FORMULAS/EQUATIONS Distance Formula If ( ) = P x y , 1 1 1 and ( ) = P x y , , 2 2 2 the distance from P1 to P2 is ( ) ( ) ( ) = − + − d P P x x y y , 1 2 2 1 2 2 1 2 Standard Equation of a Circle The standard equation of a circle of radius r with center at ( ) h k , is ( ) ( ) − + − = x h y k r 2 2 2 Slope Formula The slope m of the line containing the points ( ) = P x y , 1 1 1 and ( ) = P x y , 2 2 2 is = − − ≠ = m y y x x x x m x x if is undefined if 2 1 2 1 1 2 1 2 Point–Slope Equation of a Line The equation of a line with slope m containing the point ( ) x y , 1 1 is ( ) − = − y y m x x 1 1 Slope–Intercept Equation of a Line The equation of a line with slope m and y-intercept b is = + y mx b Quadratic Formula The solutions of the equation + + = ≠ ax bx c a 0, 0, 2 are = − ± − x b b ac a 4 2 2 If − > b ac 4 0, 2 there are two unequal real solutions. If − = b ac 4 0, 2 there is a repeated real solution. If − < b ac 4 0, 2 there are two complex solutions that are not real. GEOMETRY FORMULAS Circle r = = = r A C Radius, Area, Circumference π π = = A r C r 2 2 Triangle h b = = b h Base, Altitude (Height), = A area = A bh 1 2 Rectangle l w = = = = l w A P Length, Width, area, perimeter = = + A lw P l w 2 2 Rectangular Box (closed) h w l = = = = = l w h V S Length, Width, Height, Volume, Surface area = = + + VlwhS lw lh wh 2 2 2 Sphere r = = = r V S Radius, Volume, Surface area π π = = V r S r 4 4 3 3 2 Right Circular Cylinder (closed) h r = = = = r h V S Radius, Height, Volume, Surface area π π π = = + V r h S r rh 2 2 2 2

Need to Review? Model It! Prepare for Class: “Read the Book” Feature Description Benefit Page(s) Every Chapter begins with . . . Chapter-Opening Topic & Project Each chapter begins with a discussion of a topic of current interest and ends with a related project. In the concluding project, you will apply what you have learned to solve a problem related to the topic. 272, 381 Internet-Based Projects These projects allow for the integration of spreadsheet technology that you will need to be a productive member of the workforce. The projects give you an opportunity to collaborate and use mathematics to deal with issues of current interest. 381 Every Section begins with . . . LEARNING OBJECTIVES Each section begins with a list of objectives. Individual objectives also appear in the text where they are covered. These objectives focus your studying by emphasizing what’s most important and where to find it. 294 Sections contain . . . PREPARING FOR THIS SECTION Most sections begin with a list of key concepts to review, with page numbers. Ever forget what you’ve learned?This feature highlights previously learned material to be used in this section. Review it, and you’ll always be prepared to move forward. 294 Now Work the ‘Are You Prepared?’ Problems These problems assess whether you have the prerequisite knowledge for the upcoming section. Work the ‘AreYou Prepared?’ problems. If you get one wrong, you’ll know exactly what you need to review and where to review it! 294, 306 Now Work PROBLEMS These follow most examples and direct you to a related exercise. We learn best by doing.You’ll solidify your understanding of examples if you try a similar problem right away, to be sure you understand what you’ve just read. 302, 308 CAUTION Words of caution are provided in the text. These point out common mistakes and help you avoid them. 330 Explorations and Seeing the Concept These activities foreshadow a concept or reinforce a concept just presented. You will obtain a deeper and more intuitive understanding of theorems and definitions. 289, 301 In Words This feature provides alternative descriptions of select definitions and theorems. This feature translates math into plain English. 313 Calculus This symbol appears next to information essential for the study of calculus. Foreshadowing calculus now will make the material easier later. 68, 277, 302 These examples provide “how to” instruction by offering a guided, step-bystep approach to solving a problem. With each step presented on the left and the mathematics displayed on the right, you can immediately see how each step is employed. 206–207 Examples and Problems These examples and problems require you to build a mathematical model from either a verbal description or data. The homework Model It! problems are marked by purple problem numbers. It is rare for a problem to come in the form “ Solve the following equation.” Rather, the equation must be developed based on an explanation of the problem.These problems require you to develop models that will enable you to describe the problem mathematically and suggest a solution to the problem. 320, 352 These margin notes provide a just-intime reminder of a concept needed now, but covered in an earlier section of the book. Each note is back-referenced to the chapter, section and page where the concept was originally discussed. Sometimes as you read, you encounter a word or concept you know you’ve seen before, but don't remember exactly what it means.This feature will point you to where you first learned the word or concept. A quick review now will help you see the connection to what you are learning for the first time and make remembering easier the next time. 301 SHOWCASE EXAMPLES

Practice: “Work the Problems” Feature Description Benefit Page(s) ‘Are You Prepared?’ Problems These problems assess your retention of the prerequisite material. Answers are given at the end of the section exercises. This feature is related to the Preparing for This Section feature. Do you always remember what you’ve learned? Working these problems is the best way to find out. If you get one wrong, you’ll know exactly what you need to review and where to review it! 294, 306 Concepts and Vocabulary These short-answer questions, mainly fill-in-the-blank, multiple-choice, and true/ false items, assess your understanding of key definitions and concepts in the current section. It is difficult to learn math without knowing the language of mathematics.These problems test your understanding of the formulas and vocabulary. 306–307 Skill Building Correlated with section examples, these problems provide straightforward practice. These problems give you ample opportunity to dig in and develop your skills. 307–309 Mixed Practice These problems offer comprehensive assessment of the skills learned in the section by asking problems related to more than one concept or objective.These problems may also require you to utilize skills learned in previous sections. Learning mathematics is a building process. Many concepts build on each other and are related.These problems help you see how mathematics builds on itself and how the concepts are linked together. 309 Applications and Extensions These problems allow you to apply your skills to real-world problems.They also enable you to extend concepts learned in the section. You will see that the material learned within the section has many uses in everyday life. 309–312 Challenge Problems These problems have been added in most sections and appear at the end of the Application and Extensions exercises.They are intended to be thought-provoking, requiring some ingenuity to solve. Challenge problems can be used for group work or to challenge your students. Solutions to Challenge Problems are in the Annotated Instructor’s Edition or in the Instructor’s Solutions Manual (online). 312 Explaining Concepts These problems, colored red, can be used to support class discussion, verbalization of mathematical ideas, and writing and research projects. To verbalize an idea, or to describe it clearly in writing, shows real understanding.These problems nurture that understanding. Many are challenging, but you’ll get out what you put in. 312 Retain Your Knowledge These problems allow you to practice content learned earlier in the course. Remembering how to solve all the different kinds of problems that you encounter throughout the course is difficult.This practice helps you remember previously learned skills. 312 Now Work PROBLEMS Many examples refer you to a related homework problem.These related problems are marked by and orange problem numbers. If you get stuck while working problems, look for the closest Now Work problem, and refer to the related example to see if it helps. 302, 304, 305 Interactive Figure Exercises Exercises that require you manipulate an interactive figure to solve.These exercises are labeled with the icon . These exercises help you visualize important concepts and develop a “feel” for them. The figures are housed at bit.ly/2MibgaO and were developed in GeoGebra by author Michael Sullivan III. 306, 307, 321, 322 Review Exercises Every chapter concludes with a comprehensive list of exercises to practice. Use the list of objectives to determine what objective and examples correspond to each problem. Work these problems to ensure that you understand all the skills and concepts employed in the chapter.Think of it as a comprehensive review of the chapter. All answers to Chapter Review problems appear in the back of the text. 376–379

Review: “Study for Quizzes and Tests” Feature Description Benefit Page(s) Most Sections Contain . . . . Retain Your Knowledge Keeps what you have learned at the forefront and see how topics are connected. These problems allow content to remain fresh so you are more prepared for the final exam. 326 The Chapter Review at the end of each chapter contains . . . Things to Know A detailed list of important theorems, formulas, and definitions from the chapter. Review these and you’ll know the most important material in the chapter! 374–375 You Should Be Able to . . . A complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective. Do the recommended exercises and you’ll have mastered the key material. If you get something wrong, go back and review the example listed, and try again. 375–376 Review Exercises These provide comprehensive review and practice of key skills, matched to the Learning Objectives for each section. Practice makes perfect.These problems combine exercises from all sections, giving you a comprehensive review in one place. 376–379 Chapter Test About 15–20 problems that can be taken as a ChapterTest. Be sure to take the ChapterTest under test conditions—no notes! Be prepared.Take the sample practice test under test conditions.This will get you ready for your instructor’s test. If you get a problem wrong, you can watch the Chapter Test Prep Video. 379–380 Cumulative Review These problem sets appear at the end of each chapter, beginning with Chapter 2.They combine problems from previous chapters, providing an ongoing cumulative review. When you use them in conjunction with the RetainYour Knowledge problems, you will be ready for the final exam. These problem sets are really important. Completing them will ensure that you are not forgetting anything as you go.This will go a long way toward keeping you primed for the final exam. 380–381

Precalculus Enhanced with Graphing Utilities Michael Sullivan Chicago State University Michael Sullivan III Joliet Junior College Florida SouthWestern State College Ninth Edition

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For the Next Generation Shannon, Patrick, and Ryan (Murphy) Maeve, Sean, and Nolan (Sullivan) Michael S., Kevin, and Marissa (Sullivan) Kaleigh, Billy, and Timmy (O’Hara)

ix Contents Three Distinct Series xix The Enhanced with Graphing Utilities Series xx Preface to the Instructor xxi To the Student xxx Applications Index xxxi 2 Functions and Their Graphs 60 2.1 Functions 61 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 2.2 The Graph of a Function 77 Identify the Graph of a Function • Obtain Information from or about the Graph of a Function 1 Graphs 1 1.1 Graphing Utilities; Introduction to Graphing Equations 2 Graph Equations by Plotting Points • Graph Equations Using a Graphing Utility • Use a Graphing Utility to Create Tables • Find Intercepts from a Graph • Use a Graphing Utility to Approximate Intercepts 1.2 The Distance and Midpoint Formulas 13 Use the Distance Formula • Use the Midpoint Formula 1.3 Intercepts; Symmetry; Graphing Key Equations 20 Find Intercepts Algebraically 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 1.4 Solving Equations Using a Graphing Utility 28 Solve Equations Using a Graphing Utility 1.5 Lines 32 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 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 1.6 Circles 48 Write the Standard Form of the Equation of a Circle • Graph a Circle by Hand and by Using a Graphing Utility • Work with the General Form of the Equation of a Circle Chapter Review 56 Chapter Test 58 Chapter Project 59

x Contents 2.3 Properties of Functions 87 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 2.4 Library of Functions; Piecewise-defined Functions 100 Graph the Functions Listed in the Library of Functions • Analyze a Piecewise-defined Function 2.5 Graphing Techniques: Transformations 112 Graph Functions Using Vertical and Horizontal Shifts • Graph Functions Using Compressions and Stretches • Graph Functions Using Reflections about the x -Axis or y -Axis 2.6 Mathematical Models: Building Functions 126 Build and Analyze Functions Chapter Review 131 Chapter Test 135 Cumulative Review 136 Chapter Project 136 3 Linear and Quadratic Functions 138 3.1 Properties of Linear Functions and Linear Models 139 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 3.2 Building Linear Models from Data 149 Draw and Interpret Scatter Plots • Distinguish between Linear and Nonlinear Relations • Use a Graphing Utility to Find the Line of Best Fit 3.3 Quadratic Functions and Their Properties 157 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 Quadratic Function 3.4 Building Quadratic Models from Verbal Descriptions and from Data 171 Build Quadratic Models from Verbal Descriptions • Build Quadratic Models from Data 3.5 Inequalities Involving Quadratic Functions 179 Solve Inequalities Involving a Quadratic Function Chapter Review 184 Chapter Test 186 Cumulative Review 187 Chapter Project 188 4 Polynomial and Rational Functions 189 4.1 Polynomial Functions 190 Identify Polynomial Functions and Their Degree • Graph Polynomial Functions Using Transformations • Identify the Real Zeros of a Polynomial Function and Their Multiplicity

Contents xi 5 Exponential and Logarithmic Functions 272 5.1 Composite Functions 273 Form a Composite Function • Find the Domain of a Composite Function 5.2 One-to-One Functions; Inverse Functions 281 Determine Whether a Function Is One-to-One • Determine the Inverse of a Function Defined by a Mapping or a Set of Ordered Pairs • Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function • Verify that a Function Defined by an Equation Is an Inverse Function • Find the Inverse of a Function Defined by an Equation 5.3 Exponential Functions 294 Evaluate Exponential Functions • Graph Exponential Functions • Define the Number e • Solve Exponential Equations 5.4 Logarithmic Functions 313 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 5.5 Properties of Logarithms 327 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 • Graph a Logarithmic Function Whose Base Is Neither 10 Nor e 4.2 The Graph of a Polynomial Function; Models 205 Analyze the Graph of a Polynomial Function • Build Cubic Models from Data 4.3 The Real Zeros of a Polynomial Function 214 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 4.4 Complex Zeros; Fundamental Theorem of Algebra 229 Use the Conjugate Pairs Theorem • Find a Polynomial Function with Specified Zeros • Find the Complex Zeros of a Polynomial Function 4.5 Properties of Rational Functions 236 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 4.6 The Graph of a Rational Function 247 Analyze the Graph of a Rational Function • Solve Applied Problems Involving Rational Functions 4.7 Polynomial and Rational Inequalities 259 Solve Polynomial Inequalities Graphically and Algebraically • Solve Rational Inequalities Graphically and Algebraically Chapter Review 266 Chapter Test 269 Cumulative Review 269 Chapter Project 271

xii Contents 5.6 Logarithmic and Exponential Equations 336 Solve Logarithmic Equations• Solve Exponential Equations • Solve Logarithmic and Exponential Equations Using a Graphing Utility 5.7 Financial Models 345 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 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 355 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 5.9 Building Exponential, Logarithmic, and Logistic Models from Data 366 Build an Exponential Model from Data • Build a Logarithmic Model from Data • Build a Logistic Model from Data Chapter Review 374 Chapter Test 379 Cumulative Review 380 Chapter Project 381 6 Trigonometric Functions 382 6.1 Angles, Arc Length, and Circular Motion 383 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 6.2 Trigonometric Functions: Unit Circle Approach 397 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle • Find the Exact Values of the Trigonometric Functions of Quadrantal Angles • Find the Exact Values of the Trigonometric Functions of π = ° 4 45 • Find the Exact Values of the Trigonometric Functions of π = ° 6 30 and π = ° 3 60 • Find the Exact Values of the Trigonometric Functions for Integer Multiples of π = ° 6 30 , π = ° 4 45 , and π = ° 3 60 • Use a Calculator to Approximate the Value of a Trigonometric Function • Use a Circle of Radius r to Evaluate the Trigonometric Functions 6.3 Properties of the Trigonometric Functions 412 Determine the Domain and the Range of the Trigonometric Functions • Determine the Period of the Trigonometric Functions • Determine the Signs of the Trigonometric Functions in a Given Quadrant • Find the Values of the Trigonometric Functions Using Fundamental Identities • Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle • Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions 6.4 Graphs of the Sine and Cosine Functions 427 Graph the Sine Function = y x sin and Functions of the Form ω( ) = y A x sin • Graph the Cosine Function = y x cos and Functions of the Form ω( ) = y A x cos • Determine the Amplitude and Period of Sinusoidal Functions • Graph Sinusoidal Functions Using Key Points • Find an Equation for a Sinusoidal Graph

Contents xiii 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 443 Graph the Tangent Function = y x tan and the Cotangent Function = y x cot • Graph Functions of the Form ω( ) = + y A x B tan and ω( ) = + y A x B cot • Graph the Cosecant Function = y x csc and the Secant Function = y x sec • Graph Functions of the Form ω( ) = + y A x B csc and ω( ) = + y A x B sec 6.6 Phase Shift; Sinusoidal Curve Fitting 451 Graph Sinusoidal Functions of the Form ω φ ( ) = − + y A x B sin • Build Sinusoidal Models from Data Chapter Review 462 Chapter Test 468 Cumulative Review 469 Chapter Project 470 7 Analytic Trigonometry 471 7.1 The Inverse Sine, Cosine, and Tangent Functions 472 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 7.2 The Inverse Trigonometric Functions (Continued) 487 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 7.3 Trigonometric Equations 493 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 7.4 Trigonometric Identities 503 Use Algebra to Simplify Trigonometric Expressions • Establish Identities 7.5 Sum and Difference Formulas 511 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 7.6 Double-angle and Half-angle Formulas 524 Use Double-angle Formulas to Find Exact Values • Use Double-angle Formulas to Establish Identities • Use Half-angle Formulas to Find Exact Values 7.7 Product-to-Sum and Sum-to-Product Formulas 535 Express Products as Sums • Express Sums as Products Chapter Review 539 Chapter Test 542 Cumulative Review 543 Chapter Project 544

xiv Contents 9 Polar Coordinates; Vectors 599 9.1 Polar Coordinates 600 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 9.2 Polar Equations and Graphs 610 Identify and Graph Polar Equations by Converting to Rectangular Equations • Graph Polar Equations Using a Graphing Utility • Test Polar Equations for Symmetry • Graph Polar Equations by Plotting Points 9.3 The Complex Plane; De Moivre’s Theorem 627 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 9.4 Vectors 637 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 9.5 The Dot Product 651 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 9.6 Vectors in Space 659 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 Vector 8 Applications of Trigonometric Functions 545 8.1 Right Triangle Trigonometry; Applications 546 Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles • Use the Complementary Angle Theorem • Solve Right Triangles • Solve Applied Problems 8.2 The Law of Sines 559 Solve SAA or ASA Triangles • Solve SSA Triangles • Solve Applied Problems 8.3 The Law of Cosines 570 Solve SAS Triangles • Solve SSS Triangles • Solve Applied Problems 8.4 Area of a Triangle 577 Find the Area of SAS Triangles • Find the Area of SSS Triangles 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves 583 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 593 Chapter Test 596 Cumulative Review 597 Chapter Project 597

Contents xv 11 Systems of Equations and Inequalities 754 11.1 Systems of Linear Equations: Substitution and Elimination 755 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 Analytic Geometry 680 10.1 Conics 681 Know the Names of the Conics 10.2 The Parabola 682 Analyze Parabolas with Vertex at the Origin • Analyze Parabolas with Vertex at ( ) h k , • Solve Applied Problems Involving Parabolas 10.3 The Ellipse 692 Analyze Ellipses with Center at the Origin • Analyze Ellipses with Center at ( ) h k , • Solve Applied Problems Involving Ellipses 10.4 The Hyperbola 705 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 10.5 Rotation of Axes; General Form of a Conic 720 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 10.6 Polar Equations of Conics 728 Analyze and Graph Polar Equations of Conics • Convert the Polar Equation of a Conic to a Rectangular Equation 10.7 Plane Curves and Parametric Equations 735 Graph Parametric Equations by Hand • Graph Parametric Equations Using a Graphing Utility • 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 749 Chapter Test 752 Cumulative Review 752 Chapter Project 753 9.7 The Cross Product 669 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 675 Chapter Test 678 Cumulative Review 679 Chapter Project 679

xvi Contents 11.2 Systems of Linear Equations: Matrices 769 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 11.3 Systems of Linear Equations: Determinants 784 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 11.4 Matrix Algebra 795 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 11.5 Partial Fraction Decomposition 812 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 11.6 Systems of Nonlinear Equations 821 Solve a System of Nonlinear Equations Using Substitution • Solve a System of Nonlinear Equations Using Elimination 11.7 Systems of Inequalities 831 Graph an Inequality by Hand • Graph an Inequality Using a Graphing Utility • Graph a System of Inequalities 11.8 Linear Programming 840 Set Up a Linear Programming Problem • Solve a Linear Programming Problem Chapter Review 847 Chapter Test 851 Cumulative Review 852 Chapter Project 853 12 Sequences; Induction; the Binomial Theorem 854 12.1 Sequences 855 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 Algebraically and Using a Graphing Utility • Solve Annuity and Amortization Problems Using Recursive Formulas 12.2 Arithmetic Sequences 869 Determine Whether a Sequence Is Arithmetic • Find a Formula for an Arithmetic Sequence • Find the Sum of an Arithmetic Sequence 12.3 Geometric Sequences; Geometric Series 875 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 Using Formulas 12.4 The Limit of a Sequence; Infinite Series 887 Find the Limit of a Sequence • Define Infinite Series and Geometric Series 12.5 Mathematical Induction 894 Prove Statements Using Mathematical Induction

Contents xvii 13 Counting and Probability 910 13.1 Counting 911 Find All the Subsets of a Set • Count the Number of Elements in a Set • Solve Counting Problems Using the Multiplication Principle 13.2 Permutations and Combinations 916 Solve Counting Problems Using Permutations Involving n Distinct Objects • Solve Counting Problems Using Combinations • Solve Counting Problems Using Permutations Involving n Nondistinct Objects 13.3 Probability 925 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 935 Chapter Test 937 Cumulative Review 938 Chapter Project 938 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 940 14.1 Investigating Limits Using Tables and Graphs 941 Investigate a Limit Using a Table • Investigate a Limit Using a Graph 14.2 Algebraic Techniques for Finding Limits 946 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 14.3 One-sided Limits; Continuity 953 Find the One-sided Limits of a Function • Determine Whether a Function Is Continuous at a Number 14.4 The Tangent Problem; The Derivative 960 • 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 an Object 14.5 The Area Problem; The Integral 968 Approximate the Area under the Graph of a Function • Approximate Integrals Using a Graphing Utility Chapter Review 974 Chapter Test 977 Chapter Project 978 12.6 The Binomial Theorem 898 Evaluate ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j • Use the Binomial Theorem Chapter Review 905 Chapter Test 907 Cumulative Review 908 Chapter Project 909

xviii Contents Appendix 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 Triangles A.3 Polynomials A22 Recognize Monomials • Recognize Polynomials • Know Formulas for Special Products • Divide Polynomials Using Long Division • Factor Polynomials • Complete the Square A.4 Synthetic Division A31 Divide Polynomials Using Synthetic Division A.5 Rational Expressions A35 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.6 Solving Equations A44 Solve Linear Equations • Solve Rational Equations • Solve Quadratic Equations by Factoring • Solve Quadratic Equations Using the Square Root Method • Solve Quadratic Equations by Completing the Square • Solve Quadratic Equations Using the Quadratic Formula • Solve Equations Quadratic in Form • Solve Absolute Value Equations • Solve Equations by Factoring A.7 Complex Numbers; Quadratic Equations in the Complex Number System A58 Add, Subtract, Multiply, and Divide Complex Numbers • Solve Quadratic Equations in the Complex Number System A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications A66 Translate Verbal Descriptions into Mathematical Expressions • Solve Interest Problems • Solve Mixture Problems • Solve Uniform Motion Problems • Solve Constant Rate Job Problems A.9 Interval Notation; Solving Inequalities A76 Use Interval Notation • Use Properties of Inequalities • Solve Inequalities • Solve Combined Inequalities • Solve Inequalities Involving Absolute Value A.10 n th Roots; Rational Exponents A87 Work with n th Roots • Simplify Radicals • Rationalize Denominators and Numerators • Solve Radical Equations • Simplify Expressions with Rational Exponents Answers AN1 Subject Index I1

xix Students have different goals, learning styles, and levels of preparation. Instructors have different teaching philosophies, styles, and techniques. Rather than write one series to fit all, the Sullivans have written three distinct series. All share the same goal—to develop a high level of mathematical understanding and an appreciation for the way mathematics can describe the world around us.The manner of reaching that goal, however, differs from series to series. Enhanced with Graphing Utilities Series, Ninth Edition This series provides a thorough integration of graphing utilities into topics, allowing students to explore mathematical concepts and encounter ideas usually studied in later courses. Many examples show solutions using algebra side-by-side with graphing techniques. Using technology, the approach to solving certain problems differs from the Contemporary (Flagship) or Concepts through Functions Series, while the emphasis on understanding concepts and building strong skills is maintained. Texts in this series are College Algebra, Algebra & Trigonometry, and Precalculus. Flagship Series, Twelfth Edition The Flagship Series is the most traditional in approach, yet modern in its treatment of precalculus mathematics. In each text, needed review material is included and is referenced when it is used. Graphing utility coverage is optional and can be included or excluded at the discretion of the instructor.Texts in this series are College Algebra, Algebra & Trigonometry, Trigonometry: A Unit Circle Approach, and Precalculus. Concepts through Functions Series, Fifth Edition This series differs from the others, utilizing a functions approach that serves as the organizing principle tying concepts together. Functions are introduced early in various formats. This approach supports the Rule of Four, which states that functions are represented symbolically, numerically, graphically, and verbally. Each chapter introduces a new type of function and then develops all concepts pertaining to that particular function. The solutions of equations and inequalities, instead of being developed as stand-alone topics, are developed in the context of the underlying functions. Graphing utility coverage is optional and can be included or excluded at the discretion of the instructor. Texts in this series are College Algebra; Precalculus, with a Unit Circle Approach to Trigonometry; Precalculus, with a Right Triangle Approach to Trigonometry. Three Distinct Series to Meet Varied Instructional Needs

The Enhanced with Graphing Utilities Series College Algebra, Ninth Edition This text provides an approach to college algebra that completely integrates graphing technology without sacrificing mathematical analysis and conceptualization. The text has three chapters of review material preceding the chapter on functions. Graphing calculator usage is integrated throughout. After completing this text, a student will be prepared for trigonometry, finite mathematics, and business calculus. Algebra & Trigonometry, Ninth Edition This text contains all the material in College Algebra, but it also develops the trigonometric functions using a right triangle approach and shows how that approach is related to the unit circle approach. Graphing techniques are emphasized, including a thorough discussion of polar coordinates, parametric equations, and conics using polar coordinates. Vectors in the plane, including the dot product, sequences, induction, and the binomial theorem are also presented. After completing this text, a student will be prepared for finite mathematics, business calculus, and engineering calculus. Precalculus, Ninth Edition This text contains a review chapter before covering the traditional precalculus topics of functions and their graphs, polynomial and rational functions, and exponential and logarithmic functions. The trigonometric functions are introduced using a unit circle approach and show how it is related to the right triangle approach. Graphing techniques are emphasized, including a thorough discussion of polar coordinates, parametric equations, and conics using polar coordinates. Vectors in the plane and in space, including the dot and cross products, sequences, induction, and the binomial theorem are also presented. Graphing calculator usage is integrated throughout. The final chapter provides an introduction to calculus, with a discussion of the limit, the derivative, and the integral of a function. After completing this text, a student will be prepared for finite mathematics, business calculus, and engineering calculus. xx

xxi As professors at an urban university (Michael Sullivan) and a community college (Michael Sullivan III), we are aware of the varied needs of students in this course. Such students range from those who have little mathematical background and are fearful of mathematics courses to those with a strong mathematical education and a high level of motivation. For some of your students, this will be their last course in mathematics, whereas others will further their mathematical education. We have written this text with both groups in mind. As a teacher, and as an author of precalculus, engineering calculus, finite mathematics, and business calculus texts, Michael Sullivan understands what students must know if they are to be focused and successful in upper-level math courses. As an instructor and an author of a developmental mathematics series, Michael’s son and co-author, Michael Sullivan III, understands the trepidations and skills that students bring to the Precalculus course. As the father of recent college graduates, Michael III realizes that today’scollege students demand a variety of media to support their education. This text addresses that demand by providing technology and video support that enhances understanding without sacrificing math skills.Together, we have taken great pains to ensure that the text offers solid, student-friendly examples and problems, as well as a clear and seamless writing style. A tremendous benefit of authoring a successful series is the broad-based feedback we receive from teachers and students. We are sincerely grateful for their support. Virtually every change in this edition is the result of their thoughtful comments and suggestions.We are confident that, building on the success of the first eight editions and incorporating many of these suggestions, we have made Precalculus Enhanced with Graphing Utilities, 9th Edition, an even better tool for learning and teaching.We continue to encourage you to share with us your experiences teaching from this text. Features in the Ninth Edition A descriptive list of the many special features of Precalculus can be found on the endpapers in the front of this text. This list places the features in their proper context, as building blocks of an overall learning system that has been carefully crafted over the years to help students get the most out of the time they put into studying. Please take the time to review it and to discuss it with your students at the beginning of your course. Our experience has been that when students use these features, they are more successful in the course. • Highlighting Contemporary Mathematicians NEW! It is important for students to see themselves represented in the world around them. Who is a mathematician? What are they working on and what was their journey? A short bio is presented, near the beginning of each chapter, to highlight a selection of diverse contemporary mathematicians and scientists who have made advancements in various fields. Information is provided to encourage students to learn more about each mathematician’s work and be inspired to pursue their own mathematical and scientific journeys. • MyLab Exercises UPDATED Michael Sullivan III reviewed all MyLab exercises for this revision and edited them to better match the language and approach of the text and to make the solutions (within View an Example and Help Me Solve This) more student-friendly. • Instructional Videos NEW! Every instructional video is new, created exclusively for this product! The videos have the following features: • Author-driven—The authors reviewed every script and every video. Author Michael Sullivan III appears in many videos. • Objective-based—one video for each objective in the book • Segmented—videos divided into shorter parts for ease of navigation (Introductions, Examples, and Summary) • Interactive Figures—to help students visualize key concepts • Handwriting—our research showed that students prefer seeing examples worked out by hand • High-definition—clearly readable on phones • Accessible—all are close-captioned • Various instructors—Featured in the videos are: ■ Michael Sullivan III (Joliet Junior College and Florida SouthWestern State College) ■ Kevin Bodden (Lewis and Clark Community College) ■ Randy Gallaher (Lewis and Clark Community College) ■ Sue Glascoe (Mesa Community College) ■ Paulette Haywood-Watson (Stillman College) ■ Brian Macon (Valencia College) ■ Caleb Schroeder (Antelope Valley College) • Graphing Utility Screen Captures We have added even more Desmos and GeoGebra screen captures along with TI-84 Plus CE screen captures. We recognize that Desmos and GeoGebra simplify finding points of interest such as local extrema, intercepts, and points of intersection with systems of equations. Additionally, Desmos and GeoGebra are powerful tools when it comes to graphing implicit equations and polar curves. These updated screen captures provide alternate ways of visualizing concepts and making connections between equations, data, and graphs in full color. • Margin Notes When appropriate, a variety of notes are provided in the margins of the text. • Need To Review? comments are included as a timely reference to the chapter, section and page, where a student may need to refresh a particular skill necessary for the current conversation. Preface to the Instructor

xxii Preface to the Instructor • In Words notes are used to help students focus on a concept or a formula, definition, or theorem being presented. • A Note is used at times to further clarify reasoning or make connections to prior concepts. • Seeing the Concept margin notes are strategically provided where graphing exploration may be useful to strengthen understanding of a mathematical concept. • Career Tips and Fun Facts are provided occasionally to help students connect the topic being discussed to the world around them. • Caution margin notes are used to help students avoid common mistakes that we all make when first learning a new mathematical skill. • Retain Your Knowledge Problems These problems are based on the article “To Retain New Learning, Do the Math,” published in the Edurati Review. In this article, Kevin Washburn suggests that “the more students are required to recall new content or skills, the better their memory will be.” In most sections, problems were chosen that preview skills required to succeed in subsequent sections. All answers to Retain Your Knowledge problems are given in the back of the text and all are assignable in MyMathLab for School. • NEW! Video Note-Taking Guide Ideal for online, emporium/redesign courses, flipped classrooms, or traditional lecture classrooms. The note-taking guide helps students take thorough, organized, and understandable notes as they watch the new instructional videos. They ask students to complete definitions, procedures, and examples based on the content of the videos and text. In addition, experience suggests that students learn by doing and understanding the why/how of the concept or property. Therefore, many sections will have an expoloration activity to motivate student learning. These explorations introduce the topic and/or connect it to either a real-world application or a previous section. For example, when the vertical-line test is discussed in Section 2.2, after the theorem statement, the notes ask the students to explain why the vertical-line test works by using the definitions of a function. This challenge helps students process the information at a higher level of understanding. • Chapter Projects apply the concepts of each chapter to a real-world situation that is current and relevant today. Many of these projects embrace the spirit of active learning by requiring the student to research information online in order to solve problems. • Exercise Sets The exercises in the text have been reviewed and analyzed. Some have been removed and new ones have been added. All time-sensitive problems, espcially those involving data, have been updated to the most recent information available. The problem sets remain classified according to purpose. The ‘Are You Prepared?’ problems serve their purpose as a just-in-time review of concepts that the student will need to apply in the upcoming section. The Concepts and Vocabulary problems cover each objective of the section. These multiple-choice, fill-in-the-blank, and True/False exercises have been written to also serve as reading quizzes. Skill Building problems develop the student’s computational skills with a large selection of exercises that are directly related to the objectives of the section. Often the Skill Building problems begin with Interactive Figure Exercises. Mixed Practice problems offer a comprehensive assessment of skills that relate to more than one objective. Often these require skills learned earlier in the course which helps students retain their knowledge. Applications and Extensions problems have been updated. Further, many new application-type exercises have been added, especially ones involving information and data drawn from sources the student will recognize, to improve relevance and timeliness. At the end of Applications and Extensions, we have a collection of one or more Challenge Problems . These problems, as the title suggests, are intended to be thought-provoking, requiring some ingenuity to solve. They can be used for group work or to challenge students. At the end of the Annotated Instructor’s Edition and in the online Instructor’s Solutions Manual, we have provided solutions to all these problems. The Explaining Concepts exercises provide opportunity for classroom discussion and group projects. • Diversity, Equity, and Inclusion— Pearson conducted an external review of the text’s content to determine how it could be improved to address issues related to diversity, equity, and inclusion.The results of that review informed this revision. • Key to Exercise Types— To help you navigate the features of the exercise sets, we’ve included a key at the bottom of the first page of each section’s exercises. 1. Now Work 1. Modeling 1. Explaining Concepts Calculus Preview 1. Interactive Figure Content Changes to the 9 th edition Chapter 1 • Exercises have been updated with current and relevant scenarios such as credit card debt and revenue earned by social media influencers. • NEW Desmos and GeoGebra screenshots added to illustrate how to graph equations, create tables, and solve equations. Chapter 2 • NEW Desmos and GeoGebra screenshots added to illustrate the use of function notation and finding points of interest such as local extrema. • Continued updates to exercises with current data and relevance.

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