Polynomial and Rational Functions


senior contributing author Jay abramson, arizona state University



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Reviewers Nina Alketa, Cecil College Kiran Bhutani, Catholic University of America Brandie Biddy, Cecil College Lisa Blank, Lyme Central School Bryan Blount, Kentucky Wesleyan College Jessica Bolz, The Bryn Mawr School Sheri Boyd, Rollins College Sarah Brewer, Alabama School of Math and Science Charles Buckley, St. Gregory’s University Michael Cohen, Hofstra University Kenneth Crane, Texarkana College Rachel Cywinski, Alamo Colleges Nathan Czuba Srabasti Dutta, Ashford University Kristy Erickson, Cecil College Nicole Fernandez, Georgetown University / Kent State University David French, Tidewater Community College Douglas Furman, SUNY Ulster Lance Hemlow, Raritan Valley Community College Erinn Izzo, Nicaragua Christian Academy John Jaffe Jerry Jared, Blue Ridge School Stan Kopec, Mount Wachusett Community College Kathy Kovacs Cynthia Landrigan, Erie Community College Sara Lenhart, Christopher Newport University Wendy Lightheart, Lane Community College Joanne Manville, Bunker Hill Community College Karla McCavit, Albion College Cynthia McGinnis, Northwest Florida State College Lana Neal, University of Texas at Austin Rhonda Porter, Albany State University Steven Purtee, Valencia College William Radulovich, Florida State College Jacksonville Alice Ramos, Bethel College Nick Reynolds, Montgomery Community College Amanda Ross, A. A. Ross Consulting and Research, LLC Erica Rutter, Arizona State University Sutandra Sarkar, Georgia State University Willy Schild, Wentworth Institute of Technology Todd Stephen, Cleveland State University Scott Sykes, University of West Georgia Linda Tansil, Southeast Missouri State University John Thomas, College of Lake County Diane Valade, Piedmont Virginia Community College

Allen Wolmer, Atlanta Jewish Academy

About Our Team Senior Contributing Author

Jay Abramson has been teaching Precalculus for 33 years, the last 14 at Arizona State University, where he is a principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions.

Contributing Authors Valeree Falduto, Palm Beach State College Rachael Gross, Towson University David Lippman, Pierce College Melonie Rasmussen, Pierce College Rick Norwood, East Tennessee State University Nicholas Belloit, Florida State College Jacksonville Jean-Marie Magnier, Springfield Technical Community College Harold Whipple Christina Fernandez

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PRINT ISBN-10 1-938168-34-8 PRINT ISBN-13 978-1-938168-34-5 PDF ISBN-10 1-947172-06-9 PDF ISBN-13 978-1-947172-06-7 Revision PR-2015-002(03/17)-BW

Original Publication Year: 2014



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Brief ContentsBrief Contents

1 Functions 1 2 Linear Functions 125 3 Polynomial and Rational Functions 197 4 Exponential and Logarithmic Functions 327 5 Trigonometric Functions 439 6 Periodic Functions 505 7 Trigonometric Identities and Equation 559 8 Further Applications of Trigonometry 643 9 Systems of Equations and Inequalities 757 10 Analytic Geometry 863 11 Sequences, Probability and Counting Theory 937 12 Introduction to Calculus 1017




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Contents Preface xi

1 Functions 1 1.1 Functions and Function Notation 2

1.2 Domain and Range 22

1.3 Rates of Change and Behavior of Graphs 38

1.4 Composition of Functions 51

1.5 Transformation of Functions 64

1.6 Absolute Value Functions 89

1.7 Inverse Functions 100

Chapter 1 Review 113

Chapter 1 Review Exercises 118

Chapter 1 Practice Test 123

2 Linear Functions 125 2.1 Linear Functions 126

2.2 Graphs of Linear Functions 143

2.3 Modeling with Linear Functions 162

2.4 Fitting Linear Models to Data 175

Chapter 2 Review 187

Chapter 2 Review Exercises 190

Chapter 2 Practice Test 194

3 Polynomial and Rational Functions 197 3.1 Complex Numbers 198

3.2 Quadratic Functions 208

3.3 Power Functions and Polynomial Functions 224

3.4 Graphs of Polynomial Functions 239

3.5 Dividing Polynomials 257

3.6 Zeros of Polynomial Functions 266

3.7 Rational Functions 278

3.8 Inverses and Radical Functions 299

3.9 Modeling Using Variation 310

Chapter 3 Review 317

Chapter 3 Review Exercises 322

Chapter 3 Practice Test 325



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4 Exponential and Logarithmic Functions 327 4.1 Exponential Functions 328

4.2 Graphs of Exponential Functions 343

4.3 Logarithmic Functions 355

4.4 Graphs of Logarithmic Functions 363

4.5 Logarithmic Properties 380

4.6 Exponential and Logarithmic Equations 390

4.7 Exponential and Logarithmic Models 401

4.8 Fitting Exponential Models to Data 416

Chapter 4 Review 429

Chapter 4 Review Exercises 434

Chapter 4 Practice Test 437

5 Trigonometric Functions 439 5.1 Angles 440

5.2 Unit Circle: Sine and Cosine Functions 457

5.3 The Other Trigonometric Functions 473

5.4 Right Triangle Trigonometry 486

Chapter 5 Review 498

Chapter 5 Review Exercises 502

Chapter 5 Practice Test 504

6 Periodic Functions 505 6.1 Graphs of the Sine and Cosine Functions 506

6.2 Graphs of the Other Trigonometric Functions 523

6.3 Inverse Trigonometric Functions 541

Chapter 6 Review 552

Chapter 6 Review Exercises 554

Chapter 6 Practice Test 556

7 Trigonometric Identities and Equations 559 7.1 Solving Trigonometric Equations with Identities 560

7.2 Sum and Difference Identities 570

7.3 Double-Angle, Half-Angle, and Reduction Formulas 584

7.4 Sum-to-Product and Product-to-Sum Formulas 596

7.5 Solving Trigonometric Equations 603

7.6 Modeling with Trigonometric Equations 617

Chapter 7 Review 634

Chapter 7 Review Exercises 639

Chapter 7 Practice Test 642



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8 Further Applications of Trigonometry 643 8.1 Non-right Triangles: Law of Sines 644

8.2 Non-right Triangles: Law of Cosines 658

8.3 Polar Coordinates 670

8.4 Polar Coordinates: Graphs 681

8.5 Polar Form of Complex Numbers 697

8.6 Parametric Equations 708

8.7 Parametric Equations: Graphs 719

8.8 Vectors 729

Chapter 8 Review 747

Chapter 8 Review Exercises 752

Chapter 8 Practice Test 755

9 Systems of Equations and Inequalities 757 9.1 Systems of Linear Equations: Two Variables 758

9.2 Systems of Linear Equations: Three Variables 774

9.3 Systems of Nonlinear Equations and Inequalities: Two Variables 785

9.4 Partial Fractions 795

9.5 Matrices and Matrix Operations 805

9.6 Solving Systems with Gaussian Elimination 816

9.7 Solving Systems with Inverses 829

9.8 Solving Systems with Cramer’s Rule 843

Chapter 9 Review 854

Chapter 9 Review Exercises 858

Chapter 9 Practice Test 861

10 Analytic Geometry 863 10.1 The Ellipse 864

10.2 The Hyperbola 879

10.3 The Parabola 896

10.4 Rotation of Axis 909

10.5 Conic Sections in Polar Coordinates 922

Chapter 10 Review 931

Chapter 10 Review Exercises 934

Chapter 10 Practice Test 936




11 Sequences, Probability and Counting Theory 937 11.1 Sequences and Their Notations 938

11.2 Arithmetic Sequences 951

11.3 Geometric Sequences 961

11.4 Series and Their Notations 969

11.5 Counting Principles 982

11.6 Binomial Theorem 992

11.7 Probability 999

Chapter 11 Review 1008

Chapter 11 Review Exercises 1012

Chapter 11 Practice Test 1015

12 Introduction to Calculus 1017 12.1 Finding Limits: Numerical and Graphical Approaches 1018

12.2 Finding Limits: Properties of Limits 1028

12.3 Continuity 1037

12.4 Derivatives 1051

Chapter 12 Review 1070

Chapter 12 Review Exercises 1073

Chapter 12 Practice Test 1075

Appendix A-1

Try It Answer Section B-1

Odd Answer Section C-1

Index D-1



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Preface Welcome to Precalculus, an OpenStax resource. This textbook was w ritten to i ncrease student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost.

About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learning outcomes through personalized educational paths, is being piloted in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

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Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.



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About Precalculus Precalculus is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course. The content is organized by clearly-defined learning objectives, and includes worked examples that demonstrate problem-solving approaches in an accessible way. Coverage and Scope

Precalculus contains twelve chapters, roughly divided into three groups.

Chapters 1-4 discuss various types of functions, providing a foundation for the remainder of the course. Chapter 1: Functions Chapter 2: Linear Functions Chapter 3: Polynomial and Rational Functions Chapter 4: Exponential and Logarithmic Functions

Chapters 5-8 focus on Trigonometry. In Precalculus, we approach trigonometry by first introducing angles and the unit circle, as opposed to the right triangle approach more commonly used in College Algebra and Trigonometry courses.

Chapter 5: Trigonometric Functions Chapter 6: Periodic Functions Chapter 7: Trigonometric Identities and Equations Chapter 8: Further Applications of Trigonometry

Chapters 9-12 present some advanced Precalculus topics that build on topics introduced in chapters 1-8. Most Precalculus syllabi include some of the topics in these chapters, but few include all. Instructors can select material as needed from this group of chapters, since they are not cumulative.

Chapter 9: Systems of Equations and Inequalities Chapter 10: Analytic Geometry Chapter 11: Sequences, Probability and Counting Theory Chapter 12: Introduction to Calculus

All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.

Development Overview Precalculus is the product of a collaborative effort by a group of dedicated authors, editors, and instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision for the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading numerous drafts, and assimilating field reviews into actionable revision plans for our authors and editors.

The first eight chapters are built on the foundation of Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. Chapters 9-12 were written and developed from by our expert and highly experienced author team. All twelve chapters follow a new and innovative instructional design, and great care has been taken to maintain a consistent voice from cover to cover. New features have been introduced to flesh out the instruction, all of the graphics have been redone in a more contemporary style, and much of the content has been revised, replaced, or supplemented to bring the text more in line with mainstream approaches to teaching precalculus.

Accuracy of the Content We have taken great pains to ensure the validity and accuracy of this text. Each chapter’s manuscript underwent at least two rounds of review and revision by a panel of active precalculus instructors. Then, prior to publication, a separate team of experts checked all text, examples, and graphics for mathematical accuracy; multiple reviewers were assigned to each chapter to minimize the chances of any error escaping notice. A third team of experts was responsible for the accuracy of the Answer Key, dutifully reworking every solution to eradicate any lingering errors. Finally, the editorial team conducted a multi-round post-production review to ensure the integrity of the content in its final form was written and developed after the Student Edition, has also been rigorously checked for accuracy following a process similar to that described above. Incidentally, the act of writing out solutions step-by-step served as yet another round of validation for the Answer Key in the back of the Student Edition.



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Pedagogical Foundations and Features Learning Objectives Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives. The learning objectives are listed explicitly at the beginning of each section and are the focal point of every instructional element.

Narrative Text Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide transitions between topics and examples. Throughout this book, we rely on a few basic conventions to highlight the most important ideas:

• Key terms are boldfaced, typically when first introduced and/or when formally defined. • Key concepts and definitions are called out in a blue box for easy reference.

Example Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective in order to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. All told, there are more than 650 Examples, or an average of about 55 per chapter.

All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the Solution, spelling out the steps along the way. Finally (for select Examples), we conclude with an Analysis reflecting on the broader implications of the Solution just shown.

Figures Precalculus contains more than 2000 figures and illustrations, the vast majority of which are graphs and diagrams. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish between the different functions or features of a graph.



(h, k − a)

(h, k − c)

(h, k) (h + b, k)

(h, k + a)

(h, k + c)

(h − b, k)

Major Axis

Minor Axis


–6 –4 2 4 6








y y = x2 + 4x + 3

Vertex Axis of symmetry

x-intercepts –6 –4 –2 2 4 6









(1, 6)

(0, 1) (6, 1)

(1, 0)

f (x) = 5×3 + 1

y = x

Supporting Features Four small but important features, each marked by a distinctive icon, serve to support Examples.

A “How To” is a list of steps necessary to solve a certain type of problem. A How To typically precedes an Example that proceeds to demonstrate the steps in action.

A “Try It” exercise immediately follows an Example or a set of related Examples, providing the student with an immediate opportunity to solve a similar problem. In the Online version of the text, students can click an Answer link directly below the question to check their understanding. In other versions, answers to the Try-It exercises are located in the Answer Key.

A “Q & A…” may appear at any point in the narrative, but most often follows an Example. This feature pre-empts misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation.

The “Media” links appear at the conclusion of each section, just prior to the Section Exercises. These are a list of links to online video tutorials that reinforce the concepts and skills introduced in the section

While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany Precalculus.



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Section Exercises Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. With over 5,900 exercises across the 12 chapters, instructors should have plenty to choose from[i].

Section Exercises are organized by question type, and generally appear in the following order: Verbal questions assess conceptual understanding of key terms and concepts. Algebraic problems require students to apply algebraic manipulations demonstrated in the section. Graphical problems assess students’ ability to interpret or produce a graph. Numeric problems require the student perform calculations or computations. Technology problems encourage exploration through use of a graphing utility, either to visualize or verify algebraic results or to solve problems via an alternative to the methods demonstrated in the section. Extensions pose problems more challenging than the Examples demonstrated in the section. They require students to synthesize multiple learning objectives or apply critical thinking to solve complex problems. Real-World Applications present realistic problem scenarios from fields such as physics, geology, biology, finance, and the social sciences.

Chapter Review Features Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.x

Key Terms provides a formal definition for each bold-faced term in the chapter. Key Equations presents a compilation of formulas, theorems, and standard-form equations. Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant Example(s) in case students need to review. Chapter Review Exercises include 40-80 practice problems that recall the most important concepts from each section. Practice Test includes 25-50 problems assessing the most important learning objectives from the chapter. Note that the practice test is not organized by section, and may be more heavily weighted toward cumulative objectives as opposed to the foundational objectives covered in the opening sections.

Additional Resources

Student and Instructor Resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book.

Partner Resources OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on openstax.org.

Online Homework XYZ Homework is built using the fastest-growing mathematics cloud platform. XYZ Homework gives instructors access to the Precalculus aligned problems, organized in the Precalculus Course Template. Instructors have access to thousands of additional algorithmically-generated questions for unparalleled course customization. For one low annual price, students can take multiple classes through XYZ Homework. Learn more at www.xyzhomework.com/openstax.

WebAssign is an independent online homework and assessment solution first launched at North Carolina State University in 1997. Today, WebAssign is an employee-owned benefit corporation and participates in the education of over a million students each year. WebAssign empowers faculty to deliver fully customizable assignments and high quality content to their students in an interactive online environment. WebAssign supports Precalculus with hundreds of problems covering every concept in the course, each containing algorithmically-generated values and links directly to the eBook providing a completely integrated online learning experience.

i. 5,924 total exercises. Includes Chapter Reviews and Practice Tests.






1,500 P




1970 1975 1980 1985 1990 1995 2000 2005 2010 0

Figure 1 Standard and Poor’s Index with dividends reinvested (credit “bull”: modification of work by Prayitno Hadinata; credit “graph”: modification of work by MeasuringWorth)

Introduction Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. Figure 1 tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business

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