Preface xi
Resources for Success xviii
1 Graphs, Functions, and Models 1
1.1 Introduction to Graphing 2
Graphs
Solutions of Equations
Graphs of Equations
The Distance Formula
Midpoints of Segments
Circles
Visualizing the Graph 13
1.2 Functions and Graphs 17
Functions
Notation for Functions
Graphs of Functions
Finding Domains of Functions
Visualizing Domain and Range
Applications of Functions
1.3 Linear Functions, Slope, and Applications 32
Linear Functions
The Linear Function f(x)=mx+bf(x)=mx+b and Slope
Applications of Slope
Slope–Intercept Equations of Lines
Graphing f(x)=mx+bf(x)=mx+b Using m and b
Applications of Linear Functions
Visualizing the Graph 43
Mid-Chapter Mixed Review 48
1.4 Equations of Lines and Modeling 50
Point–Slope Equations of Lines
Parallel Lines
Perpendicular Lines
Mathematical Models
Curve Fitting
1.5 Linear Equations, Functions, Zeros, and Applications 62
Linear Equations
Special Cases
Applications Using Linear Models
Zeros of Linear Functions
1.6 Solving Linear Inequalities 79
Linear Inequalities
Compound Inequalities
An Application
Study Guide 85
Review Exercises, 91
Chapter Test 95
2 More on Functions 97
2.1 Increasing, Decreasing, and Piecewise Functions; Applications 98
Increasing, Decreasing, and Constant Functions
Relative Maximum and Minimum Values
Functions Defined Piecewise
2.2 The Algebra of Functions 111
The Algebra of Functions: Sums, Differences, Products, and Quotients
Difference Quotients
2.3 The Composition of Functions 118
The Composition of Functions
Decomposing a Function as a Composition
Mid-Chapter Mixed Review 125
2.4 Symmetry 127
Symmetry
Even Functions and Odd Functions
2.5 Transformations 133
Transformations of Functions
Vertical Translations and Horizontal Translations
Reflections
Vertical and Horizontal Stretchings and Shrinkings
Visualizing the Graph 143
2.6 Variation and Applications 147
Direct Variation
Inverse Variation
Combined Variation
Study Guide 155
Review Exercises 162
Chapter Test 165
3 Quadratic Functions and Equations; Inequalities 167
3.1 The Complex Numbers 168
The Complex-Number System
Addition and Subtraction
Multiplication
Conjugates and Division
3.2 Quadratic Equations, Functions, Zeros, and Models 174
Quadratic Equations and Quadratic Functions
Completing the Square
Using the Quadratic Formula
The Discriminant
Equations Reducible to Quadratic
Applications
3.3 Analyzing Graphs of Quadratic Functions 189
Graphing Quadratic Functions of the Type f(x)=a(x−h)2+kf(x)=a(x−h)2+k
Graphing Quadratic Functions of the Type f(x)=ax2+bx+c, a≠0f(x)=ax2+bx+c, a≠0
Visualizing the Graph 198
Mid-Chapter Mixed Review 202
3.4 Solving Rational Equations and Radical Equations 203
Rational Equations
Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value 211
Equations with Absolute Value
Inequalities with Absolute Value
Study Guide 214
Review Exercises 220
Chapter Test 223
4 Polynomial Functions and Rational Functions 225
4.1 Polynomial Functions and Models 226
The Leading-Term Test
Finding Zeros of Polynomial Functions
Polynomial Models
4.2 Graphing Polynomial Functions 238
Graphing Polynomial Functions
The Intermediate Value Theorem
Visualizing the Graph 246
4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem 248
Division and Factors
The Remainder Theorem and Synthetic Division
Finding Factors of Polynomials
Mid-Chapter Mixed Review 256
4.4 Theorems about Zeros of Polynomial Functions 257
The Fundamental Theorem of Algebra
Finding Polynomials with Given Zeros
Zeros of Polynomial Functions with Real Coefficients
Rational Coefficients
Integer Coefficients and the Rational Zeros Theorem
Descartes’ Rule of Signs
4.5 Rational Functions 266
The Domain of a Rational Function
Asymptotes
Visualizing the Graph 280
4.6 Polynomial Inequalities and Rational Inequalities 284
Polynomial Inequalities
Rational Inequalities
Study Guide 295
Review Exercises 305
Chapter Test 309
5 Exponential Functions and Logarithmic Functions 311
5.1 Inverse Functions 312
Inverses
Inverses and One-to-One Functions
Finding Formulas for Inverses
Inverse Functions and Composition
Restricting a Domain
5.2 Exponential Functions and Graphs 323
Graphing Exponential Functions
The Number e
Graphs of Exponential Functions, Base e
5.3 Logarithmic Functions and Graphs 333
Logarithmic Functions
Finding Certain Logarithms
Converting Between Exponential Equations and Logarithmic Equations
Finding Logarithms on a Calculator
Natural Logarithms
Changing Logarithmic Bases
Graphs of Logarithmic Functions
Visualizing the Graph 344
Mid-Chapter Mixed Review 347
5.4 Properties of Logarithmic Functions 349
Logarithms of Products
Logarithms of Powers
Logarithms of Quotients
Applying the Properties
Simplifying Expressions of the Type loga axloga ax and aloga xaloga x
5.5 Solving Exponential Equations and Logarithmic Equations 356
Solving Exponential Equations
Solving Logarithmic Equations
5.6 Applications and Models: Growth and Decay; Compound Interest 367
Population Growth
Interest Compounded Continuously
Models of Limited Growth
Exponential Decay
Study Guide 381
Review Exercises 388
Chapter Test 391
6 The Trigonometric Functions 393
6.1 Trigonometric Functions of Acute Angles 394
The Trigonometric Ratios
The Six Functions Related
Function Values of 30°, 45°, and 60°
Function Values of Any Acute Angle
Cofunctions and Complements
6.2 Applications of Right Triangles 405
Solving Right Triangles
6.3 Trigonometric Functions of Any Angle 417
Angles, Rotations, and Degree Measure
Trigonometric Functions of Angles or Rotations
Terminal Side on an Axis
Reference Angles: 30°, 45°, and 60°
Function Values for Any Angle
Mid-Chapter Mixed Review 431
6.4 Radians, Arc Length, and Angular Speed 433
Distances on the Unit Circle
Radian Measure
Arc Length and Central Angles
Linear Speed and Angular Speed
6.5 Circular Functions: Graphs and Properties 448
Reflections on the Unit Circle
Finding Function Values
Graphs of the Sine and Cosine Functions
Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
6.6 Graphs of Transformed Sine and Cosine Functions 462
Variations of Basic Graphs
Graphs of Sums: Addition of Ordinates
Damped Oscillation: Multiplication of Ordinates
Visualizing the Graph 475
Study Guide 479
Review Exercises 488
Chapter Test 491
7 Trigonometric Identities, Inverse Functions, and Equations 493
7.1 Identities: Pythagorean and Sum and Difference 494
Pythagorean Identities
Simplifying Trigonometric Expressions
Sum and Difference Identities
7.2 Identities: Cofunction, Double-Angle, and Half-Angle 507
Cofunction Identities
Double-Angle Identities
Half-Angle Identities
7.3 Proving Trigonometric Identities 515
The Logic of Proving Identities
Proving Identities
Product-to-Sum and Sum-to-Product Identities
Mid-Chapter Mixed Review 521
7.4 Inverses of the Trigonometric Functions 522
Restricting Ranges to Define Inverse Functions
Composition of Trigonometric Functions and Their Inverses
7.5 Solving Trigonometric Equations 533
Visualizing the Graph 541
Study Guide 545
Review Exercises 551
Chapter Test 554
8 Applications of Trigonometry 555
8.1 The Law of Sines 556
Solving Oblique Triangles
The Law of Sines
Solving Triangles (AAS and ASA)
Solving Triangles (SSA)
The Area of a Triangle
8.2 The Law of Cosines 568
The Law of Cosines
Solving Triangles (SAS)
Solving Triangles (SSS)
8.3 Complex Numbers: Trigonometric Notation 578
Graphical Representation
Trigonometric Notation for Complex Numbers
Multiplication and Division with Trigonometric Notation
Powers of Complex Numbers
Roots of Complex Numbers
Mid-Chapter Mixed Review 588
8.4 Polar Coordinates and Graphs 589
Polar Coordinates
Polar Equations and Rectangular Equations
Graphing Polar Equations
Visualizing the Graph 596
8.5 Vectors and Applications 598
Vectors
Vector Addition
Components
8.6 Vector Operations 606
Position Vectors
Operations on Vectors
Unit Vectors
Direction Angles
Angle Between Vectors
Forces in Equilibrium
Study Guide 619
Review Exercises 630
Chapter Test 633
9 Systems of Equations and Matrices 635
9.1 Systems of Equations in Two Variables 636
Solving Systems of Equations Graphically
The Substitution Method
The Elimination Method
Visualizing the Graph 646
9.2 Systems of Equations in Three Variables 651
Solving Systems of Equations in Three Variables
Mathematical Models and Applications
9.3 Matrices and Systems of Equations 660
Matrices and Row-Equivalent Operations
Gaussian Elimination with Matrices
Gauss–Jordan Elimination
9.4 Matrix Operations 667
Matrix Addition and Subtraction
Scalar Multiplication
Products of Matrices
Matrix Equations
Mid-Chapter Mixed Review 677
9.5 Inverses of Matrices 678
The Identity Matrix
The Inverse of a Matrix
Solving Systems of Equations
9.6 Determinants and Cramer’s Rule 684
Determinants of Square Matrices
Evaluating Determinants Using Cofactors
Cramer’s Rule
9.7 Systems of Inequalities and Linear Programming 691
Graphs of Linear Inequalities
Systems of Linear Inequalities
Applications: Linear Programming
9.8 Partial Fractions 703
Partial Fraction Decompositions
Study Guide 708
Review Exercises 714
Chapter Test 717
10 Analytic Geometry Topics 719
10.1 The Parabola 720
Parabolas
Finding Standard Form by Completing the Square
10.2 The Circle and the Ellipse 728
Ellipses
10.3 The Hyperbola 738
Standard Equations of Hyperbolas
10.4 Nonlinear Systems of Equations and Inequalities 746
Nonlinear Systems of Equations
Modeling and Problem Solving
Nonlinear Systems of Inequalities
Visualizing the Graph 754
Mid-Chapter Mixed Review 758
10.5 Rotation of Axes 759
Rotation of Axes
10.6 Polar Equations of Conics 768
Polar Equations of Conics
Converting from Polar Equations to Rectangular Equations
Finding Polar Equations of Conics
10.7 Parametric Equations 774
Graphing Parametric Equations
Determining a Rectangular Equation for Given Parametric Equations
Determining Parametric Equations for a Given Rectangular Equation
Study Guide 780
Review Exercises 787
Chapter Test 789
11 Sequences, Series, and Combinatorics 791
11.1 Sequences and Series 792
Sequences
Finding the General Term
Sums and Series
Sigma Notation
Recursive Definitions
11.2 Arithmetic Sequences and Series 799
Arithmetic Sequences
Sum of the First n Terms of an Arithmetic Sequence
11.3 Geometric Sequences and Series 807
Geometric Sequences
Sum of the First n Terms of a Geometric Sequence
Infinite Geometric Series
Visualizing the Graph 813
11.4 Mathematical Induction 816
Proving Infinite Sequences of Statements
Mid-Chapter Mixed Review 820
11.5 Combinatorics: Permutations 822
Permutations
Factorial Notation
Permutations of n Objects Taken k at a Time
Permutations of Sets with Nondistinguishable Objects
11.6 Combinatorics: Combinations 830
Combinations
11.7 The Binomial Theorem 836
Binomial Expansion Using Pascal’s Triangle
Binomial Expansion Using Factorial Notation
Finding a Specific Term
Total Number of Subsets
11.8 Probability 843
Experimental Probability and Theoretical Probability
Computing Experimental Probabilities
Theoretical Probability
Study Guide 852
Review Exercises 856
Chapter Test 859
Just-In-Time Review 861
Answers A-1
Photo Credits A-73
Index of Applications I-1
Index I-5