**The Rule of Four—A Balanced Approach**

A principal feature of this textbook is the balance among the algebraic, numerical, graphical, and verbal methods of representing problems—the rule of four. For instance, we obtain solutions algebraically when that is the most appropriate technique to use, and we obtain solutions graphically or numerically when algebra is difficult to use. We urge students to solve problems by one method and then to support or confirm their solutions using another method. We believe that students must learn the value of each of these methods or representations and must learn to choose the one most appropriate for solving the problem at hand. This approach reinforces the idea that to understand a problem fully, students need to understand it algebraically as well as graphically and numerically.**Problem-Solving Approach**

We introduce systematic problem solving in Section 1.2 and then emphasize it in the examples throughout the text. We use the following variation of Pólya’s problem-solving process that emphasizes modeling with mathematics.:

- Understand the problem.
- Develop a mathematical model of the problem.
- Solve the mathematical model and support or confirm the solution.
- Present the answer in the context of the problem.

Students are encouraged to use this process throughout the text.**Focus on Functions**

Another distinctive feature of this text is that we introduce students to the full vocabulary of functions in Chapter 1. Students encounter the concepts of domain and range, change in tandem, increasing and decreasing, concavity, points of inflections, boundedness, extrema, symmetry, continuity, asymptotes, and end behavior. Students then meet the following twelve basic functions:

- The Identity Function
- The Squaring Function
- The Cubing Function
- The Reciprocal Function
- The Square Root Function
- The Exponential Function
- The Natural Logarithm Function
- The Sine Function
- The Cosine Function
- The Absolute Value Function
- The Greatest Integer Function
- The Logistic Function

Armed with this diverse set of examples, students compare and contrast their graphical properties to gain an understanding of the vocabulary of functions, which would be difficult to appreciate if the only examples a teacher could point to were polynomials.

Consequently, students can characterize functions by their behavior within the first month of classes. Once students have a comfortable understanding of functions in general, the rest of the course consists of studying the various types of functions in greater depth, particularly with respect to their algebraic properties and modeling applications. We include a complete gallery of basic functions inside the back cover of the text for handy reference.**Applications, Data, and Modeling**

Most of the applications in the text are based on real data from cited sources. As they work through the applications, students see functions as tools for modeling real-life problems. They learn to analyze and model data, represent data graphically, interpret graphs, and fit curves. Additionally, the tabular representation of data presented in this text highlights the concept that a function is a correspondence between numerical variables. This helps students build the connection between numerical quantities and graphs and recognize the importance of a full graphical, numerical, and algebraic understanding of a problem. For convenience we provide links to large data sets and a list of applications by category in the Applications Index.**Technology and Exercises**

The authors of this text have advocated for the use of technology in mathematics education for more than three decades. Our approach to problem solving (page 58[JD1] ) distinguishes between solving the problem and supporting or confirming the solution, and it emphasizes how technology figures in each of these processes.

Advances in technology and increased familiarity with calculators have gradually blurred some of the distinctions between solving and supporting a solution. Therefore, and in line with the AP® exam, we mark some exercises to be solved without using a calculator; the numbers of these exercises have a red underline . These usually are exercises that develop algebraic fluency or demonstrate how various functions behave algebraically. Application problems typically have no restrictions, in keeping with our emphasis on modeling and on using all representations when confronting real-world problems.

Incidentally, we continue to encourage the use of calculators to support answers graphically or numerically after the problems have been solved with pencil and paper. Any time students can make connections among the graphical, analytical, and numerical representations, they are doing worthwhile mathematics.

As a final note, we will freely admit that teachers use our text in different ways, and some of you will override our no-calculator recommendations to fit with your pedagogical purposes. In the end, you know what is best for your students, and we are just here to help.**Alignment with the AP® Framework****Mathematical Practices**

A key feature of the AP® Precalculus Course Framework that we especially like are the Mathematical Practices. These practices are high-level skills that build on one another:

- Procedural and Symbolic Fluency. This first core competency for students involves algebraic expertise: Students learn how to manipulate formulas, equations, and expressions—with and without technology. This algebraic fluency provides a foundation for success in calculus. Since the first edition, we have designed examples and exercises to develop such fluency throughout this textbook.
- Multiple Representations. This practice is so central to our approach that we call multiple representations by name in the title of the book. This AP® Precalculus skill is the ability to interpret and translate across mathematical models: graphical, numerical, algebraic, and verbal. Students read tables and graphs, parse equations, and make sense of written problem statements, and then convert the given information into other forms to solve the problem at hand.
- Communication and Reasoning. This practice involves students making their thinking seen and heard by others. To help students achieve this goal we have included Explorations, Writing to Learn exercises, Helping Learners of English margin notes, and Chapter Modeling Projects. Plus, for this edition we have added Chapter Challenges, Group Activities, and Answer and Explain exercises to give students even more opportunities to explain what they did, how they did it, and why it works. Our in-depth treatments of functions and modeling help students understand attributes of families of functions and learn when to use them to model real-world situations. Students learn how to articulate their underlying assumptions and to recognize the assumptions within a given problem.

In the AP® Framework, these three practices are divided into eight AP® Skills. At the beginning of every section in Chapters 1–7 we provide lists of the AP® Skills and the AP® Learning Objectives addressed within the section.**Organization of the Book**

If you page through the table of contents, you will notice that we have organized the chapters of the text into units that align with the AP® Precalculus Course Framework. Chapter 0 and the Appendix focus on Expected Prior Knowledge and Skills. Chapters 1–5 address Units 1–3 of the AP® Framework and constitute the material currently tested by the College Board on the AP® Precalculus Exam. Chapters 6 and 7 focus on Unit 4, which at least initially is not being tested but addresses topics that the AP® Precalculus development team deemed worthwhile. Chapter 8 contains an introduction to calculus, which you can use after the AP® exam. Alternatively, after the exam you can pick and choose topics from Chapters 6 and 7.**Pacing Guides**

We provide two pacing guides. One is for teachers and schools focusing only on Units 1–3. A second pacing guide is for those teaching Units 1–4. If you are only teaching Units 1–3, you have lots of options for in-depth student engagement: Chapter Challenges, Group Activities, Extending the Ideas exercises, and Chapter Modeling Projects. If you elect to teach Units 1–4, you will need to proceed at a brisk pace. In either case, we hope that you engage your students in the Explorations.**Teacher’s Guide**

Because the AP® Precalculus is a brand-new course, we have included a detailed Teacher’s Guide to walk you through every section of the book. For each section you will find (a) key ideas and AP themes; (b) notes on explorations, challenges, and activities; and (c) notes on examples. (For each section we provide Notes on Exercises as a margin note before the exercises.) We encourage you to use the Teacher’s Guide in tandem with one of the Pacing Guides to help you plan your students’ path through the book.**New to This Edition**

Joining the author team in this endeavor are two amazing new authors: Rachael Gorsuch and Steve Phelps. Each brings a wealth of classroom and mathematical experience to the project. Their fresh perspectives and creativity have greatly enhanced this edition!

The new aspects of the text and accompanying MyMathLab® for School course make the product the best possible resource for AP® Precalculus. We’ve organized these elements into the categories Features, Content, and Technology to make it easier for you to navigate.**New Features**

- We have added resources to make AP® test preparation a regular part of the student experience:
- AP® Test Prep in the section-level homework. Here we include multiple-choice and “Answer and Explain” exercises. The latter exercises are not full-blown Free Response Questions (FRQs) like those on the AP® exam, but we designed them to build up to FRQs by getting students in the habit of explaining their thinking.
- AP® Test Prep Quick Quiz. These appear every 2–3 sections and include both multiple-choice and free-response questions.
- AP® Test Prep Chapter Quiz. These end-of-chapter quizzes call upon all skills learned to that point and include both multiple-choice and free-response questions.
- Chapter Challenge is a multi-part investigative activity that is introduced on the opening page of the chapter and builds part by part throughout the chapter. The Challenges are in Chapters 1–7.
- Group Activities have been added to all sections in Chapters 1–7. These appear right before the exercise sets. These are excellent opportunities to support collaborative learning among your students.
- The Function Spotlight feature provides all key information about a given function in a single location, including new concepts such as concavity that are called for in the AP Precalculus Guidelines.
- Restructured exercise sets
- For ease of use, we moved the Quick Review exercises within the exercise sets.
- AP® Test Prep exercises are a part of every exercise set in Chapters 1–7.
- New to this Teacher’s Edition
- Table of Contents contains “Post-It Notes” in the margin to explain key aspects of the revised contents.
- AP® Precalculus Pacing Guides. One is for teachers and schools focusing only on Units 1–3. A second pacing guide is for those teaching Units 1–4.
- Concept Outline for AP® Precalculus. This outline shows which sections concept which AP® Learning Objectives and AP® Essential Knowledge.
- Teacher’s Guide. Notes for each section (in addition to those in the side margin).

**Changes to Content**

This 11th edition is the most substantial revision in the book’s history! In addition to the many new features just described, to conform with the AP® Precalculus Course Framework, we have added a great deal of content and increased our emphases on mathematical modeling and on student collaboration and communication. Not only have we added many new topics and revised still more, but we have also trimmed material on power and logistics functions, tightened our treatment of trigonometry, condensed and split Systems and Matrices, and removed entire chapters on Discrete Mathematics and on Statistics and Probability. The result is a leaner and livelier Precalculus: a course that focuses squarely on functions and modeling and allows students to pursue fewer topics in greater depth.

Many of the examples are totally new; others have been replaced or have new data or new contexts. Still others have been enhanced or clarified in some way. Many of the exercises are new, or we have improved them in some way. To keep the applications of mathematics relevant to students, we have included the most current data available to us at the time of publication.

The chapter-by-chapter changes include the following:

**Chapter 0: Prerequisites**replaces Chapter P. In keeping with AP’s Expected Prior Knowledge and Skills, this chapter now includes a section on Solving Systems of Equations.**Chapter 1: Functions and Their Properties.**Our approach to problem solving and modeling in Section 1.1 has been updated. Sections 1.2, 1.4, 1.5, and 1.7 all have new subsections: We added content on change in tandem, concavity and points of inflection, parametric functions, piecewise-defined models, and assumption articulation. We give the concept of a removable discontinuity (or hole) greater emphasis.**Chapter 2: Polynomials and Rational Functions.**In Sections 1.1 and 1.2, we have trimmed our treatment of power functions while expanding material on linear and quadratic functions to include more on transformations of graphs and totally new content on local linearity, second order rates of change, and residual analysis. There is a new subsection on Association, Correlation, and Linear Models in Section 2.1 and a new subsection on The Complex Plane in Section 2.5.**Chapter 3: Exponential and Logarithmic Functions.**Section 3.1: Arithmetic and Geometric Sequences and Section 3.6: Logarithmic Scaling are new. Logistic functions are downplayed, but overall, there is a greater emphasis on modeling, and students get to explore connections between arithmetic and geometric sequences and their continuous counterparts—linear and exponential functions.**Chapters 4: Trigonometric Functions.**We have streamlined this chapter somewhat by incorporating applications of right triangle trigonometry and simple harmonic motion earlier in the chapter rather than placing them as a separate section at the end.**Chapters 5: Equivalent Trigonometric Representations and Polar Functions.**This new Chapter 5 treats trigonometric identities as a way to obtain equivalent representations. We have retained Section 5.2: Proving Trigonometric Identities, although it is not required by AP®, to give student opportunities to practice using equivalent trigonometric representations. By trimming a bit, we have included double-angle identities in Section 5.3: Sum and Difference Identities and combined The Laws of Sines and Cosines into Section 5.4. This makes room for Sections 5.5 and 5.6 on polar functions, which are part of AP® Unit 3.**Chapter 6: Vectors, Parametric Functions, and Conic Sections.**This chapter combines and reconfigures material that was in various places in the 10th edition. Section 6.2: Parametric and Implicit Functions Revisited contains much new material. We treat conic sections traditionally and as parametric functions.**Chapters 7: Linear Transformations and Matrix-Vector Functions.**Section 7.1: Matrix Algebra comes mostly from Section 7.2 of the 10th edition. The remaining three sections are totally new material on functions involving vectors and matrices.**Chapters 8: Limits, Derivatives, and Integrals.**This material, which is designed for use after the AP® exam, includes only a few tweaks. We left this pretty much as it was.

Other content changes include the following:

**Revised Accuracy.**In keeping with AP® guidelines, we adjusted the default numerical accuracy level to be 3 decimal places.**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. We used the results of that review to improve this revision.

**New Technology Features**

- GeoGebra activities are sprinkled throughout the text. These provide students with a chance to interactively manipulate mathematical objects and discover key ideas and relationships. We include a QR code and a short URL for each GeoGebra activity to make it easy for students to access this content.
- Online data sets are included for larger data arrays in the text. Teachers and students can access these data sets in *.xls format, which they can then paste into the spreadsheet software of choice.
- Graphing calculator screens have all been updated to the TI-84 Plus Color Edition (CE).

- Each chapter opens with the first part of the Chapter Challenge, which is a series of related activities that build throughout the chapter. These Challenges are focused on motivating applications that can be solved with the concepts learned in the chapter.
- Chapter Overviews give students a sense of what they are going to learn. They provide a roadmap of the chapter and indicate how the topics are connected under one big idea. It is always helpful to remember that mathematics isn’t modular, but interconnected, and that the skills and concepts learned throughout the course build on one another to help students understand more complicated processes and relationships.
- Similarly, the What you’ll learn . . . and why feature presents the big ideas in each section and explains their purpose.
- A Group Activity is provided immediately before each exercise set. These activities give students a way to develop their communication and collaborative skills and help you assess their progress as problem solvers.
- Throughout the text, vocabulary is highlighted in yellow for easy reference. Additionally, Properties, Definitions, Theorems, and Procedures are boxed to stand out.
- Each example ends with a suggestion to Now try Exercise X. Working the suggested exercise is an easy way for students to check their comprehension of the material while reading each section.
- Explorations appear throughout the text and provide students with the perfect opportunity to become active learners and to discover mathematics on their own. This will help them hone their critical-thinking and problem-solving skills. Some are based on technology; others involve exploring mathematical ideas and connections.
- Margin Notes offer additional information to enrich the learning experience. Some of these provide practical advice on using a grapher to obtain the best, most accurate results. Other notes include historical information, address conceptual subtleties, or provide insight to help students avoid common pitfalls and errors.
- Teacher Margin Notes on various topics appear throughout the Annotated Instructor’s Edition. These just-for-teacher annotations are set in a blue font. Each section contains the Objective of the section, AP® Skills, AP® Learning Objectives, Helping Learners of English, and Notes of Exercises. Other instructor annotations offer Alerts, Remarks, Notes of Examples, Exploration Extensions, as well as other general and specific Teaching Notes.
- The review material at the end of each chapter consists of sections dedicated to helping students review the chapter concepts.
- Key Ideas are broken into parts: Properties, Theorems, and Formulas; Procedures; and Gallery of Functions.
- The Review Exercises represent the full range of exercises covered in the chapter and give students additional practice with the ideas developed in the chapter.
- Each AP® Test Prep Chapter Quiz calls upon all skills learned to that point and includes both multiple-choice and free-response questions.
- Each Chapter Modeling Project requires students to analyze data. It can be assigned as either individual or group work. Each project expands upon concepts and ideas taught in the chapter and engages students in modeling with mathematics.

**Exercise Sets**

- Some exercises are designed to be solved without a calculator; the numbers of these exercises are underlined in red. Students are urged to support the answers to these (and all) exercises graphically or numerically, but only after they have solved them with pencil and paper. These are good preparation for an AP® Exam that includes major sections on which a calculator is not permitted.
- There are more than 5,000 exercises. The section exercises progress from routine to challenging. The following types of skills are developed in each exercise set:
- Algebraic understanding and procedures
- Applications of mathematics
- Connecting algebra to geometry
- Interpretation of graphs
- Graphical and numerical representations of functions
- Data analysis
- The exercise sets include distinctive kinds of exercises:
**Quick Review**exercises at the beginning of each section’s exercises help students review skills needed in the exercise set. Often there is a reference to where students can go for a refresher earlier in the book or in the Appendix.**AP® Test Prep**include multiple-choice and “Answer and Explain” exercises. The latter exercises are not full-blown free-response questions like those on the exam, but they are designed to build up to such questions by getting students in the habit of explaining their work. Free-response questions are included in the AP quick quizzes and chapter quizzes.**Writing to Learn**exercises give students opportunities to communicate their thinking and demonstrate their understanding of important ideas.**Extending the Idea**s exercises go beyond what is presented in the section. These exercises are challenging extensions of the material in the text.

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