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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2023
The easy way to understand and retain all the concepts taught in pre-calculus classes Pre-Calculus All-in-One For Dummies is a great resource if you want to do you best in Pre-Calculus. Packed with lessons, examples, and practice problems in the book, plus extra chapter quizzes online, it gives you absolutely everything you need to succeed in pre-calc. Unlike your textbook, this book presents the essential topics clearly and concisely, so you can really understand the stuff you learn in class, score high on your tests (including the AP Pre-Calculus exam!), and get ready to confidently move ahead to upper-level math courses. And if you need a refresher before launching into calculus, look no further--this book has your back. * Review what you learned in algebra and geometry, then dig into pre-calculus * Master logarithms, exponentials, conic sections, linear equations, and beyond * Get easy-to-understand explanations that match the methods your teacher uses * Learn clever shortcuts, test-taking tips, and other hacks to make your life easier Pre-Calculus All-in-One For Dummies is the must-have resource for students who need to review for exams or just want a little (or a lot of!) extra help understanding what's happening in class.
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Table of Contents
Cover
Title Page
Copyright
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Unit 1: Getting Started with Pre-Calculus
Chapter 1: Preparing for Pre-Calculus
Recapping Pre-Calculus: An Overview
Checking in on Number Basics and Processes
Looking at Visual Statements: When Math Follows Form with Function
Getting Yourself a Graphing Calculator
Practice Questions Answers and Explanations
Whaddya Know? Chapter 1 Quiz
Answers to Chapter 1 Quiz
Chapter 2: Operating with Real Numbers
Describing Numbers on the Number Line
Solving Inequalities
Working with Radicals and Exponents
Practice Questions Answers and Explanations
Whaddya Know? Chapter 2 Quiz
Answers to Chapter 2 Quiz
Chapter 3: Cementing the Building Blocks of Pre-Calculus Functions
Identifying Special Function Types and Their Graphs
Dealing with Parent Functions and Their Graphs
Setting the Stage for Rational Functions
Putting the Results to Work: Graphing Rational Functions
Practice Questions Answers and Explanations
Whaddya Know? Chapter 3 Quiz
Answers to Chapter 3 Quiz
Chapter 4: Operating on Functions
Transforming the Parent Graphs
Sharpen Your Scalpel: Operating on Functions
Turning Inside Out with Inverse Functions
Practice Questions Answers and Explanations
Whaddya Know? Chapter 4 Quiz
Answers to Chapter 4 Quiz
Unit 2: Getting the Grip on Graphing
Chapter 5: Graphing Polynomial Functions
Understanding Degrees and Roots
Factoring a Polynomial Expression
Finding the Roots of a Factored Equation
Cracking a Quadratic Equation When It Won’t Factor
Solving Polynomial Equations with a Degree Higher Than Two
Put It in Reverse: Using Solutions to Find Factors
Graphing Polynomials
Practice Questions Answers and Explanations
Whaddya Know? Chapter 5 Quiz
Answers to Chapter 5 Quiz
Chapter 6: Exponential and Logarithmic Functions
Exploring Exponential Functions
Logarithms: The Inverse of Exponential Functions
Base Jumping to Simplify and Solve Equations
Growing Exponentially: Word Problems in the Kitchen
Practice Questions Answers and Explanations
Whaddya Know? Chapter 6 Quiz
Answers to Chapter 6 Quiz
Chapter 7: Piece-Wise and Greatest-Integer Functions
Looking at Functions with More Than One Rule: Piece-Wise Functions
Grappling with the Greatest-Integer Function
Practice Questions Answers and Explanations
Whaddya Know? Chapter 7 Quiz
Answers to Chapter 7 Quiz
Unit 3: The Essentials of Trigonometry
Chapter 8: Circling In on Angles
Introducing Radians and Relating to Degrees
Trig Ratios: Taking Right Triangles a Step Further
Understanding How Trig Ratios Work on the Coordinate Plane
Digesting Special Triangle Ratios
Triangles and the Unit Circle: Working Together for the Common Good
Practice Questions Answers and Explanations
Whaddya Know? Chapter 8 Quiz
Answers to Chapter 8 Quiz
Chapter 9: Homing In on the Friendliest Angles
Building on the Unit Circle
Solving Equations Using Inverse Functions
Measuring Arcs: When the Circle Is Put in Motion
Practice Questions Answers and Explanations
Whaddya Know? Chapter 9 Quiz
Answers to Chapter 9 Quiz
Chapter 10: Picturing Basic Trig Functions and Reciprocal Functions
Drafting the Sine and Cosine Parent Graphs
Graphing Tangent and Cotangent Parent Graphs
Putting Secant and Cosecant in Parent Graphs Pictures
Practice Questions Answers and Explanations
Whaddya Know? Chapter 10 Quiz
Answers to Chapter 10 Quiz
Chapter 11: Graphing and Transforming Trig Functions
Transforming Trig Graphs
Practice Questions Answers and Explanations
Whaddya Know? Chapter 11 Quiz
Answers to Chapter 11 Quiz
Unit 4: Identities and Special Triangles
Chapter 12: Identifying with Trig Identities: The Basics
Keeping the End in Mind: A Quick Primer on Identities
Lining Up the Means to the End: Basic Trig Identities
Tackling Difficult Trig Proofs: Some Techniques to Know
Practice Questions Answers and Explanations
Whaddya Know? Chapter 12 Quiz
Answers to Chapter 12 Quiz
Chapter 13: Advancing with Advanced Identities
Finding Trig Functions of Sums and Differences
Doubling an Angle and Finding Its Trig Value
Taking Trig Functions of Common Angles Divided in Two
A Glimpse of Calculus: Traveling from Products to Sums and Back
Eliminating Exponents with Power-Reducing Formulas
Practice Questions Answers and Explanations
Whaddya Know? Chapter 13 Quiz
Answers to Chapter 13 Quiz
Chapter 14: Getting the Slant on Oblique Triangles
Solving a Triangle with the Law of Sines
Conquering a Triangle with the Law of Cosines
Filling in the Triangle by Calculating Area
Making Triangles Work for You
Practice Questions Answers and Explanations
Whaddya Know? Chapter 14 Quiz
Answers to Chapter 14 Quiz
Unit 5: Analytic Geometry
Chapter 15: Coordinating with Complex Numbers
Understanding Real versus Imaginary
Combining Real and Imaginary: The Complex Number System
Graphing Complex Numbers
Grasping the usefulness of complex numbers
Practice Questions Answers and Explanations
Whaddya Know? Chapter 15 Quiz
Answers to Chapter 15 Quiz
Chapter 16: Warming Up to Polar Coordinates
Plotting around a Pole: Polar Coordinates
Picturing polar equations
Practice Questions Answers and Explanations
Whaddya Know? Chapter 16 Quiz
Answers to Chapter 16 Quiz
Chapter 17: Relating Conics to Sliced Cones
Cone to Cone: Identifying the Four Conic Sections
Going Round and Round: Graphing Circles
Riding the Ups and Downs with Parabolas
The Fat and the Skinny on the Ellipse
Join Two Curves and Get a Hyperbola
Expressing Conics Outside the Realm of Cartesian Coordinates
Practice Questions Answers and Explanations
Whaddya Know? Chapter 17 Quiz
Answers to Chapter 17 Quiz
Unit 6: Systems, Sequences, and Series
Chapter 18: Streamlining Systems of Equations
A Primer on Your System-Solving Options
Graphing Equations to Find Solutions
Algebraic Solutions of Two-Equation Systems
Solving Systems with More Than Two Equations
Decomposing Partial Fractions
Surveying Systems of Inequalities
Practice Questions Answers and Explanations
Whaddya Know? Chapter 18 Quiz
Answers to Chapter 18 Quiz
Chapter 19: Making Matrices Work
Introducing Matrices: The Basics
Simplifying Matrices to Ease the Solving Process
Making Matrices Work for You
Practice Questions Answers and Explanations
Whaddya Know? Chapter 19 Quiz
Answers to Chapter 19 Quiz
Chapter 20: Sequences and Series
Speaking Sequentially: Grasping the General Method
Finding the Difference between Terms: Arithmetic Sequences
Looking at Ratios and Consecutive Paired Terms: Geometric Sequences
Creating a Series: Summing Terms of a Sequence
Practice Questions Answers and Explanations
Whaddya Know? Chapter 20 Quiz
Answers to Chapter 20 Quiz
Chapter 21: Expanding Binomials for the Real World
Expanding with the Binomial Theorem
Working Backwards
Practice Questions Answers and Explanations
Whaddya Know? Chapter 21 Quiz
Answers to Chapter 21 Quiz
Unit 7: Onward to Calculus
Chapter 22: Lining Up the Tools
Scoping Out the Changes: Pre-Calculus to Calculus
Understanding Your Limits
Finding the Limit of a Function
Operating on Limits: The Limit Laws
Calculating the Average Rate of Change
Exploring Continuity in Functions
Practice Questions Answers and Explanations
Whaddya Know? Chapter 22 Quiz
Answers to Chapter 22 Quiz
Chapter 23: Proceeding with Successful Procedures
Figure Out What the Problem Is Asking
Draw Pictures (the More the Better)
Plan Your Attack: Identify Your Targets
Write Down Any Formulas
Show Each Step of Your Work
Know When to Quit
Check Your Answers
Practice Plenty of Problems
Keep Track of the Order of Operations
Use Caution When Dealing with Fractions
Practice Questions Answers and Explanations
Whaddya Know? Chapter 23 Quiz
Answers to Chapter 23 Quiz
Index
About the Author
Advertisement Page
Connect with Dummies
End User License Agreement
List of Tables
Chapter 5
Table 5-1 Using the Roots to Frame the Graph
Chapter 6
Table 6-1 The Values of
x
in Two Exponential Functions
Chapter 9
Table 9-1 Function Values for Basic Angles
Chapter 10
TABLE 10-1 Finding Out Where tan Is Undefined
TABLE 10-2 Spotting Where cot Is Undefined
Chapter 17
Table 17-1 Ellipse Parts
Table 17-2 Plug and Chug
t
Values from the Interval
Chapter 21
Table 21-1 Binomial Coefficients
Chapter 22
Table 22-1 Finding a Limit Analytically
List of Illustrations
Chapter 1
FIGURE 1-1: The graph of .
Chapter 2
FIGURE 2-1: Number lines.
FIGURE 2-2: The solutions to and .
FIGURE 2-3: The solutions to and on a number line.
FIGURE 2-4: The graph of on a number line.
FIGURE 2-5: The graph of the OR statement OR .
Chapter 3
FIGURE 3-1: Graphing a quadratic function.
FIGURE 3-2: Graphing the parent square-root function .
FIGURE 3-3: Staying positive with the graph of an absolute-value function.
FIGURE 3-4: Putting the cubic parent function in graph form.
FIGURE 3-5: The graph of a cube-root function.
FIGURE 3-6: The graph of
f
(
x
) with asymptotes and intercepts filled in.
FIGURE 3-7: The final graph of
f
(
x
).
FIGURE 3-8: The graph of
g
(
x
), which is a rational function with equal degrees ...
FIGURE 3-9: The graph of
h
(
x
), which has an oblique asymptote.
Chapter 4
FIGURE 4-1: Graphing the transformations and .
FIGURE 4-2: The graph of two transformations: and .
FIGURE 4-3: The graph of two horizontal shifts: 3 to the right and 2 to the lef...
FIGURE 4-4: The graph of vertical shifts: and .
FIGURE 4-5: A reflection over a horizontal line mirrors up and down.
FIGURE 4-6: A reflection over a vertical line mirrors side to side.
FIGURE 4-7: A view of multiple transformations.
FIGURE 4-8: Graphing the function .
FIGURE 4-9: The graphs of (a) and (b) .
FIGURE 4-10: The graph of and its inverse.
Chapter 5
FIGURE 5-1: Illustrating the leading-coefficient test.
FIGURE 5-2: Graphing the polynomial
FIGURE 5-3: Graphing the polynomial
Chapter 6
FIGURE 6-1: The graphs of the exponential functions and .
FIGURE 6-2: The horizontal shift (a) plus the vertical shift (b).
FIGURE 6-3: The snail rule helps you remember how to change exponentials and lo...
FIGURE 6-4: Graphing the inverse function .
FIGURE 6-5: Graphing the logarithm .
FIGURE 6-6: The transformed exponential function.
FIGURE 6-7: You change the domain and range to get the inverse function (log).
Chapter 7
FIGURE 7-1: This piece-wise function is discontinuous.
FIGURE 7-2: The greatest-integer (floor) function.
Chapter 8
FIGURE 8-1: Angles co-terminal with a angle.
FIGURE 8-2: Find the sine with two sides given.
FIGURE 8-3: One ladder plus one building equals one cosine problem.
FIGURE 8-4: Using tangent to solve a word problem.
FIGURE 8-5: Finding the hypotenuse of a right triangle when given a point on th...
FIGURE 8-6: A –– right triangle.
FIGURE 8-7: A square with a diagonal.
FIGURE 8-8: A –– right triangle.
FIGURE 8-9: Finding the other sides of a 30–60–90 triangle when you know the hy...
FIGURE 8-10: A –– triangle drawn on the unit circle.
FIGURE 8-11: The whole unit circle.
FIGURE 8-12: A 45er triangle positioned in the third quadrant.
Chapter 9
FIGURE 9-1: Finding the solution angle, given the reference angle.
FIGURE 9-2: A angle on the coordinate plane.
FIGURE 9-3: The variables involved in computing arc length.
FIGURE 9-4: The arc length for an angle measurement of .
Chapter 10
FIGURE 10-1: The parent graph of sine, .
FIGURE 10-2: The parent graph of cosine, .
FIGURE 10-3: The parent graph of tangent, .
FIGURE 10-4: The parent graph of cotangent, .
FIGURE 10-5: The graph of cosine reveals the asymptotes of secant.
FIGURE 10-6: The parent graph of secant, .
FIGURE 10-7: The graphs of and its reciprocal .
Chapter 11
FIGURE 11-1: The sinusoidal axis and amplitude of a trig function graph.
FIGURE 11-2: The graphs of transformations of sine.
FIGURE 11-3: Creating period changes on function graphs.
FIGURE 11-4: The graphs of and its reflection.
FIGURE 11-5: Shifting the parent graph of sine to the right by .
FIGURE 11-6: Moving the graph of sine up by 3 after moving to the right.
FIGURE 11-7: Changing the amplitude and flipping the cosine.
FIGURE 11-8: Changing the amplitude, flipping, and changing the period.
FIGURE 11-9: The graph of
FIGURE 11-10: The amplitude is half as tall in the changed graph.
FIGURE 11-11: The transformed graph of
FIGURE 11-12: The graph of shows a period of .
FIGURE 11-13: The transformed graph of .
FIGURE 11-14: Graphing the cosine function first.
FIGURE 11-15: The transformed secant graph
FIGURE 11-16: A transformed sine graph.
FIGURE 11-17: The transformed cosecant graph, based on the sine graph.
Chapter 12
FIGURE 12-1: The angle in quadrant II.
Chapter 13
FIGURE 13-1: The unit circle showing angles in radians with common denominators...
FIGURE 13-2: Drawing pictures helps you visualize the missing pieces of info.
Chapter 14
FIGURE 14-1: A labeled ASA triangle.
FIGURE 14-2: A labeled AAS triangle.
FIGURE 14-3: Non-congruent triangles that follow the SSA format.
FIGURE 14-4: Two possible representations of an SSA triangle.
FIGURE 14-5: The two possible triangles overlapping.
FIGURE 14-6: The setup of an SSA triangle with only one solution set.
FIGURE 14-7: A triangle with no solution.
FIGURE 14-8: Determining angles when you know three side lengths.
FIGURE 14-9: An SSA triangle that calls for the Law of Cosines.
Chapter 15
FIGURE 15-1: Comparing the graphs of a real and an imaginary number.
FIGURE 15-2: Complex numbers plotted on the complex coordinate plane.
Chapter 16
FIGURE 16-1: A polar coordinate plane.
FIGURE 16-2: Visualizing simple and complex polar coordinates.
FIGURE 16-3: A polar and (
x
,
y
) coordinate mapped in the same plane.
FIGURE 16-4: The same points, A and B, as rectangular and polar.
FIGURE 16-5: Graphing .
FIGURE 16-6: The Archimedes spiral and three-petal rose.
Chapter 17
FIGURE 17-1: Cutting cones with a plane to get conic sections.
FIGURE 17-2: Graphing a circle centered at the origin.
FIGURE 17-3: Graphing a circle not centered at the origin.
FIGURE 17-4: Vertical and horizontal parabolas based at the origin.
FIGURE 17-5: Graphing transformed horizontal parabolas.
FIGURE 17-6: Identifying the parts of a vertical parabola.
FIGURE 17-7: Finding all the parts of the parabola .
FIGURE 17-8: The graph of a horizontal parabola.
FIGURE 17-9: Graphing a parabola to find a maximum value from a word problem.
FIGURE 17-10: An ellipse drawn with a pencil.
FIGURE 17-11: The parts of a horizontal ellipse and a vertical ellipse.
FIGURE 17-12: The many points and parts of a horizontal ellipse.
FIGURE 17-13: A horizontal and a vertical hyperbola.
FIGURE 17-14: Using a rectangle to graph a hyperbola and its asymptotes.
FIGURE 17-15: Graphing a parametric curve.
FIGURE 17-16: The graph of an ellipse in polar coordinates.
Chapter 18
FIGURE 18-1: The intersection of two lines.
FIGURE 18-2: The intersection of a line and parabola.
FIGURE 18-3: The intersection of a polynomial and parabola.
FIGURE 18-4: Intersections of curves.
FIGURE 18-5: Graphing a nonlinear system of inequalities.
Chapter 19
FIGURE 19-1: Computing the determinant of a matrix.
Chapter 21
FIGURE 21-1: Determining coefficients with Pascal’s triangle.
FIGURE 21-2: Binomial coefficients of powers of through .
Chapter 22
FIGURE 22-1: Functions with limit at 3 and with no limit at 0.
FIGURE 22-2: Observing the limit of a function graphically.
FIGURE 22-3: The graph of the function .
FIGURE 22-4: Slopes changing with the curvature of the function.
FIGURE 22-5: Finding the slope at a point using a tangent line.
FIGURE 22-6: The graph of a removable discontinuity is indicated with a hole.
FIGURE 22-7: A graph showing both a removable and non-removable discontinuity.
Chapter 23
FIGURE 23-1: The original garden and the new, larger garden.
FIGURE 23-2: Labelling the figure with equations.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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Pre-Calculus All-in-One For Dummies®
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2024 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Introduction
Here you are: ready to take on these challenging pre-calculus topics — possibly on your way to calculus! Believe it or not, it was calculus that was responsible for my switching majors and taking on the exciting world of mathematics!
Pre-calculus books and classes are wonderful ways of taking the mathematics you’ve studied in the past and bolstering the experience with new, exciting, and challenging material. Some of what is presented in pre-calculus is review, but studying it and adding on to the topics is what will make you even more of a success in your next endeavor.
Maybe some of the concepts you’ve already covered in pre-calculus have given you a hard time, or perhaps you just want more practice. Maybe you’re deciding whether you even want to take pre-calculus and then calculus at all. This book fits the bill to help you with your decision for all those reasons. And it’s here to encourage you on your pre-calculus adventure.
You’ll find this book has many examples, valuable practice problems, and complete explanations. In instances where you feel you may need a more thorough explanation, please refer to Pre-Calculus For Dummies or Pre-Calculus Workbook For Dummies by Mary Jane Sterling (Wiley). This book, however, is a great stand-alone resource if you need extra practice or want to just brush up in certain areas.
About This Book
Pre-calculus can be a starting point, a middle point, and even a launching point. When you realize that you already know a whole bunch from Algebra I and Algebra II, you’ll see that pre-calculus allows you to use that information in a new way. Before you get ready to start this new adventure, you need to know a few things about this book.
This book isn’t a novel. It’s not meant to be read in order from beginning to end. You can read any topic at any time, but it’s structured in such a way that it follows a “typical” curriculum. Not everyone agrees on exactly what makes pre-calculus pre-calculus. So this book works hard at meeting the requirements of all those curriculums; hopefully, this is a good representation of any pre-calculus course.
Here are two different suggestions for using this book:
Look up what you need to know when you need to know it. The index and the table of contents direct you where to look.
Start at the beginning and read straight through. This way, you may be reminded of an old topic that you had forgotten (anything to get those math wheels turning inside your head). Besides, practice makes perfect, and the problems in this book are a great representation of the problems found in pre-calculus textbooks.
For consistency and ease of navigation, this book uses the following conventions:
Math terms are
italicized
when they’re introduced or defined in the text.
Variables are
italicized
to set them apart from letters.
The symbol used when writing imaginary numbers is a lowercase
i.
Foolish Assumptions
I don’t assume that you love math the way I do, but I do assume that you picked this book up for a reason special to you. Maybe you want a preview of the course before you take it, or perhaps you need a refresher on the topics in the course, or maybe your kid is taking the course and you’re trying to help them to be more successful.
It has to be assumed that you’re willing to put in some time and effort here. Pre-calculus topics include lots of algebraic equations, geometric theorems and rules, and trigonometry. You will see how these topics are used and intertwined, but you may need to go deeper into one or more of the topics than what is presented here.
And it’s pretty clear that you are a dedicated and adventurous person, just by the fact that you’re picking up this book and getting serious about what it has to offer. If you’ve made it this far, you’ll go even farther!
Icons Used in This Book
Throughout this book you’ll see icons in the margins to draw your attention to something important that you need to know.
You see this icon when I present an example problem whose solution I walk you through step by step. You get a problem and a detailed answer.
Tips are great, especially if you wait tables for a living! These tips are designed to make your life easier, which are the best tips of all!
The material following this icon is wonderful mathematics; it’s closely related to the topic at hand, but it’s not absolutely necessary for your understanding of the material being presented. You can take it or leave it — you’ll be fine just taking note and leaving it behind as you proceed through the section.
This icon is used in one way: It asks you to remember old material from a previous math course.
Warnings are big red flags that draw your attention to common mistakes that may trip you up.
When you see this icon, it’s time to tackle some practice questions. Answers and explanations appear in a separate section near the end of the chapter.
Beyond the Book
No matter how well you understand the concepts of algebra, you’ll likely come across a few questions where you don’t have a clue. Be sure to check out the free Cheat Sheet for a handy guide that covers tips and tricks for answering pre-calculus questions. To get this Cheat Sheet, simply go to www.dummies.com and type Pre-Calculus All In One For Dummies in the Search box.
The online quiz that comes free with this book contains over 300 questions so you can really hone your pre-calculus skills! To gain access to the online practice, all you have to do is register. Just follow these simple steps:
Register your book or ebook at Dummies.com to get your PIN. Go to
www.dummies.com/go/getaccess
.
Select your product from the dropdown list on that page.
Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.
If you do not receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.
Now you’re ready to go! You can come back to the practice material as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Your registration is good for one year from the day you activate your PIN.
Where to Go from Here
Pick a starting point in the book and go practice the problems there. If you’d like to review the basics first, start at Chapter 1. If you feel comfy enough with your algebra skills, you may want to skip that chapter and head over to Chapter 2. Most of the topics there are reviews of Algebra II material, but don’t skip over something because you think you have it under control. You’ll find in pre-calculus that the level of difficulty in some of these topics gets turned up a notch or two. Go ahead — dive in and enjoy the world of pre-calculus!
If you’re ready for another area of mathematics, look for a couple more of my titles: Trigonometry For Dummies and Linear Algebra For Dummies.
Unit 1
Getting Started with Pre-Calculus
IN THIS UNIT …
Sharpening algebraic skills.
Reviewing number systems and their uses.
Describing basic function types.
Performing operations on real numbers and functions.
Chapter 1
Preparing for Pre-Calculus
IN THIS CHAPTER
Refreshing your memory on numbers and variables
Accepting the importance of graphing
Preparing for pre-calculus by understanding the vocabulary
Pre-calculus is the bridge (drawbridge, suspension bridge, covered bridge) between Algebra II and calculus. In its scope, you review concepts you’ve seen before in math, but then you quickly build on them. You see some brand-new ideas, and even those build on the material you’ve seen before; the main difference is that the problems now get even more interesting and challenging (for example, going from linear systems to nonlinear systems). You keep on building until the end of the bridge span, which doubles as the beginning of calculus. Have no fear! What you find here will help you cross the bridge (toll free).
Because you’ve probably already taken Algebra I, Algebra II, and geometry, it’s assumed throughout this book that you already know how to do certain things. Just to make sure, though, I address some particular items in this chapter in a little more detail before moving on to the material that is pre-calculus.
If there is any topic in this chapter that you’re not familiar with, don’t remember how to do, or don’t feel comfortable doing, I suggest that you pick up another For Dummies math book and start there. If you need to do this, don’t feel like a failure in math. Even pros have to look up things from time to time. Use these books like you use encyclopedias or the Internet — if you don’t know the material, just look it up and get going from there.
Recapping Pre-Calculus: An Overview
Don’t you just love movie previews and trailers? Some people show up early to movies just to see what’s coming out in the future. Well, consider this section a trailer that you see a couple months before the Pre-Calculus For Dummies movie comes out! The following list presents some items you’ve learned before in math, and some examples of where pre-calculus will take you next.
Algebra I and II: Dealing with real numbers and solving equations and inequalities.
Pre-calculus: Expressing inequalities in a new way called interval notation.
You may have seen solutions to inequalities in set notation, such as . This is read in inequality notation as . In pre-calculus, you often express this solution as an interval: . (For more, see Chapter 2.)
Geometry: Solving right triangles, whose sides are all positive.
Pre-calculus: Solving non-right triangles, whose sides aren’t always represented by positive numbers.
You’ve learned that a length can never be negative. Well, in pre-calculus you sometimes use negative numbers for the lengths of the sides of triangles. This is to show where these triangles lie in the coordinate plane (they can be in any of the four quadrants).
Geometry/trigonometry: Using the Pythagorean Theorem to find the lengths of a triangle’s sides.
Pre-calculus: Organizing some frequently used angles and their trig function values into one nice, neat package known as the unit circle (see Unit 3).
In this book, you discover a handy shortcut to finding the sides of triangles — a shortcut that is even handier for finding the trig values for the angles in those triangles.
Algebra I and II: Graphing equations on a coordinate plane.
Pre-calculus: Graphing in a brand-new way with the polar coordinate system (see Chapter 16).
Say goodbye to the good old days of graphing on the Cartesian coordinate plane. You have a new way to graph, and it involves goin’ round in circles. I’m not trying to make you dizzy; actually, polar coordinates can make you some pretty pictures.
Algebra II: Dealing with imaginary numbers.
Pre-calculus: Adding, subtracting, multiplying, and dividing complex numbers gets boring when the complex numbers are in rectangular form . In pre-calculus, you become familiar with something new called polar form and use that to find solutions to equations you didn’t even know existed.
Checking in on Number Basics and Processes
When entering pre-calculus, you should be comfy with sets of numbers (natural, integer, rational, and so on). By this point in your math career, you should also know how to perform operations with numbers. You can find a quick review of these concepts in this section. Also, certain properties hold true for all sets of numbers, and it’s helpful to know them by name. I review them in this section, too.
Understanding the multitude of number types: Terms to know
Mathematicians love to name things simply because they can; it makes them feel special. In this spirit, mathematicians attach names to many sets of numbers to set them apart and cement their places in math students’ heads for all time.
The set of natural or counting numbers: {1, 2, 3 …}
. Notice that the set of natural numbers doesn’t include 0.
The set of whole numbers: {0, 1, 2, 3 …}
. The set of whole numbers consists of all the natural numbers plus the number 0.
The set of integers: {… –3, –2, –1, 0, 1, 2, 3 …}. The set of integers includes all positive and negative natural numbers and 0.
Dealing with integers is like dealing with money: Think of positives as having it and negatives as owing it. This becomes important when operating on numbers (see the next section).
The set of rational numbers: The numbers that can be expressed as a fraction where the numerator and the denominator are both integers. The word rational comes from the idea of a ratio (fraction or division) of two integers.
Examples of rational numbers include (but in no way are limited to) , , and 0.23. A rational number is any number in the form where p and q are integers, but q is never 0. If you look at any rational number in decimal form, you notice that the decimal either stops or repeats.
The set of irrational numbers: All numbers that can’t be expressed as fractions. Examples of irrational numbers include , , and .
The decimal value of an irrational number never ends and never repeats.
The set of all real numbers: All the sets of numbers previously discussed. For an example of a real number, think of a number … any number. Whatever it is, it’s real. Any number from the previous bullets works as an example. The numbers that aren’t real numbers are imaginary.
Like telemarketers and pop-up ads on the web, real numbers are everywhere; you can’t get away from them — not even in pre-calculus. Why? Because they include all numbers except the following.
A fraction with a zero as the denominator:
Such numbers don’t exist and are called
undefined
.
The square root of a negative number:
These numbers are part of
complex numbers
; the negative root is the
imaginary
part (see
Chapter 15
). And this extends to any even root of a negative number.
Infinity:
Infinity is a concept, not an actual number. It describes a behavior.
The set of imaginary numbers: square roots of negative numbers.
Imaginary numbers have an imaginary unit, like
i,
4
i,
and –2
i.
Imaginary numbers used to be considered to be made-up numbers, but mathematicians soon realized that these numbers pop up in the real world. They are still called imaginary because they’re square roots of negative numbers, but they are a part of the language of mathematics. The imaginary unit is defined as . (For more on these numbers, head to
Chapter 15
.)
The set of complex numbers: the sum or difference of a real number and an imaginary number. Complex numbers include these examples: ,, and . However, they also cover all the previous lists, including the real numbers (3 is the same thing as ) and imaginary numbers (2i is the same thing as ).
The set of complex numbers is the most complete set of numbers in the math vocabulary because it includes real numbers (any number you can possibly think of), imaginary numbers (i), and any combination of the two.
In Questions 1 to 4, determine the different ways you can refer to each number. Your choices are: natural, whole, integer, rational, irrational, and real. Select all that apply.
1
2
3
4
Looking at the fundamental operations you can perform on numbers
From positives and negatives to fractions, decimals, and square roots, you should know how to perform all the basic operations on all real numbers. These operations include adding, subtracting, multiplying, dividing, taking powers of, and taking roots of numbers. The order of operations is the way in which you perform these operations.
The mnemonic device used most frequently to remember the order is PEMDAS, which stands for
P
arentheses (and other grouping devices such as brackets and division lines)
E
xponents (and roots, which can be written as exponents)
M
ultiplication and
D
ivision (done in order from left to right)
A
ddition and
S
ubtraction (done in order from left to right)
One type of operation that is often overlooked or forgotten about is absolute value. Absolute value gives you the distance from zero on the number line. Absolute value should be included with the parentheses step because you have to consider what’s inside the absolute-value bars first (because the bars are a grouping device). Don’t forget that absolute value is always positive or zero. Hey, even if you’re walking backward, you’re still walking!
Q. Simplify the expression using the order of operations: .
A.. First, simplify the numerator. Do this by squaring the binomial, multiplying through by , and then combining like terms.
Next, factor the from the terms in the numerator and simplify the denominator.
Divide the numerator and denominator by 3. And, finally, rationalize the denominator.
In Questions 5 and 6, simplify each expression using the order of operations.
5
6
Knowing the properties of numbers: Truths to remember
Remembering the properties of numbers is important because you use them consistently in pre-calculus. You may not often use these properties by name in pre-calculus, but you do need to know when to use them. The following list presents properties of numbers.
Reflexive property:
.
For example, .
Symmetric property: If
,
then
.
For example, if , then .
Transitive property: If
and
,
then
.
For example, if and , then .
Commutative property of addition:
.
For example, .
Commutative property of multiplication:
.
For example, .
Associative property of addition:
.
For example, .
Associative property of multiplication:
.
For example, .
Additive identity:
.
For example, .
Multiplicative identity:
.
For example, .
Additive inverse property:
.
For example, .
Multiplicative inverse property:
.
For example, . (Remember: .)
Distributive property:
.
For example, .
Multiplicative property of zero:
.
For example, .
Zero-product property: If
, then
or
.
For example, if , then or .
If you’re trying to perform an operation that isn’t on the previous list, then the operation probably isn’t correct. After all, algebra has been around since 1600 B.c., and if a property exists, someone has probably already discovered it. For example, it may look inviting to say that , but that’s incorrect. The correct process and answer is . Knowing what you can’t do is just as important as knowing what you can do.
Q. Use the associative and commutative properties to simplify the expression: .
A. 11x + 4y + 9. Rewrite with the variable terms grouped followed by the numbers: . Now combine the like terms: .
Q. Use the distributive property to simplify the expression: .
A. –17x2 – 3x + 1. First, distribute the and over their respective parentheses; then combine like terms: .
In Questions 7 and 8, simplify each using the distributive, associative, and commutative properties.
7
8
Looking at Visual Statements: When Math Follows Form with Function
Graphs are great visual tools. They’re used to display what’s going on in math problems, in companies, and in scientific experiments. For instance, graphs can be used to show how something (like the price of real estate) changes over time. Surveys can be taken to get facts or opinions, the results of which can be displayed in a graph. Open up the newspaper on any given day and you can find a graph in there somewhere.
Hopefully, the preceding paragraph answers the question of why you need to understand how to construct graphs. Even though in real life you don’t walk around with graph paper and a pencil to make the decisions you face, graphing is vital in math and in other walks of life. Regardless of the absence of graph paper, graphs are indeed everywhere.
For example, when scientists go out and collect data or measure things, they often arrange the data as x and y values. Typically, scientists are looking for some kind of general relationship between these two values to support their hypotheses. These values can then be graphed on a coordinate plane to show trends in data. For example, a good scientist may show with a graph that the more you read this book, the more you understand pre-calculus! (Another scientist may show that people with longer arms have bigger feet. Boring!)
Using basic terms and concepts
Graphing equations is a huge part of pre-calculus, and eventually calculus, so it’s good to review the basics of graphing before getting into the more complicated and unfamiliar graphs that you will see later in the book.
Although some of the graphs in pre-calculus will look very familiar, some will be new — and possibly intimidating. This book will get you more familiar with these graphs so that you will be more comfortable with them. However, the information in this chapter is mostly information that you remember from Algebra II. You did pay attention then, right?
Each point on the coordinate plane on which you construct graphs — that is, a plane made up of the horizontal (x-) axis and the vertical (y-) axis, creating four quadrants — is called a coordinate pair (x,y), also often referred to as a Cartesian coordinate pair.
The name Cartesian coordinates comes from the French mathematician and philosopher who invented all this graphing stuff, René Descartes. Descartes worked to merge algebra and Euclidean geometry (flat geometry), and his work was influential in the development of analytic geometry, calculus, and cartography.
A
relation
is a set (which can be empty, but in this book I only consider nonempty sets) of ordered pairs that can be graphed on a coordinate plane. Each relation is kind of like a computer that expresses
x
as input and
y
as output. You know you’re dealing with a relation when the set is given in those curly brackets (like these: { }) and has one or more points inside. For example, is a relation with three ordered pairs. Think of each point as (input, output) just like from a computer.
The
domain
of a relation is the set of all the input values, usually listed from least to greatest. For example, the domain of set R is . The
range
is the set of all the output values, also often listed from least to greatest. For example, the range of R is . If any value in the domain or range is repeated, you don’t have to list it twice. Usually, the domain is the
x-
variable and the range is
y.
If different variables appear, such as m and n, input (domain) and output (range) usually go alphabetically, unless you’re told otherwise. In this case, m would be your input/domain and n would be your output/range. But when written as a point, a relation is always (input, output).
Graphing linear equalities and inequalities
When you first figured out how to graph a line on the coordinate plane, you learned to pick domain values (x) and plug them into the equation to solve for the range values (y). Then, you went through the process multiple times, expressed each pair as a coordinate point, and connected the dots to make a line. Some mathematicians call this the ol’ plug-and-chug method.
After a bit of that tedious work, somebody said to you, “Hold on! You can use a shortcut.” That shortcut involves an equation called slope-intercept form, and it’s expressed as . The variable m stands for the slope of the line (see the next section, “Gathering information from graphs”), and b stands for the y-intercept (or where the line crosses the y-axis). You can change equations that aren’t written in slope-intercept form to that form by solving for y. For example, graphing requires you to subtract 2x from both sides first to get . Then you divide every term by –3 to get .
In the first quadrant, this graph starts at –4 on the y-axis; to find the next point, you move up 2 and right 3 (using the slope). Slope is often expressed as a fraction because it’s rise over run — in this case .
Q. Write the linear equation in slope-intercept form.
. First, subtract 4x and add 100 to each side.
A. Then, divide each term by –5: . The line has a slope of and a y-intercept at .
Inequalities are used for comparisons, which are a big part of pre-calculus. They show a relationship between two expressions (greater-than, less-than, greater-than-or-equal-to, and less-than-or-equal-to). Graphing inequalities starts exactly the same as graphing equalities, but at the end of the graphing process (you still put the equation in slope-intercept form and graph), you have two decisions to make:
Is the line
dashed,
indicating
y
< or
y
>, or is the line
solid,
indicating or ?
Do you shade under the line for y < or , or do you shade above the line for y > or ? Simple inequalities (like x < 3) express all possible answers. For inequalities, you show all possible answers by shading the side of the line that works in the original equation.
For example, when graphing , you follow these steps (refer to Figure 1-1):
Graph the line by starting off at –5 on the
y-
axis, marking that point, and then moving up 2 and right 1 to find a second point.
When connecting the dots, you produce a straight dashed line through the points.
Shade on the bottom half of the graph to show all possible points in the solution.
Check your choice on the shading by checking a point. You have (5,0) in your shaded area, so inserting it into , you have , which is a true statement. It checks!
FIGURE 1-1: The graph of .
Q. Write the inequality represented by the graph.
A.. The graph of the shaded area shows all the points below the graph of . Write the inequality , using less-than-or-equal to because the line is solid. Select a point in the shaded area such as (0,0) and check to be sure the inequality is correct. Substituting, you have , which is a true statement.
9 Write the equation in slope-intercept form.
10 Write the inequality represented by the graph.
Gathering information from graphs
After getting you used to coordinate points and graphing equations of lines on the coordinate plane, typical math books and teachers begin to ask you questions about the points and lines that you’ve been graphing. The three main things you’ll be asked to find are the distance between two points, the midpoint of the segment connecting two points, and the exact slope of a line that passes through two points.
Calculating distance
Knowing how to calculate distance by using the information from a graph comes in handy in a big way, so allow this quick review of a few things first. Distance is how far apart two objects, or two points, are. To find the distance, d, between the two points and on a coordinate plane, for example, use the following formula:
You can use this equation to find the length of the segment between two points on a coordinate plane whenever the need arises. For example, to find the distance between A(–6,4) and B(2,1), first identify the parts: and ; and . Plug these values into the distance formula: . This problem simplifies to .
Finding the midpoint
Finding the midpoint of a segment pops up in pre-calculus topics like conics (see Chapter 17). To find the midpoint of the segment connecting two points, you just average their x values and y values and express the answer as an ordered pair: .
You can use this formula to find the center of various graphs on a coordinate plane, but for now you’re just finding the midpoint. You find the midpoint of the segment connecting the two points A(–6,4) and B(2,1) by using this formula. You have .
Figuring a line’s slope
When you graph a linear equation, slope plays a role. The slope of a line tells how steep the line is on the coordinate plane. When you’re given two points and and are asked to find the slope of the line between them, you use the following formula: .
If you use the points A(–6, 4) and B(2, 1) and plug the values into the formula, the slope is .
Positive slopes always move up when going to the right or move down going to the left on the plane. Negative slopes move either down when going right or up when going left. (Note that if you moved the slope down and left, it would be negative divided by negative, which has a positive result.) Horizontal lines have zero slope, and vertical lines have undefined slope.
If you ever get the different types of slopes confused, remember the skier on the ski-slope:
When they’re going uphill, they’re doing a lot of work (+ slope).
When they’re going downhill, the hill is doing the work for them (– slope).
When they’re standing still on flat ground, they’re not doing any work at all (0 slope).
When they hit a wall (the vertical line), they’re done for and they can’t ski anymore (undefined slope)!
Q. Find the distance between the points and .
A.. Using the distance formula, .
Q. Find the midpoint of the two points and .
A.. Using the midpoint formula, .
Q. Find the slope of the line through the two points and .
A.. Using the slope formula, .
Given the two points and :
11 Find the distance between the points and the midpoint of the two points.
12 Find the slope of the line that goes through the points.
Getting Yourself a Graphing Calculator
It’s highly recommended that you purchase a graphing calculator for pre-calculus work. Since the invention of the graphing calculator, the emphasis and time spent on calculations in the classroom and when doing homework have changed because the grind-it-out computation isn’t necessary. Many like doing most of the work with the calculator, but others prefer not to use one. A graphing calculator does so many things for you, and, even if you don’t use it for every little item, you can always use one to check your work on the big problems.
Many different types of graphing calculators are available, and their inner workings are all different. To figure out which one to purchase, ask for advice from someone who has already taken a pre-calculus class, and then look around on the Internet for the best deal.
Many of the more theoretical concepts in this book, and in pre-calculus in general, are lost when you use your graphing calculator. All you’re told is, “Plug in the numbers and get the answer.” Sure, you get your answer, but do you really know what the calculator did to get that answer? Nope. For this reason, this book goes back and forth between using the calculator and doing complicated problems longhand. But whether you’re allowed to use the graphing calculator or not, be smart with its use. If you plan on moving on to calculus after this course, you need to know the theory and concepts behind each topic.
The material found here can’t even begin to teach you how to use your unique graphing calculator, but the good For Dummies folks at Wiley supply you with entire books on the use of them, depending on the type you own. However, I can give you some general advice on their use. Here’s a list of hints that should help you use your graphing calculator:
Always double-check that the mode (degrees versus radians) in your calculator is set according to the problem you’re working on.
Look for a button somewhere on the calculator that says
mode.
Depending on the brand of calculator, this button allows you to change things like degrees or radians, or
f
(
x
) or . For example, if you’re working in degrees, you must make sure that your calculator knows that before you use it to solve a problem. The same goes for working in radians. Some calculators have more than ten different modes to choose from. Be careful!
Make sure you can solve for y before you try to construct a graph. You can graph anything in your graphing calculator as long as you can solve for y to write it as a function. The calculators are set up to accept only equations that have been solved for y.
Equations that you have to solve for x often aren’t true functions and aren’t studied in pre-calculus — except conic sections, and students generally aren’t allowed to use graphing calculators for this material because it’s entirely based on graphing (see Chapter 17).
