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To the Student xix
Three Distinct Series xx
The Concepts through Functions Series xxi
Preface to the Instructor xxii
Resources for Success xxviii
Applications Index xxx
F Foundations: A Prelude to Functions 1
F.1 The Distance and Midpoint Formulas 2
Use the Distance Formula • Use the Midpoint Formula
F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 9
Graph Equations by Plotting Points • Find Intercepts from a Graph • Find
Intercepts from an Equation • Test an Equation for Symmetry • Know How
to Graph Key Equations
F.3 Lines 19
Calculate and Interpret the Slope of a Line • Graph Lines Given a Point
and the Slope • Find the Equation of a Vertical Line • Use the Point-Slope
Form of a Line; Identify Horizontal Lines • Write the Equation of a Line in
Slope-Intercept Form • Identify the Slope and y-Intercept of a Line from
Its Equation • Find the Equation of a Line Given Two Points • Graph Lines
Written in General Form Using Intercepts • Find Equations of Parallel Lines
• Find Equations of Perpendicular Lines
F.4 Circles 34
Write the Standard Form of the Equation of a Circle • Graph a Circle
• Work with the General Form of the Equation of a Circle
Chapter Project 42
1 Functions and Their Graphs 43
1.1 Functions 44
Determine Whether a Relation Represents a Function • Find the Value of a
Function • Find the Difference Quotient of a Function • Find the Domain of a
Function Defined by an Equation • Form the Sum, Difference, Product, and
Quotient of Two Functions
1.2 The Graph of a Function 59
Identify the Graph of a Function • Obtain Information from or about the
Graph of a Function
1.3 Properties of Functions 68
Determine Even and Odd Functions from a Graph • Determine Even and
Odd Functions from the Equation • Use a Graph to Determine Where a
Function is Increasing, Decreasing, or Constant • Use a Graph to Locate Local
Maxima and Local Minima • Use a Graph to Locate the Absolute Maximum
and the Absolute Minimum • Use a Graphing Utility to Approximate Local
Maxima and Local Minima and to Determine Where a Function is Increasing
or Decreasing • Find the Average Rate of Change of a Function
Contents
ixx CONTENTS
1.4 Library of Functions; Piecewise-defined Functions 82
Graph the Functions Listed in the Library of Functions • Graph
Piecewise-defined Functions
1.5 Graphing Techniques: Transformations 93
Graph Functions Using Vertical and Horizontal Shifts • Graph Functions
Using Compressions and Stretches • Graph Functions Using Reflections
about the x-Axis and the y-Axis
1.6 Mathematical Models: Building Functions 106
Build and Analyze Functions
1.7 Building Mathematical Models Using Variation 111
Construct a Model Using Direct Variation • Construct a Model Using
Inverse Variation • Construct a Model Using Joint or Combined Variation
Chapter Review 117
Chapter Test 121
Chapter Projects 122
2 Linear and Quadratic Functions 124
2.1 Properties of Linear Functions and Linear Models 125
Graph Linear Functions • Use Average Rate of Change to Identify Linear
Functions • Determine Whether a Linear Function Is Increasing, Decreasing,
or Constant • Find the Zero of a Linear Function • Build Linear Models
from Verbal Descriptions
2.2 Building Linear Models from Data 136
Draw and Interpret Scatter Diagrams • Distinguish between Linear and
Nonlinear Relations • Use a Graphing Utility to Find the Line of Best Fit
2.3 Quadratic Functions and Their Zeros 144
Find the Zeros of a Quadratic Function by Factoring • Find the Zeros of a
Quadratic Function Using the Square Root Method • Find the Zeros of a
Quadratic Function by Completing the Square • Find the Zeros of a Quadratic
Function Using the Quadratic Formula • Find the Point of Intersection of Two
Functions • Solve Equations That Are Quadratic in Form
2.4 Properties of Quadratic Functions 155
Graph a Quadratic Function Using Transformations • Identify the Vertex
and Axis of Symmetry of a Quadratic Function • Graph a Quadratic
Function Using Its Vertex, Axis, and Intercepts • Find a Quadratic
Function Given Its Vertex and One Other Point • Find the Maximum or
Minimum Value of a Quadratic Function
2.5 Inequalities Involving Quadratic Functions 167
Solve Inequalities Involving a Quadratic Function
2.6 Building Quadratic Models from Verbal
Descriptions and from Data 171
Build Quadratic Models from Verbal Descriptions • Build Quadratic Models
from Data
2.7 Complex Zeros of a Quadratic Function 181
Find the Complex Zeros of a Quadratic Function
2.8 Equations and Inequalities Involving the Absolute
Value Function 185
Solve Absolute Value Equations • Solve Absolute Value Inequalities
Chapter Review 190
Chapter Test 193CONTENTS xi
Cumulative Review 194
Chapter Projects 195
3 Polynomial and Rational Functions 197
3.1 Polynomial Functions and Models 198
Identify Polynomial Functions and Their Degree • Graph Polynomial
Functions Using Transformations • Identify the Real Zeros of a Polynomial
Function and Their Multiplicity • Analyze the Graph of a Polynomial
Function • Build Cubic Models from Data
3.2 The Real Zeros of a Polynomial Function 219
Use the Remainder and Factor Theorems • Use Descartes’ Rule of Signs to
Determine the Number of Positive and the Number of Negative Real Zeros
of a Polynomial Function • Use the Rational Zeros Theorem to List the
Potential Rational Zeros of a Polynomial Function • Find the Real Zeros of
a Polynomial Function • Solve Polynomial Equations • Use the Theorem for
Bounds on Zeros • Use the Intermediate Value Theorem
3.3 Complex Zeros; Fundamental Theorem of Algebra 234
Use the Conjugate Pairs Theorem • Find a Polynomial Function with
Specified Zeros • Find the Complex Zeros of a Polynomial Function
3.4 Properties of Rational Functions 240
Find the Domain of a Rational Function • Find the Vertical Asymptotes
of a Rational Function • Find the Horizontal or Oblique Asymptote of a
Rational Function
3.5 The Graph of a Rational Function 251
Analyze the Graph of a Rational Function • Solve Applied Problems
Involving Rational Functions
3.6 Polynomial and Rational Inequalities 266
Solve Polynomial Inequalities • Solve Rational Inequalities
Chapter Review 273
Chapter Test 277
Cumulative Review 277
Chapter Projects 278
4 Exponential and Logarithmic Functions 280
4.1 Composite Functions 281
Form a Composite Function • Find the Domain of a Composite Function
4.2 One-to-One Functions; Inverse Functions 289
Determine Whether a Function Is One-to-One • Determine the Inverse of a
Function Defined by a Map or a Set of Ordered Pairs • Obtain the Graph of
the Inverse Function from the Graph of the Function • Find the Inverse of a
Function Defined by an Equation
4.3 Exponential Functions 302
Evaluate Exponential Functions • Graph Exponential Functions • Define
the Number e • Solve Exponential Equations
4.4 Logarithmic Functions 318
Change Exponential Statements to Logarithmic Statements and Logarithmic
Statements to Exponential Statements • Evaluate Logarithmic Expressions
• Determine the Domain of a Logarithmic Function • Graph Logarithmic
Functions • Solve Logarithmic Equationsxii CONTENTS
4.5 Properties of Logarithms 332
Work with Properties of Logarithms • Write a Logarithmic Expression as
a Sum or Difference of Logarithms • Write a Logarithmic Expression as a
Single Logarithm • Evaluate a Logarithm Whose Base Is Neither 10 Nor e
• Graph a Logarithmic Function Whose Base Is Neither 10 Nor e
4.6 Logarithmic and Exponential Equations 341
Solve Logarithmic Equations • Solve Exponential Equations • Solve
Logarithmic and Exponential Equations Using a Graphing Utility
4.7 Financial Models 349
Determine the Future Value of a Lump Sum of Money • Calculate Effective Rates
of Return • Determine the Present Value of a Lump Sum of Money • Determine
the Rate of Interest or the Time Required to Double a Lump Sum of Money
4.8 Exponential Growth and Decay Models; Newton’s Law;
Logistic Growth and Decay Models 359
Find Equations of Populations That Obey the Law of Uninhibited Growth
• Find Equations of Populations That Obey the Law of Decay • Use
Newton’s Law of Cooling • Use Logistic Models
4.9 Building Exponential, Logarithmic, and Logistic
Models from Data 369
Build an Exponential Model from Data • Build a Logarithmic Model from
Data • Build a Logistic Model from Data
Chapter Review 378
Chapter Test 383
Cumulative Review 384
Chapter Projects 385
5 Trigonometric Functions 386
5.1 Angles and Their Measure 387
Convert between Decimal and Degree, Minute, Second Measures for Angles
• Find the Length of an Arc of a Circle • Convert from Degrees to Radians
and from Radians to Degrees • Find the Area of a Sector of a Circle • Find
the Linear Speed of an Object Traveling in Circular Motion
5.2 Right Triangle Trigonometry 401
Find the Values of Trigonometric Functions of Acute Angles • Use Fundamental
Identities • Find the Values of the Remaining Trigonometric Functions, Given
the Value of One of Them • Use the Complementary Angle Theorem
5.3 Computing the Values of Trigonometric Functions
of Acute Angles 413
Find the Exact Values of the Trigonometric Functions of p
4
= 45° • Find the
Exact Values of the Trigonometric Functions of p
6 = 30° and p3 = 60° • Use
a Calculator to Approximate the Values of the Trigonometric Functions of
Acute Angles • Model and Solve Applied Problems Involving Right Triangles
5.4 Trigonometric Functions of any Angle
Find the Exact Values of the Trigonometric Functions for Any Angle • Use
Coterminal Angles to Find the Exact Value of a Trigonometric Function •
Determine the Signs of the Trigonometric Functions of an Angle in a Given
Quadrant • Find the Reference Angle of an Angle • Use a Reference Angle to
Find the Exact Value of a Trigonometric Function • Find the Exact Values of
Trigonometric Functions of an Angle, Given Information about the Function
5.5 Unit Circle Approach; Properties of the Trigonometric Functions
Find the Exact Values of the Trigonometric Functions Using the Unit Circle •
Know the Domain and Range of the Trigonometric Functions • Use Periodic
Properties to Find the Exact Values of the Trigonometric Functions • Use EvenOdd Properties to Find the Exact Values of the Trigonometric Functions
424
435CONTENTS xiii
5.6 Graphs of the Sine and Cosine Functions 447
Graph Functions of the Form y = A sin (vx) Using Transformations • Graph
Functions of the Form y = A cos (vx) Using Transformations • Determine the
Amplitude and Period of Sinusoidal Functions • Graph Sinusoidal Functions
Using Key Points • Find an Equation for a Sinusoidal Graph
5.7 Graphs of the Tangent, Cotangent, Cosecant,
and Secant Functions 462
Graph Functions of the Form y = A tan(vx) + B and y = A cot (vx) + B
• Graph Functions of the Form y = A csc (vx) + B and y = A sec (vx) + B
5.8 Phase Shift; Sinusoidal Curve Fitting 470
Graph Sinusoidal Functions of the Form y = A sin (vx - f) + B
• Build Sinusoidal Models from Data
Chapter Review 482
Chapter Test 487
Cumulative Review 488
Chapter Projects 489
6 Analytic Trigonometry 491
6.1 The Inverse Sine, Cosine, and Tangent Functions 492
Find the Exact Value of an Inverse Sine Function • Find an Approximate
Value of an Inverse Sine Function • Use Properties of Inverse Functions
to Find Exact Values of Certain Composite Functions • Find the Inverse
Function of a Trigonometric Function • Solve Equations Involving Inverse
Trigonometric Functions
6.2 The Inverse Trigonometric Functions (Continued) 505
Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and
Tangent Functions • Define the Inverse Secant, Cosecant, and Cotangent
Functions • Use a Calculator to Evaluate sec-1 x, csc-1 x, and cot-1 x • Write
a Trigonometric Expression as an Algebraic Expression
6.3 Trigonometric Equations 511
Solve Equations Involving a Single Trigonometric Function • Solve
Trigonometric Equations Using a Calculator • Solve Trigonometric Equations
Quadratic in Form • Solve Trigonometric Equations Using Fundamental
Identities • Solve Trigonometric Equations Using a Graphing Utility
6.4 Trigonometric Identities 521
Use Algebra to Simplify Trigonometric Expressions • Establish Identities
6.5 Sum and Difference Formulas 529
Use Sum and Difference Formulas to Find Exact Values • Use Sum and
Difference Formulas to Establish Identities • Use Sum and Difference
Formulas Involving Inverse Trigonometric Functions • Solve Trigonometric
Equations Linear in Sine and Cosine
6.6 Double-angle and Half-angle Formulas 542
Use Double-angle Formulas to Find Exact Values • Use Double-angle Formulas
to Establish Identities • Use Half-angle Formulas to Find Exact Values
6.7 Product-to-Sum and Sum-to-Product Formulas 552
Express Products as Sums • Express Sums as Products
Chapter Review 556
Chapter Test 560
Cumulative Review 560
Chapter Projects 561xiv CONTENTS
7 Applications of Trigonometric Functions 562
7.1 Applications Involving Right Triangles 563
Solve Right Triangles • Solve Applied Problems
7.2 The Law of Sines 569
Solve SAA or ASA Triangles • Solve SSA Triangles • Solve Applied
Problems
7.3 The Law of Cosines 579
Solve SAS Triangles • Solve SSS Triangles • Solve Applied Problems
7.4 Area of a Triangle 586
Find the Area of SAS Triangles • Find the Area of SSS Triangles
7.5 Simple Harmonic Motion; Damped Motion; Combining Waves 593
Build a Model for an Object in Simple Harmonic Motion • Analyze Simple
Harmonic Motion • Analyze an Object in Damped Motion • Graph the Sum
of Two Functions
Chapter Review 602
Chapter Test 604
Cumulative Review 605
Chapter Projects 605
8 Polar Coordinates; Vectors 607
8.1 Polar Coordinates 608
Plot Points Using Polar Coordinates • Convert from Polar Coordinates to
Rectangular Coordinates • Convert from Rectangular Coordinates to Polar
Coordinates • Transform Equations between Polar and Rectangular Forms
8.2 Polar Equations and Graphs 617
Identify and Graph Polar Equations by Converting to Rectangular
Equations • Test Polar Equations for Symmetry • Graph Polar Equations by
Plotting Points
8.3 The Complex Plane; De Moivre’s Theorem 632
Plot Points in the Complex Plane • Convert a Complex Number between
Rectangular Form and Polar Form • Find Products and Quotients of
Complex Numbers in Polar Form • Use De Moivre’s Theorem • Find
Complex Roots
8.4 Vectors 640
Graph Vectors • Find a Position Vector • Add and Subtract Vectors
Algebraically • Find a Scalar Multiple and the Magnitude of a Vector • Find
a Unit Vector • Find a Vector from Its Direction and Magnitude • Model
with Vectors
8.5 The Dot Product 654
Find the Dot Product of Two Vectors • Find the Angle between Two Vectors
• Determine Whether Two Vectors Are Parallel • Determine Whether Two
Vectors Are Orthogonal • Decompose a Vector into Two Orthogonal Vectors
• Compute Work
8.6 Vectors in Space 661
Find the Distance between Two Points in Space • Find Position Vectors in
Space • Perform Operations on Vectors • Find the Dot Product • Find the
Angle between Two Vectors • Find the Direction Angles of a VectorCONTENTS xv
8.7 The Cross Product 671
Find the Cross Product of Two Vectors • Know Algebraic Properties of the
Cross Product • Know Geometric Properties of the Cross Product • Find a
Vector Orthogonal to Two Given Vectors • Find the Area of a Parallelogram
Chapter Review 677
Chapter Test 680
Cumulative Review 681
Chapter Projects 681
9 Analytic Geometry 682
9.1 Conics 683
Know the Names of the Conics
9.2 The Parabola 684
Analyze Parabolas with Vertex at the Origin • Analyze Parabolas with
Vertex at (h, k) • Solve Applied Problems Involving Parabolas
9.3 The Ellipse 693
Analyze Ellipses with Center at the Origin • Analyze Ellipses with Center at
(h, k) • Solve Applied Problems Involving Ellipses
9.4 The Hyperbola 704
Analyze Hyperbolas with Center at the Origin • Find the Asymptotes of
a Hyperbola • Analyze Hyperbolas with Center at (h, k) • Solve Applied
Problems Involving Hyperbolas
9.5 Rotation of Axes; General Form of a Conic 716
Identify a Conic • Use a Rotation of Axes to Transform Equations • Analyze an
Equation Using Rotation of Axes • Identify Conics without a Rotation of Axes
9.6 Polar Equations of Conics 724
Analyze and Graph Polar Equations of Conics • Convert the Polar Equation
of a Conic to a Rectangular Equation
9.7 Plane Curves and Parametric Equations 730
Graph Parametric Equations • Find a Rectangular Equation for a Curve
Defined Parametrically • Use Time as a Parameter in Parametric Equations
• Find Parametric Equations for Curves Defined by Rectangular Equations
Chapter Review 743
Chapter Test 745
Cumulative Review 746
Chapter Projects 747
10 Systems of Equations and Inequalities 748
10.1 Systems of Linear Equations: Substitution and Elimination 749
Solve Systems of Equations by Substitution • Solve Systems of Equations
by Elimination • Identify Inconsistent Systems of Equations Containing
Two Variables • Express the Solution of a System of Dependent Equations
Containing Two Variables • Solve Systems of Three Equations Containing
Three Variables • Identify Inconsistent Systems of Equations Containing
Three Variables • Express the Solution of a System of Dependent Equations
Containing Three Variables
10.2 Systems of Linear Equations: Matrices 764
Write the Augmented Matrix of a System of Linear Equations • Write the
System of Equations from the Augmented Matrix • Perform Row Operations on
a Matrix • Solve a System of Linear Equations Using Matricesxvi CONTENTS
10.3 Systems of Linear Equations: Determinants 779
Evaluate 2 by 2 Determinants • Use Cramer’s Rule to Solve a System of Two
Equations Containing Two Variables • Evaluate 3 by 3 Determinants
• Use Cramer’s Rule to Solve a System of Three Equations Containing Three
Variables • Know Properties of Determinants
10.4 Matrix Algebra 789
Find the Sum and Difference of Two Matrices • Find Scalar Multiples of a
Matrix • Find the Product of Two Matrices • Find the Inverse of a Matrix
Solve a System of Linear Equations Using an Inverse Matrix
10.5 Partial Fraction Decomposition 806
Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors
• Decompose P/Q, Where Q Has Repeated Linear Factors • Decompose
P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor •
Decompose P/Q, Where Q Has a Repeated Irreducible Quadratic Factor
10.6 Systems of Nonlinear Equations 814
Solve a System of Nonlinear Equations Using Substitution • Solve a System
of Nonlinear Equations Using Elimination
10.7 Systems of Inequalities 823
Graph an Inequality • Graph a System of Inequalities
10.8 Linear Programming 830
Set up a Linear Programming Problem • Solve a Linear Programming
Problem
Chapter Review 837
Chapter Test 841
Cumulative Review 842
Chapter Projects 842
11 Sequences; Induction; the Binomial Theorem 844
11.1 Sequences 845
Write the First Several Terms of a Sequence • Write the Terms of a Sequence
Defined by a Recursive Formula • Use Summation Notation • Find the Sum
of a Sequence
11.2 Arithmetic Sequences 855
Determine Whether a Sequence Is Arithmetic • Find a Formula for an
Arithmetic Sequence • Find the Sum of an Arithmetic Sequence
11.3 Geometric Sequences; Geometric Series 861
Determine Whether a Sequence Is Geometric • Find a Formula for a
Geometric Sequence • Find the Sum of a Geometric Sequence • Determine
Whether a Geometric Series Converges or Diverges • Solve Annuity
Problems
11.4 Mathematical Induction 872
Prove Statements Using Mathematical Induction
11.5 The Binomial Theorem 876
Evaluate a nj b • Use the Binomial Theorem
Chapter Review 883
Chapter Test 885
Cumulative Review 885
Chapter Projects 886CONTENTS xvii
12 Counting and Probability 887
12.1 Counting 888
Find All the Subsets of a Set • Count the Number of Elements in a Set
• Solve Counting Problems Using the Multiplication Principle
12.2 Permutations and Combinations 893
Solve Counting Problems Using Permutations Involving n Distinct Objects
• Solve Counting Problems Using Combinations • Solve Counting Problems
Using Permutations Involving n Nondistinct Objects
12.3 Probability 902
Construct Probability Models • Compute Probabilities of Equally Likely
Outcomes • Find Probabilities of the Union of Two Events • Use the
Complement Rule to Find Probabilities
Chapter Review 912
Chapter Test 914
Cumulative Review 914
Chapter Projects 915
13 A Preview of Calculus: The Limit, Derivative,
and Integral of a Function 916
13.1 Finding Limits Using Tables and Graphs 917
Find a Limit Using a Table • Find a Limit Using a Graph
13.2 Algebra Techniques for Finding Limits 922
Find the Limit of a Sum, a Difference, and a Product • Find the Limit of
a Polynomial • Find the Limit of a Power or a Root • Find the Limit of a
Quotient • Find the Limit of an Average Rate of Change
13.3 One-sided Limits; Continuous Functions 929
Find the One-sided Limits of a Function • Determine Whether
a Function Is Continuous
13.4 The Tangent Problem; The Derivative 936
Find an Equation of the Tangent Line to the Graph of a Function • Find the
Derivative of a Function • Find Instantaneous Rates of Change • Find the
Instantaneous Velocity of a Particle
13.5 The Area Problem; The Integral 944
Approximate the Area Under the Graph of a Function • Approximate
Integrals Using a Graphing Utility
Chapter Review 951
Chapter Test 953
Chapter Projects 954
A Review A1
A.1 Algebra Essentials A1
Work with Sets • Graph Inequalities • Find Distance on the Real Number
Line • Evaluate Algebraic Expressions • Determine the Domain of a
Variable • Use the Laws of Exponents • Evaluate Square Roots • Use a
Calculator to Evaluate Exponents
A.2 Geometry Essentials A14
Use the Pythagorean Theorem and Its Converse • Know Geometry
Formulas • Understand Congruent Triangles and Similar Trianglesxviii CONTENTS
A.3 Polynomials A23
Recognize Monomials • Recognize Polynomials • Add and Subtract
Polynomials • Multiply Polynomials • Know Formulas for Special
Products • Divide Polynomials Using Long Division • Work with
Polynomials in Two Variables
A.4 Factoring Polynomials A33
Factor the Difference of Two Squares and the Sum and Difference of Two
Cubes • Factor Perfect Squares • Factor a Second-Degree Polynomial:
x2 + Bx + C • Factor by Grouping • Factor a Second-Degree Polynomial:
Ax2 + Bx + C, A ? 1 • Complete the Square
A.5 Synthetic Division A43
Divide Polynomials Using Synthetic Division
A.6 Rational Expressions A47
Reduce a Rational Expression to Lowest Terms • Multiply and Divide
Rational Expressions • Add and Subtract Rational Expressions • Use the
Least Common Multiple Method • Simplify Complex Rational Expressions
A.7 nth Roots; Rational Exponents A57
Work with nth Roots • Simplify Radicals • Rationalize Denominators
• Simplify Expressions with Rational Exponents
A.8 Solving Equations A66
Solve Linear Equations • Solve Rational Equations • Solve Equations by
Factoring • Solve Radical Equations
A.9 Problem Solving: Interest, Mixture, Uniform Motion,
Constant Rate Job Applications A75
Translate Verbal Descriptions into Mathematical Expressions • Solve
Interest Problems • Solve Mixture Problems • Solve Uniform Motion
Problems • Solve Constant Rate Job Problems
A.10 Interval Notation; Solving Inequalities A84
Use Interval Notation • Use Properties of Inequalities • Solve Inequalities
• Solve Combined Inequalities
A.11 Complex Numbers A92
Add, Subtract, Multiply, and Divide Complex Numbers
B Graphing Utilities B1
B.1 The Viewing Rectangle B1
B.2 Using a Graphing Utility to Graph Equations B3
B.3 Using a Graphing Utility to Locate Intercepts and Check
for Symmetry B5
B.4 Using a Graphing Utility to Solve Equations B6
B.5 Square Screens B8
B.6 Using a Graphing Utility to Graph Inequalities B9
B.7 Using a Graphing Utility to Solve Systems of Linear Equations B9
B.8 Using a Graphing Utility to Graph a Polar Equation B11
B.9 Using a Graphing Utility to Graph Parametric Equations B11
Answers AN1
Subject Index I1
Photo Credits C1