ANCILLARIES

Solutions Builder
This is an electronic version of the complete solutions manual available via the
PowerLecture or Instructor’s Companion Website. It provides instructors with an
efficient method for creating solution sets to homework or exams that can then be
printed or posted.
Enhanced WebAssign®
Enhanced WebAssign is designed for students to do their homework online. This
proven and reliable system uses pedagogy and content found in Stewart, Redlin,
Watson, and Panman’s text and enhances it to help students learn college algebra
more effectively. Automatically graded homework allows students to focus on their
learning and get interactive study assistance outside of class.
Electronic Test Bank (0-495-38793-2)
April Strom—Scottsdale Community College
The Test Bank includes every problem that comes loaded in ExamView in easy-toedit Word® documents.

TO THE STUDENT

This textbook was written for you. With this book you will learn how you can use algebra in your daily life and in your other courses. Here are some suggestions to help
you get the most out of your course.
This book tells the story of how algebra explains many things in the real-world.
So make sure you start from the beginning, and don’t miss any of the topics that your
teacher assigns. You should read the appropriate section of the book before you attempt your homework exercises. You may find that you need to reread a passage several times before you understand it. Pay special attention to the examples, and work
them out yourself with pencil and paper as you read. Then do the linked exercises referred to in the “Now Try Exercise . . .” at the end of each example. You may want to
obtain the Study Guide that accompanies this book. This guide shows you how to
read and understand the examples and explains the purpose of each step. The guide
also contains worked-out solutions to many of the exercises that are linked to the
examples.
To learn anything well requires practice. In studying algebra, a little practice
goes a long way. This is because the concepts learned in one situation apply to many
others. Pay special attention to the Context exercises (word problems); these exercises explain why we study algebra in the first place.
Answers to odd-numbered exercises as well as to all Concepts exercises, Algebra Checkpoints, and Chapter Tests appear at the back of the book.
Have a great semester.
The authors

xix


ABBREVIATIONS

Cal
cm
dB
ft

g
gal
h
Hz
in.
kg
km
L
lb
lm
M
m
mg

xx

Calorie
centimeter
decibel
foot
gram
gallon
hour
Hertz
inch
kilogram
kilometer
liter
pound
lumen

mole of solute
per liter of solution
meter
milligram

MHz
MW
mi
min
mL
mm
N
qt
oz
s
⍀
V
W
yd
yr
°C
°F
K

megahertz
megawatt
mile
minute
milliliter
millimeter

Newton
quart
ounce
second
ohm
volt
watt
yard
year
degree Celsius
degree Fahrenheit
Kelvin


PROLOGUE

Algebra and Alcohol
Algebra helps us better understand many real-world situations. In this prologue we
preview how the topics we learn in this book can help us to analyze a major social
issue: the overconsumption of alcohol.
People have been drinking alcoholic beverages since prehistoric times to enliven
social occasions—but frequently also to ill effect. Overconsumption of alcohol is
widely perceived as a major social problem on college campuses. How can we predict the effects of different levels of drinking? How can guidelines for responsible
drinking be established? The answers to these questions involve a combination of
science, data collection, and algebra. Let’s examine the process.

Investigating the Science
Biomedical scientists study the chemical and physiological changes in the body that
result from alcohol consumption. They have found that the reaction in the human
body occurs in two stages: a fairly rapid process of absorption and a more gradual

one of metabolism.
The term absorption refers to the physical process by which alcohol passes from
the stomach to the small intestine and then into the bloodstream. After one standard
drink (defined as 12 ounces of beer, 5 ounces of wine, or 1.5 ounces of 80-proof distilled spirits, which contain equivalent amounts of alcohol), the blood alcohol concentration (BAC) peaks within 30 to 45 minutes. Several factors influence the rate of
absorption; the presence and type of food before drinking, medication, and the gender and ethnicity of the drinker all play a role.
The term metabolism refers to chemical processes in the body through which ingested substances are converted to other compounds. One of these processes is oxidation, in which alcohol is detoxified and removed from the blood (primarily in the
liver), preventing the alcohol from accumulating and destroying cells. Alcohol is oxidized to acetaldehyde by the enzyme ADH (alcohol dehydrogenase). Usually, acetaldehyde is itself metabolized quite rapidly and doesn’t accumulate. But when a
person drinks large amounts of alcohol, the accumulation of acetaldehyde can cause
headaches, nausea, and dizziness, contributing to a hangover. The rate of alcohol metabolism depends on the amounts of certain enzymes in the liver, and these amounts
vary from person to person.

Collecting the Data
To predict the effect of different amounts of alcohol consumption, we need to know
the rate at which alcohol is absorbed and metabolized. The starting point is to experiment and collect data. For example, in a medical study, researchers measured
the BAC of eight fasting adult male subjects after rapid consumption of different
amounts of alcohol. Table 1 on the next page shows the data they obtained after averaging the measurements from the eight subjects.
P1


P2

PROLOGUE

table 1
Mean fasting ethanol concentration (mg/mL) at indicated
sampling times following the oral administration of four
different doses of ethanol to eight adult male subjects*
Concentration (mg/mL) after 95%
ethanol oral dose of:
Time (h)

15 mL

30 mL

45 mL

60 mL

0.0
0.067
0.133
0.167
0.2
0.267
0.333
0.417
0.5
0.667
0.75
0.833
1.0
1.167
1.25
1.33
1.5
1.75
2.0
2.25
2.5

2.75
3.0
3.5
3.75
4.0
4.25
4.5
4.75
5.0
5.25
5.5
5.75
6.0
6.25
6.5
6.75
7.0

0.0
0.032
0.096
—
0.13
0.17
0.16
0.17
0.16
—
0.12
—

0.090
—
0.062
—
0.033
0.020
0.012
0.0074
0.0052
0.0034
0.0024
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—

0.0
0.071
0.019

—
0.25
0.30
0.31
—
0.41
—
0.40
—
0.33
—
0.29
—
0.24
0.22
0.18
0.15
0.12
—
0.069
0.034
0.017
0.010
0.0068
0.0052
0.0037
—
—
—
—

—
—
—
—
—

0.0
—
—
0.28
—
—
0.42
—
0.51
0.61
—
0.65
0.63
0.59
—
0.53
0.50
0.43
0.40
—
0.32
—
0.28
0.22

—
0.15
—
0.081
0.059
0.042
0.021
0.014
0.0099
0.0056
—
—
—
—

0.0
—
—
0.30
—
—
0.46
—
0.59
0.66
—
0.71
0.77
0.75
—

0.70
0.71
0.72
0.64
—
0.57
—
0.45
0.43
—
0.36
—
0.27
0.22
0.18
0.15
0.11
0.079
0.050
0.037
0.020
0.017
0.012

*P. Wilkinson, A. Sedman, E. Sakmar, D. Kay, and J. Wagner,
“Pharmacokinetics of Ethanol After Oral Administration in the
Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics,
5(3): 207–224, 1977.