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Unformatted text preview: College Algebra senior contributing author Jay Abramson, Arizona State University About Our Team Senior Contributing Author Jay Abramson has been teaching College Algebra 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 The following faculty contributed to the development of OpenStax Precalculus, the text from which this product was updated and derived. Honorable Mention 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 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 Erinn Izzo, Nicaragua Christian Academy John Jaffe Jerry Jared, Blue Ridge School Stan Kopec, Mount Wachusett Community College Kathy Kovacs Sara Lenhart, Christopher Newport University Joanne Manville, Bunker Hill Community College Karla McCavit, Albion College Cynthia McGinnis, Northwest Florida State College Lana Neal, University of Texas at Austin Steven Purtee, Valencia College 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 Reviewers Phil Clark, Scottsdale Community College Michael Cohen, Hofstra University Matthew Goodell, SUNY Ulster Lance Hemlow, Raritan Valley Community College Dongrin Kim, Arizona State University Cynthia Landrigan, Erie Community College Wendy Lightheart, Lane Community College Carl Penziul, Tompkins-Cortland Community College Sandra Nite, Texas A&M University Eugenia Peterson, Richard J. Daley College Rhonda Porter, Albany State University Michael Price, University of Oregon William Radulovich, Florida State College Jacksonville Camelia Salajean, City Colleges of Chicago Katy Shields, Oakland Community College Nathan Schrenk, ECPI University Pablo Suarez, Delaware State University Allen Wolmer, Atlanta Jewish Academy OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit . Individual print copies and bulk orders can be purchased through our website. © 2017 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). 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The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo, OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and Rice University logo are not subject to the license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this licensing, please contact [email protected] PRINT ISBN-10 1-93816 8-38-0 PRINT ISBN-13 978-1-938168-38-3 PDF ISBN-10 1-947172-12-3 PDF ISBN-13 978-1-947172-12-8 Revision CA-2015-002(03/17)-BW Original Publication Year: 2015 OpenStax OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced Placement® courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech initiative based at Rice University, we’re committed to helping students access the tools they need to complete their courses and meet their educational goals. 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In the United States, it seeks to signifi antly improve education so that all young people have the opportunity to reach their full potential. Based in Seattle, Washington, the foundation is led by CEO Jeff Raikes and Co-chair William H. Gates Sr., under the direction of Bill and Melinda Gates and Warren Buffett. The Maxfield Foundation supports projects with potential for high impact in science, education, sustainability, and other areas of social importance. THE MICHELSON 20MM FOUNDATION Our mission at The Michelson 20MM Foundation is to grow access and success by eliminating unnecessary hurdles to affordability. We support the creation, sharing, and proliferation of more effective, more affordable educational content by leveraging disruptive technologies, open educational resources, and new models for collaboration between for-profi , nonprofi , and public entities. Calvin K. Kazanjian was the founder and president of Peter Paul (Almond Joy), Inc. He firmly believed that the more people understood about basic economics the happier and more prosperous they would be. Accordingly, he established the Calvin K. Kazanjian Economics Foundation Inc, in 1949 as a philanthropic, nonpolitical educational organization to support efforts that enhanced economic understanding. The Bill and Stephanie Sick Fund supports innovative projects in the areas of Education, Art, Science and Engineering. iii new school year. new classes. new books. free books. new assignments. free app. new app. OpenStax + SE Get free textbooks for over 30 college courses in the free OpenStax + SE app. Download it now on the App Store or get it on Google Play. Brief Contents 1  Prerequisites 1 2  Equations and Inequalities  73 3  Functions 159 4  Linear Functions  279 5  Polynomial and Rational Functions  343 6  Exponential and Logarithmic Functions  463 7  Systems of Equations and Inequalities  575 8  Analytic Geometry  681 9  Sequences, Probability and Counting Theory  755 v Contents Preface xi 1  Prerequisites 1 1.1  Real Numbers: Algebra Essentials  2 1.2  Exponents and Scientific Notation  17 1.3  Radicals and Rational Expressions  31 1.4  Polynomials 41 1.5  Factoring Polynomials  49 1.6  Rational Expressions  58 Chapter 1 Review 66 Chapter 1 Review Exercises 70 Chapter 1 Practice Test 72 2  Equations and Inequalities  73 2.1  The Rectangular Coordinate Systems and Graphs  74 2.2  Linear Equations in One Variable  87 2.3  Models and Applications  102 2.4  Complex Numbers  111 2.5  Quadratic Equations  119 2.6  Other Types of Equations  131 2.7  Linear Inequalities and Absolute Value Inequalities  142 Chapter 2 Review 151 Chapter 2 Review Exercises 155 Chapter 2 Practice Test 158 3  Functions 159 3.1  Functions and Function Notation  160 3.2  Domain and Range  180 3.3  Rates of Change and Behavior of Graphs  196 3.4  Composition of Functions  209 3.5  Transformation of Functions  222 3.6  Absolute Value Functions  247 3.7  Inverse Functions  254 Chapter 3 Review 267 Chapter 3 Review Exercises 272 Chapter 3 Practice Test 277 vii 4  Linear Functions  279 4.1  Linear Functions  280 4.2  Modeling with Linear Functions  309 4.3  Fitting Linear Models to Data  322 Chapter 4 Review 334 Chapter 4 Review Exercises 336 Chapter 4 Practice Test 340 5  Polynomial and Rational Functions  343 5.1  Quadratic Functions  344 5.2  Power Functions and Polynomial Functions  360 5.3  Graphs of Polynomial Functions  375 5.4  Dividing Polynomials  393 5.5  Zeros of Polynomial Functions  402 5.6  Rational Functions  414 5.7  Inverses and Radical Functions  435 5.8  Modeling Using Variation  446 Chapter 5 Review 453 Chapter 5 Review Exercises 458 Chapter 5 Practice Test 461 6  Exponential and ­Logarithmic Functions  6.1  Exponential Functions  464 6.2  Graphs of Exponential Functions  479 6.3  Logarithmic Functions  491 6.4  Graphs of Logarithmic Functions  499 6.5  Logarithmic Properties  516 6.6  Exponential and Logarithmic Equations  526 6.7  Exponential and Logarithmic Models  537 6.8  Fitting Exponential Models to Data  552 Chapter 6 Review 565 Chapter 6 Review Exercises 570 Chapter 6 Practice Test 573 v iii 463 7  Systems of Equations and Inequalities  575 7.1  Systems of Linear Equations: Two Variables  576 7.2  Systems of Linear Equations: Three Variables  592 7.3  Systems of Nonlinear Equations and Inequalities: Two Variables  603 7.4  Partial Fractions 613 7.5  Matrices and Matrix Operations  623 7.6  Solving Systems with Gaussian Elimination  634 7.7  Solving Systems with Inverses  647 7.8  Solving Systems with Cramer's Rule  661 Chapter 7 Review 672 Chapter 7 Review Exercises 676 Chapter 7 Practice Test 679 8  Analytic Geometry  681 8.1  The Ellipse  682 8.2  The Hyperbola  697 8.3  The Parabola  714 8.4  Rotation of Axis  727 8.5  Conic Sections in Polar Coordinates  740 Chapter 8 Review 749 Chapter 8 Review Exercises 752 Chapter 8 Practice Test 754 9  Sequences, Probability and Counting Theory  755 9.1  Sequences and Their Notations   756 9.2  Arithmetic Sequences  769 9.3  Geometric Sequences  779 9.4  Series and Their Notations  787 9.5  Counting Principles  800 9.6  Binomial Theorem  810 9.7  Probability 817 Chapter 9 Review 826 Chapter 9 Review Exercises 830 Chapter 9 Practice Test 833 Try It Answer Section  A-1 Odd Answer Section  B-1 Index C-1 ix Preface Welcome to College Algebra, an OpenStax resource. This textbook was written to increase 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 20 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. About OpenStax’s Resources Customization College Algebra is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book. Faculty also have the option of creating a customized version of their OpenStax book through the OpenStax Custom platform. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on openstax.org for a link to your book on OpenStax Custom. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on openstax.org. Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print. xi About College Algebra College Algebra provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra course. The modular approach and richness of content ensure that the book meets the needs of a variety of courses. College Algebra offe s a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned. Coverage and Scope In determining the concepts, skills, and topics to cover, we engaged dozens of highly experienced instructors with a range of student audiences. The resulting scope and sequence proceeds logically while allowing for a significant amount of flexibility in instruction. Chapters 1 and 2 provide both a review and foundation for study of Functions that begins in Chapter 3. The authors recognize that while some institutions may find this material a prerequisite, other institutions have told us that they have a cohort that need the prerequisite skills built into the course. Chapter 1: Prerequisites Chapter 2: Equations and Inequalities Chapters 3-6: The Algebraic Functions Chapter 3: Functions Chapter 4: Linear Functions Chapter 5: Polynomial and Rational Functions Chapter 6: Exponential and Logarithm Functions Chapters 7-9: Further Study in College Algebra Chapter 7: Systems of Equations and Inequalities Chapter 8: Analytic Geometry Chapter 9: Sequences, Probability and Counting Theory All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents. Development Overview College Algebra is the product of a collaborative eff rt by a group of dedicated authors, editors, and instructors whose collective passion for this project has resulted in a text that is remarkably unifi d 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 fi ld reviews into actionable revision plans for our authors and editors. The collective experience of our author team allowed us to pinpoint the subtopics, exceptions, and individual connections that give students the most trouble. The textbook is therefore replete with well-designed features and highlights, which help students overcome these barriers. As the students read and practice, they are coached in methods of thinking through problems and internalizing mathematical processes. Accuracy of the Content We understand that precision and accuracy are imperatives in mathematics, and undertook a dedicated accuracy program led by experienced faculty. 1. Each chapter’s manuscript underwent rounds of review and revision by a panel of active instructors. 2. 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. 3. A third team of experts was responsible for the accuracy of the Answer Key, dutifully re-working 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. x ii 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. • Key concepts and definitions are called out in a blue box for easy reference. Examples Each learning objective is supported by one or more worked examples, which demonstrate the problem-solving approaches that students must master. The multiple Examples model different approaches to the same type of problem, or introduce similar problems of increasing complexity. All Examples follow a simple two- or three-part format. The question clearly lays out a mathematical problem to solve. The Solution walks through the steps, usually providing context for the approa...
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