Stewart J. College Algebra: Concepts and Contexts

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Ï=b

Ï=b+mx

■

Linear Functions

A linear function is a function of the form .

■

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

f 1x2= b + mx

■

Exponential Functions

An exponential function is a function of the form .

■

The graph of f has one of the shapes shown.

■

If , then a is called the growth factor and is called the growth rate.

■

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

r = a - 1a 7 1

f 1x2= Ca

y y

Ï=a˛, 0<a<1Ï=a˛, a>1

■

Logarithmic Functions

A logarithmic function with base is a function of the form . By definition,

if and only if

■

The graph of f has the general shape shown below.

■

Basic properties: ,, ,a

log

= xlog

a = 1log

1 = 0

= xlog

x = y

f 1x2= log

xa 7 1

Ï=log

x, a>1

■

Quadratic Functions

A quadratic function is a function of the form .

■

The graph of f has the shape of a parabola.

■

The maximum or minimum value of f occurs at .

■

The function f can be expressed in the standard form .

■

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

f 1x2= a1x - h2

+ k

x =-

f 1x2= ax

+ bx + c

y y

(h, k)

y=a(x-h)™+k

a<0, h>0, k>0

y=a(x-h)™+k

a>0, h>0, k>0

(h, k)

■

Power Functions

A power function is a function of the form .

■

Graphs of some power functions are shown.

f 1x2= Cx

Fractional powers

y y y y

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∑

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Positive powers

Negative powers

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College Algebra

CONCEPTS AND CONTEXTS

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 AUTHORS

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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Library of Congress Control Number: 2009934974

ISBN-13: 978-0-495-38789-3

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1 2 3 4 5 6 7 13 12 11 10 09

PROLOGUE: Algebra and Alcohol P1

chapter 1 Data, Functions, and Models 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

CHAPTER 1 Review 113

CHAPTER 1 Test 126

■ EXPLORATIONS

1 Bias in Presenting Data 128

2 Collecting and Analyzing Data 134

3 Every Graph Tells a Story 138

chapter 2 Linear Functions and Models 141

2.1 Working with Functions: Average Rate of Change 142

Average Rate of Change of a Function • Average Speed of a Moving Object • Functions

Defined by Algebraic Expressions

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

CHAPTER 2 Review 219

CHAPTER 2 Test 228

■ EXPLORATIONS

1 When Rates of Change Change 229

2 Linear Patterns 233

3 Bridge Science 237

4 Correlation and Causation 239

5 Fair Division of Assets 242

chapter 3 Exponential Functions and Models 247

3.1 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

vi CONTENTS

■ EXPLORATIONS

1 Extreme Numbers: Scientific Notation 312

2 So You Want to Be a Millionaire? 315

3 Exponential Patterns 316

4 Modeling Radioactivity with Coins and Dice 320

chapter 4 Logarithmic Functions and

Exponential Models 323

4.1 Logarithmic Functions 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

CHAPTER 4 Review 393

CHAPTER 4 Test 400

■ EXPLORATIONS

1 Super Origami 401

2 Orders of Magnitude 402

3 Semi-Log Graphs 406

4 The Even-Tempered Clavier 409

chapter 5 Quadratic Functions and Models 413

5.1 Working with Functions: Shifting and Stretching 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

CONTENTS

vii