# Algebra and Trigonometry, Eleventh Edition PDF by Michael Sullivan

## Algebra and Trigonometry, Eleventh Edition

Michael Sullivan

Contents

Three Distinct Series xix

The Flagship Series xx

Preface to the Instructor xxi

Get the Most Out of MyLab Math xxvi

Resources for Success xxvii

Applications Index xxix

R Review 1

R.1 Real Numbers 2

Work with Sets • Classify Numbers • Evaluate Numerical Expressions

Work with Properties of Real Numbers

R.2 Algebra Essentials 17

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 • Use Scientific Notation

R.3 Geometry Essentials 30

Use the Pythagorean Theorem and Its Converse • Know Geometry

Formulas

Understand Congruent Triangles and Similar Triangles

R.4 Polynomials 39

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

R.5 Factoring Polynomials 49

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

R.6 Synthetic Division 57

Divide Polynomials Using Synthetic Division

R.7 Rational Expressions 61

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

R.8 nth Roots; Rational Exponents 72

Work with nth Roots • Simplify Radicals • Rationalize Denominators

and Numerators • Simplify Expressions with Rational Exponents

1 Equations and Inequalities 81

1.1 Linear Equations 82

Solve a Linear Equation • Solve Equations That Lead to Linear Equations

Solve Problems That Can Be Modeled by Linear Equations

Solve a Quadratic Equation by Factoring • Solve a Quadratic Equation Using

the Square Root Method • Solve a Quadratic Equation by Completing

Solve Problems That Can Be Modeled by Quadratic Equations

1.3 Complex Numbers; Quadratic Equations in the Complex

Number System 104

Equations in the Complex Number System

Factorable Equations 113

Equations

by Factoring

1.5 Solving Inequalities 119

Use Interval Notation • Use Properties of Inequalities • Solve Inequalities

Solve Combined Inequalities

1.6 Equations and Inequalities Involving Absolute Value 130

Solve Equations Involving Absolute Value • Solve Inequalities Involving

Absolute Value

1.7 Problem Solving: Interest, Mixture, Uniform

Motion, Constant Rate Job Applications 134

Translate Verbal Descriptions into Mathematical Expressions • Solve

Interest

Problems • Solve Mixture Problems • Solve Uniform Motion

Problems

Solve Constant Rate Job Problems

Chapter Review 144

Chapter Test 147

Chapter Projects 147

x Contents

2 Graphs 149

2.1 The Distance and Midpoint Formulas 150

Use the Distance Formula • Use the Midpoint Formula

2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 158

Graph Equations by Plotting Points • Find Intercepts from a Graph

Find Intercepts from an Equation • Test an Equation for Symmetry with

Respect to the x-Axis, the y-Axis, and the Origin • Know How to Graph Key

Equations

2.3 Lines 169

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 • Use the Slope-Intercept Form of

a Line • Find an 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

2.4 Circles 185

Write the Standard Form of the Equation of a Circle • Graph a Circle

Work with the General Form of the Equation of a Circle

2.5 Variation 191

Construct a Model Using Direct Variation • Construct a Model Using

Inverse

Variation • Construct a Model Using Joint Variation

Chapter Review 197

Chapter Test 200

Cumulative Review 200

Chapter Project 201

3 Functions and Their Graphs 202

3.1 Functions 203

Describe a Relation • Determine Whether a Relation Represents a Function

Use Function Notation; 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

3.2 The Graph of a Function 219

Identify the Graph of a Function • Obtain Information from or about the

Graph of a Function

3.3 Properties of Functions 229

Identify Even and Odd Functions from a Graph • Identify Even and

Odd Functions from an 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

3.4 Library of Functions; Piecewise-defined Functions 242

Graph the Functions Listed in the Library of Functions • Analyze a

Piecewise-

defined Function

3.5 Graphing Techniques: Transformations 254

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

3.6 Mathematical Models: Building Functions 267

Build and Analyze Functions

Chapter Review 273

Chapter Test 277

Cumulative Review 278

Chapter Projects 278

4 Linear and Quadratic Functions 280

4.1 Properties of Linear Functions and Linear Models 281

Graph Linear Functions • Use Average Rate of Change to Identify Linear

Functions • Determine Whether a Linear Function Is Increasing, Decreasing,

or Constant • Build Linear Models from Verbal Descriptions

4.2 Building Linear Models from Data 291

Draw and Interpret Scatter Plots. • Distinguish between Linear and

Nonlinear

Relations • Use a Graphing Utility to Find the Line of Best Fit

4.3 Quadratic Functions and Their Properties 299

Graph a Quadratic Function Using Transformations • Identify the Vertex

and Axis of Symmetry of a Parabola • 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

4.4 Building Quadratic Models from Verbal Descriptions

and from Data 312

from Data

4.5 Inequalities Involving Quadratic Functions 321

Solve Inequalities Involving a Quadratic Function

Chapter Review 325

Chapter Test 327

Cumulative Review 328

Chapter Projects 329

5 Polynomial and Rational Functions 330

5.1 Polynomial Functions 331

Identify Polynomial Functions and Their Degree • Graph Polynomial

Functions

Using Transformations • Identify the Real Zeros of a Polynomial

Function and Their Multiplicity

5.2 Graphing Polynomial Functions; Models 346

Graph a Polynomial Function • Graph a Polynomial Function Using a

Graphing Utility • Build Cubic Models from Data

5.3 Properties of Rational Functions 354

Find the Domain of a Rational Function • Find the Vertical Asymptotes

of a Rational Function • Find a Horizontal or an Oblique Asymptote of a

Rational Function

5.4 The Graph of a Rational Function 365

Graph a Rational Function • Solve Applied Problems Involving

Rational Functions

5.5 Polynomial and Rational Inequalities 380

Solve Polynomial Inequalities • Solve Rational Inequalities

5.6 The Real Zeros of a Polynomial Function 387

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 of Theorem

5.7 Complex Zeros; Fundamental Theorem of Algebra 401

Use the Conjugate Pairs Theorem • Find a Polynomial Function with

Specified

Zeros • Find the Complex Zeros of a Polynomial Function

Chapter Review 408

Chapter Test 411

Cumulative Review 412

Chapter Projects 413

6 Exponential and Logarithmic Functions 414

6.1 Composite Functions 415

Form a Composite Function • Find the Domain of a Composite Function

6.2 One-to-One Functions; Inverse Functions 423

Determine Whether a Function Is One-to-One • Obtain the Graph of the

Inverse

Function from the Graph of a One-to-One Function • Verify an

Inverse Function • Find the Inverse of a Function Defined by an Equation

6.3 Exponential Functions 435

Evaluate Exponential Functions • Graph Exponential Functions

Define the Number e • Solve Exponential Equations

6.4 Logarithmic Functions 452

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 Equations

6.5 Properties of Logarithms 465

Work with the Properties of Logarithms • Write a Logarithmic Expression

as a Sum or Difference of Logarithms • Write a Logarithmic Expression as a

Single Logarithm • Evaluate Logarithms Whose Base Is Neither 10 Nor e

6.6 Logarithmic and Exponential Equations 474

Solve Logarithmic Equations • Solve Exponential Equations • Solve

Logarithmic

and Exponential Equations Using a Graphing Utility

6.7 Financial Models 481

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

6.8 Exponential Growth and Decay Models; Newton’s Law;

Logistic Growth and Decay Models 491

Model Populations That Obey the Law of Uninhibited Growth

Model Populations That Obey the Law of Uninhibited Decay

Use Newton’s Law of Cooling • Use Logistic Models

6.9 Building Exponential, Logarithmic, and Logistic

Models from Data 502

Build an Exponential Model from Data • Build a Logarithmic Model from

Data • Build a Logistic Model from Data

Chapter Review 509

Chapter Test 514

Cumulative Review 515

Chapter Projects 516

7 Trigonometric Functions 517

7.1 Angles, Arc Length, and Circular Motion 518

Angles and Degree Measure • 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

7.2 Right Triangle Trigonometry 531

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

7.3 Computing the Values of Trigonometric Functions

of Acute Angles 543

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

p

3 = 60_

Use a Calculator to Approximate the Values of the Trigonometric

Functions

of Acute Angles • Model and Solve Applied Problems Involving

Right Triangles

7.4 Trigonometric Functions of Any Angle 555

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 the Trigonometric Functions of an Angle, Given Information about the

Functions

7.5 Unit Circle Approach; Properties of the Trigonometric

Functions 566

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 Even-Odd Properties to Find the Exact Values of the Trigonometric

Functions

7.6 Graphs of the Sine and Cosine Functions 579

Graph the Sine Function y = sin x and Functions of the Form y = A sin1vx2

Graph the Cosine Function y = cos x and Functions

of the Form

y = A cos1vx2 • Determine the Amplitude and Period of Sinusoidal

Functions

Graph Sinusoidal Functions Using Key Points • Find an

Equation

for a Sinusoidal Graph

7.7 Graphs of the Tangent, Cotangent, Cosecant,

and Secant Functions 594

Graph the Tangent Function y = tan x and the Cotangent Function y = cot x

Graph Functions of the Form y = A tan1vx2 + B and y = A cot1vx2 + B

Graph the Cosecant Function y = csc x and the Secant Function y = sec x

Graph Functions of the Form y = A csc1vx2 + B and y = A sec1vx2 + B

7.8 Phase Shift; Sinusoidal Curve Fitting 601

Graph Sinusoidal Functions of the Form y = A sin1vx – f2 + B

Build Sinusoidal Models from Data

Chapter Review 613

Chapter Test 618

Cumulative Review 619

Chapter Projects 620

8 Analytic Trigonometry 621

8.1 The Inverse Sine, Cosine, and Tangent Functions 622

Define the Inverse Sine Function • Find the Value of an Inverse Sine

Function

Define the Inverse Cosine Function • Find the Value of an

Inverse Cosine Function • Define the Inverse Tangent Function • Find the

Value of an Inverse Tangent 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

8.2 The Inverse Trigonometric Functions (Continued) 635

Define the Inverse Secant, Cosecant, and Cotangent Functions • Find the

Value of Inverse Secant, Cosecant, and Cotangent Functions • Find the

Exact Value of Composite Functions Involving the Inverse Trigonometric

Functions • Write a Trigonometric Expression as an Algebraic Expression

8.3 Trigonometric Equations 641

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

8.4 Trigonometric Identities 651

Use Algebra to Simplify Trigonometric Expressions • Establish Identities

8.5 Sum and Difference Formulas 659

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

8.6 Double-angle and Half-angle Formulas 672

Use Double-angle Formulas to Find Exact Values • Use Double-angle Formulas

to Establish Identities • Use Half-angle Formulas to Find Exact Values

8.7 Product-to-Sum and Sum-to-Product Formulas 683

Express Products as Sums • Express Sums as Products

Chapter Review 687

Chapter Test 690

Cumulative Review 691

Chapter Projects 692

9 Applications of Trigonometric Functions 693

9.1 Applications Involving Right Triangles 694

Solve Right Triangles • Solve Applied Problems

9.2 The Law of Sines 700

Solve SAA or ASA Triangles • Solve SSA Triangles • Solve Applied Problems

9.3 The Law of Cosines 711

Solve SAS Triangles • Solve SSS Triangles • Solve Applied Problems

9.4 Area of a Triangle 718

Find the Area of SAS Triangles • Find the Area of SSS Triangles

9.5 Simple Harmonic Motion; Damped Motion; Combining Waves 724

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 734

Chapter Test 736

Cumulative Review 737

Chapter Projects 737

10 Polar Coordinates; Vectors 739

10.1 Polar Coordinates 740

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

10.2 Polar Equations and Graphs 749

Identify and Graph Polar Equations by Converting to Rectangular Equations

Test Polar Equations for Symmetry • Graph Polar Equations by Plotting Points

10.3 The Complex Plane; De Moivre’s Theorem 764

Plot Points in the Complex Plane • Convert a Complex Number between

Rectangular Form and Polar Form or Exponential Form • Find Products

and Quotients of Complex Numbers • Use De Moivre’s Theorem • Find

Complex Roots

10.4 Vectors 773

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

10.5 The Dot Product 788

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

Chapter Review 795

Chapter Test 798

Cumulative Review 799

Chapter Projects 799

11 Analytic Geometry 800

11.1 Conics 801

Know the Names of the Conics

11.2 The Parabola 802

Analyze Parabolas with Vertex at the Origin • Analyze Parabolas with

Vertex

at 1h, k2 • Solve Applied Problems Involving Parabolas

11.3 The Ellipse 811

Analyze Ellipses with Center at the Origin • Analyze Ellipses with

Center at 1h, k2 • Solve Applied Problems Involving Ellipses

11.4 The Hyperbola 821

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

11.5 Rotation of Axes; General Form of a Conic 834

Identify a Conic • Use a Rotation of Axes to Transform Equations • Analyze

an Equation Using a Rotation of Axes • Identify Conics without Rotating

the Axes

11.6 Polar Equations of Conics 842

Analyze and Graph Polar Equations of Conics • Convert the Polar Equation

of a Conic to a Rectangular Equation

11.7 Plane Curves and Parametric Equations 849

Graph Parametric Equations • Find a Rectangular Equation for a Plane

Curve Defined Parametrically • Use Time as a Parameter in Parametric

Equations • Find Parametric Equations for Plane Curves Defined by

Rectangular

Equations

Chapter Review 862

Chapter Test 864

Cumulative Review 865

Chapter Projects 865

12 Systems of Equations and Inequalities 867

12.1 Systems of Linear Equations: Substitution and Elimination 868

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

12.2 Systems of Linear Equations: Matrices 882

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 Matrices

12.3 Systems of Linear Equations: Determinants 896

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

12.4 Matrix Algebra 907

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

12.5 Partial Fraction Decomposition 924

Where Q Has Only Nonrepeated Linear Factors

Where Q Has Repeated Linear Factors • Decompose

Where Q Has a Nonrepeated Irreducible Quadratic Factor • Decompose

Where Q Has a Repeated Irreducible Quadratic Factor

12.6 Systems of Nonlinear Equations 933

Solve a System of Nonlinear Equations Using Substitution • Solve a

System of Nonlinear Equations Using Elimination

12.7 Systems of Inequalities 942

Graph an Inequality • Graph a System of Inequalities

12.8 Linear Programming 949

Set Up a Linear Programming Problem • Solve a Linear Programming

Problem

Chapter Review 957

Chapter Test 960

Cumulative Review 961

Chapter Projects 962

13 Sequences; Induction; the Binomial Theorem 963

13.1 Sequences 964

List the First Several Terms of a Sequence • List the Terms of a Sequence

Defined by a Recursive Formula • Use Summation Notation • Find the Sum

of a Sequence

13.2 Arithmetic Sequences 974

Determine Whether a Sequence Is Arithmetic • Find a Formula for an

Arithmetic

Sequence • Find the Sum of an Arithmetic Sequence

13.3 Geometric Sequences; Geometric Series 981

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

13.4 Mathematical Induction 993

Prove Statements Using Mathematical Induction

13.5 The Binomial Theorem 997

b • Use the Binomial Theorem

Chapter Review 1003

Chapter Test 1006

Cumulative Review 1006

Chapter Projects 1007

14 Counting and Probability 1008

14.1 Counting 1009

Find All the Subsets of a Set • Count the Number of Elements in a Set

Solve Counting Problems Using the Multiplication Principle

14.2 Permutations and Combinations 1014

Solve Counting Problems Using Permutations Involving n Distinct Objects

Solve Counting Problems Using Combinations • Solve Counting Problems

Using Permutations Involving n Nondistinct Objects

14.3 Probability 1023

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 1033

Chapter Test 1035

Cumulative Review 1036

Chapter Projects 1036

Appendix Graphing Utilities A1

A.1 The Viewing Rectangle A1

A.2 Using a Graphing Utility to Graph Equations A3

A.3 Using a Graphing Utility to Locate Intercepts and

Check for Symmetry A5

A.4 Using a Graphing Utility to Solve Equations A6

A.5 Square Screens A8

A.6 Using a Graphing Utility to Graph Inequalities A9

A.7 Using a Graphing Utility to Solve Systems of Linear Equations A9

A.8 Using a Graphing Utility to Graph a Polar Equation A11

A.9 Using a Graphing Utility to Graph Parametric Equations A11