College Algebra: Concepts and Contexts

821 Pages·2010·12.39 MB·English

by  James Stewart

■ Linear Functions
A linear functionis a function of the form f x = b + mx.
1 2
■ The graph of fis a line with slope mand y-intercept b.
y y
b
b
x x
Ï=b Ï=b+mx
■ Exponential Functions
An exponential functionis a function of the form f x = Cax.
1 2
■ The graph of fhas one of the shapes shown.
■ Ifa 7 1,thenaiscalledthegrowthfactorandr = a - 1iscalledthegrowthrate.
■ If a 6 1, then ais called the decay factorand r = a - 1is called the decay rate.
y y
1 1
0 x 0 x
Ï=a˛, a>1 Ï=a˛, 0<a<1
■ Logarithmic Functions
A logarithmic functionwith base a 7 1is a function of the form f x = log x. By definition,
1 2 a
log x = y if and only if ay = x
a
■ The graph of fhas the general shape shown below.
■ Basic properties: loga1 = 0, logaa = 1, logaax = x, alogax = x

y
0 1 x
Ï=log x, a>1
a
■ Quadratic Functions
A quadratic functionis a function of the form f x = ax2 + bx + c.
1 2
■ The graph of fhas the shape of a parabola.
b
■ The maximum or minimum value of foccurs at x = - .
2a
■ The function fcan be expressed in the standard formf 1x2 = a1x - h22 + k.
■ The vertexof the graph of fis at the point (h, k).
y y
(h, k)
0 x
(h, k)
0 x
y=a(x-h)™+k y=a(x-h)™+k
a<0, h>0, k>0 a>0, h>0, k>0
■ Power Functions
A power functionis a function of the form f x = Cxp.
1 2
■ Graphs of some power functions are shown.
Positive powers
y y y y
x x x x
Ï=≈ Ï=x£ Ï=x¢ Ï=x∞
Fractional powers
y y y y
x x x x
Ï=œx∑ Ï=œ£x∑ Ï=œ¢x∑ Ï=œ∞x∑
Negative powers
y y y y
x x x x
1 1 1 1
Ï= Ï= Ï= Ï=
x ≈ x£ x¢
College Algebra
CONCEPTS AND CONTEXTS
ABOUT THE AUTHORS
JAMESSTEWARTreceived 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.
LOTHARREDLINgrew 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.
SALEEMWATSONreceived 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.
PHYLLISPANMANreceived 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
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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
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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
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1 2 3 4 5 6 7 13 12 11 10 09
CONTENTS
PROLOGUE: Algebra and Alcohol P1
1 Data, Functions, and Models 1
chapter
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
CHAPTER1 Review 113
CHAPTER 1Test 126
■ EXPLORATIONS
1 Bias in Presenting Data 128
2 Collecting and Analyzing Data 134
3 Every Graph Tells a Story 138
2 Linear Functions and Models 141
chapter
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
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
CHAPTER 2 Review 219
CHAPTER 2Test 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
3 Exponential Functions and Models 247
chapter
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 aor 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 3Test 311
CONTENTS vii
■ 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
4 Logarithmic Functions and
chapter
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 4Test 400
■ EXPLORATIONS
1 Super Origami 401
2 Orders of Magnitude 402
3 Semi-Log Graphs 406
4 The Even-Tempered Clavier 409
5 Quadratic Functions and Models 413
chapter
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