Pre-Calculus For Dummies, 2nd Edition (2012)

Introduction

Welcome to Pre-Calculus For Dummies, a nondiscriminatory, equal-opportunity book. You’re welcome to participate whether you are a genius or (like us) you need a recipe to make ice. Don’t let the title throw you. If you’ve gotten this far in math, in no way are you a dummy! You may be reading this book for a few perfectly great reasons. Maybe you need a reference book that you can actually understand (we’ve never met a pre-calc text that we liked). Perhaps your guidance counselor told you that taking pre-calc would look good on your college application, but you couldn’t care less about the subject and just want to get a good grade. Or, maybe you’re contemplating buying this book and you want to check us out to see if we’re a good match (not unlike looking at your blind date through the window before you walk into the restaurant). Regardless of why you opened up this book, it will help you navigate the tricky path that is pre-calc.

You may also be wondering, “When will I ever really use pre-calculus?” You’re not alone. Some of our students have referred to it as pre-calc-uselessness. Well, they quickly found out how wrong they were. The concepts throughout this book are used in many real-world applications.

This book has one goal and one goal only — to teach you pre-calculus in as painless a way as possible. If you thought that you could never tackle this subject and you end up with a decent grade in this class, would you mind sending us a letter? E-mail’s good too. We love to hear our students’ success stories!

About This Book

This book is not necessarily meant to be read from beginning to end. It’s structured in a way that you can flip to a particular chapter and get your needs met (those pesky needs we all have). Sometimes we may tell you to look in another chapter to get a more in-depth explanation, but we have tried to allow each chapter to stand on its own.

All vocabulary is mathematically correct and clear. We have taken liberties at some points throughout this book to make the language more approachable and likable. It’s just more fun that way.

Pre-calc is its own special math topic. You see, some states, like California, don’t have any set standards that students need to learn to officially master pre- calculus. As a result, the subject of pre-calc varies between districts, schools, and individual teachers. Because we don’t know what your teacher is going to want you to take away from this course, we’ve covered pretty much every concept in pre-calc. We may have covered areas that you’ll never be required to tackle. That’s okay. Just use this book according to your individual needs.

If you use this book only to prop open a door or as a bug smoosher, you won’t get what you need from it. We suggest two alternatives:

Look up only what you need to know when you need to know it. This book is handy for this. Use the index, the table of contents, or better yet, the quickie contents at a glance found in the very front of this book to find what you need.

Start at the beginning and read through the book, chapter by chapter. This approach is a good way to tackle this subject because the topics sometimes build on previous ones. Even if you’re a math god and you want to skim through a section that you feel you know, you may be reminded of something that you forgot. We recommend starting at the beginning and slowly working your way through the material. The more practice you have, the better.

Conventions Used in This Book

For consistency and ability to navigate easily, this book uses the following conventions:

Math terms are italicized to indicate their introduction and to help you find their definition.

Variables are also italicized to distinguish them from common letters.

The step-by-step problems are always bold to help you identify them more easily.

The symbol for imaginary numbers is a lowercase i.

Foolish Assumptions

We can’t assume that just because we absolutely love math that you share the same enthusiasm for the subject. We can assume, however, that you opened this book for a reason: You need a refresher on the subject, need to learn it for the first time, are trying to relearn it for college, or have to help your kid understand it at home. We can also assume that you have been exposed, at least in part, to many of the concepts found in this subject because pre-calc really takes geometry and Algebra II concepts to the next level.

We also assume that you’re willing to do some work. Although pre-calculus isn’t the end-all to math courses out there, it’s still a higher level math course. You’re going to have to work a bit, but you knew that, didn’t you?

We also are pretty sure that you’re an adventurous soul and have chosen to take this class, because pre-calculus is not necessarily a required high school course (in most U.S. high schools, anyway). Maybe it’s because you love math like we do, or you have nothing better to do with your life, again like us, or because the course will enhance your college application. Obviously, you managed to get through some pretty complex concepts in geometry and Algebra II. We can assume that if you made it this far, you’ll make it even farther. We’re going to help!

How This Book Is Organized

This book is broken down into four sections dealing with the most frequently taught and studied concepts in pre-calc.

Part I: Set It Up, Solve It, Graph It

The chapters in Part I begin with a review of material you should already know from Algebra II. Then we review real numbers and how to operate with them. From there we cover functions, including polynomial, rational, exponential, and logarithmic functions and graphing them, solving them, and ­performing operations on them.

Part II: The Essentials of Trigonometry

The chapters in Part II begin with a review of angles, right triangles, and trig ratios. Then we build the glorious unit circle. Graphing trig functions may or may not be a review, depending on the Algebra II course you’ve taken, so we show you how to graph the parent graph of the six basic trig functions and then explain how to transform those graphs to get to the more complicated ones.

This part also covers the harder formulas and identities for trig functions, breaking them down methodically so you can internalize each identity and truly understand them. We then move right along into simplifying trig expressions and solving for an unknown variable using those formulas and identities. Finally, this part covers how to solve triangles that are not right triangles using the Law of Sines and Law of Cosines.

Part III: Analytic Geometry and System Solving

Part III covers a multitude of pre-calc topics. It begins with understanding complex numbers and how to perform operations with them. Next comes polar-coordinate graphing and finally conics. Systems of equations live in this part, as do sequences and series and binomial expansion. Finally, this part concludes with calculus and the study of limits and continuity of functions.

Part IV: The Part of Tens

After you’ve covered everything up to this point in the book, you may be eyeing the next big math challenge: calculus. (And if you decide to stop with pre-calc, that’s okay, too.) But before you head off for even more complex concepts, you need to do two things: Pick up some good math habits to take into calculus and break any bad habits you’ve developed along the way. This part helps you with those tasks. Both ends of this spectrum are critical for success because the problems get longer and teachers’ patience for algebra errors gets shorter.

Icons Used in This Book

Throughout the book you’ll find little drawings (called icons) that are meant to draw your attention to something important or interesting to know.

Pre-calc rules are the basic rules of pre-calc. The rules marked with this icon must be observed every time to make problems come out correctly.

Math Mumbo Jumbo alerts you to information that is helpful but not required to gain full knowledge of the concept in that section.

We love Tips! When you see this icon, you know that it points out a way to make your life a lot easier. Easier is good.

You’ll see this icon when we mention an old idea that you should never forget. We use it when we want you to recall a previously learned concept or a concept from a lower math course.

Think of Warnings as big stop signs. The presence of this icon alerts you to common errors or points out something that can be a bit tricky.

Where to Go from Here

If you have a really firm background in basic algebra skills, feel free to skip Chapter 1 and head right over to Chapter 2. If you want a brush-up, we suggest reading Chapter 1. In fact, everything in Chapter 2 is also a review except interval notation. So if you’re really impatient or are a math genius, ignore everything until interval notation in Chapter 2. As you work through this book, keep in mind that many concepts in pre-calc are take-offs from Algebra II, so don’t make the mistake of completely skipping chapters because they sound familiar. They may sound familiar but are likely to include some brand-spanking-new material. We also didn’t sit next to you when you took Algebra II, so we can’t be sure what your teacher covered. So here’s a brief list of sections that may sound familiar but include new concepts that you should pay attention to:

Translating common functions

Solving polynomials

All the trig information

Complex numbers

Matrices

So where do you go from here? You get going straight into pre-calc! Good luck.