Algebra II Essentials For Dummies E-Book

Mary Jane Sterling

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2010

Seitenzahl: 221

Beschreibung

Passing grades in two years of algebra courses are required forhigh school graduation. Algebra II Essentials For Dummiescovers key ideas from typical second-year Algebra coursework tohelp students get up to speed. Free of ramp-up material, AlgebraII Essentials For Dummies sticks to the point, with contentfocused on key topics only. It provides discrete explanations ofcritical concepts taught in a typical Algebra II course, frompolynomials, conics, and systems of equations to rational,exponential, and logarithmic functions. This guide is also aperfect reference for parents who need to review critical algebraconcepts as they help students with homework assignments, as wellas for adult learners headed back into the classroom who just needa refresher of the core concepts. The Essentials For Dummies Series Dummies is proud to present our new series, The Essentials ForDummies. Now students who are prepping for exams, preparing tostudy new material, or who just need a refresher can have aconcise, easy-to-understand review guide that covers an entirecourse by concentrating solely on the most important concepts. Fromalgebra and chemistry to grammar and Spanish, our expert authorsfocus on the skills students most need to succeed in a subject.

Leseprobe

Algebra II Essentials For Dummies®

Table of Contents

Introduction

About This Book

Conventions Used in This Book

Foolish Assumptions

Icons Used in This Book

Where to Go from Here

Chapter 1: Making Advances in Algebra

Bringing Out the Best in Algebraic Properties

Making short work of the basic properties

Organizing your operations

Enumerating Exponential Rules

Multiplying and dividing exponents

Rooting out exponents

Powering up exponents

Working with negative exponents

Assigning Factoring Techniques

Making two terms factor

Factoring three terms

Factoring four or more terms by grouping

Chapter 2: Lining Up Linear Equations

Getting the First Degree: Linear Equations

Solving basic linear equations

Eliminating fractions

Lining Up Linear Inequalities

Solving basic inequalities

Introducing interval notation

Absolute Value: Keeping Everything in Line

Solving absolute value equations

Seeing through absolute value inequality

Chapter 3: Making Quick Work of Quadratic Equations

Using the Square Root Rule When Possible

Solving Quadratic Equations by Factoring

Factoring quadratic binomials

Factoring quadratic trinomials

The Quadratic Formula to the Rescue

Realizing rational solutions

Investigating irrational solutions

Promoting Quadratic-like Equations

Solving Quadratic Inequalities

Keeping it strictly quadratic

Signing up for fractions

Increasing the number of factors

Chapter 4: Rolling Along with Rational and Radical Equations

Rounding Up Rational Equations and Eliminating Fractions

Making your least common denominator work for you

Proposing proportions for solving rational equations

Reasoning with Radicals

Squaring both sides of the equation

Taking on two radicals

Dealing with Negative Exponents

Factoring out a negative exponent as a greatest common factor

Solving quadratic-like trinomials

Fiddling with Fractional Exponents

Solving equations by factoring fractional exponents

Promoting techniques for working with fractional exponents

Chapter 5: Forging Function Facts

Describing Function Characteristics

Denoting function notation

Using function notation to evaluate functions

Determining Domain and Range

Delving into domain

Wrangling with range

Counting on Even and Odd Functions

Determining whether even or odd

Using even and odd functions in graphs

Taking on Functions One-to-One

Defining which functions are one-to-one

Testing for one-to-one functions

Composing Functions

Composing yourself with functions

Composing with the difference quotient

Getting Into Inverse Functions

Finding which functions are inverses

Finding an inverse of a function

Chapter 6: Graphing Linear and Quadratic Functions

Identifying Some Graphing Techniques

Finding x- and y-intercepts

Reflecting on a graph’s symmetry

Mastering the Graphs of Lines

Determining the slope of a line

Describing two line equations

Identifying parallel and perpendicular lines

Coming to Terms with the Standard Form of a Quadratic

Starting with “a” in the standard form

Following “a” with “b” and “c”

Eyeing a Quadratic’s Intercepts

Finding the one and only y-intercept

Getting at the x-intercepts

Finding the Vertex of a Parabola

Computing vertex coordinates

Linking up with the axis of symmetry

Sketching a Graph from the Available Information

Chapter 7: Pondering Polynomials

Sizing Up a Polynomial Equation

Identifying Intercepts and Turning Points

Interpreting relative value and absolute value

Dealing with intercepts and turning points

Solving for y-intercepts and x-intercepts

Determining When a Polynomial Is Positive or Negative

Incorporating a sign line

Recognizing a sign change rule

Solving Polynomial Equations

Factoring for roots

Taking sane steps with the rational root theorem

Putting Descartes in charge of signs

Finding Roots Synthetically

Using synthetic division when searching for roots

Synthetically dividing by a binomial

Chapter 8: Being Respectful of Rational Functions

Examining Rational Functions

Deliberating on domain

Investigating intercepts

Assigning Roles to Asymptotes

Validating vertical asymptotes

Finding equations for horizontal asymptotes

Taking vertical and horizontal asymptotes to graphs

Getting the scoop on oblique (slant) asymptotes

Discounting Removable Discontinuities

Finding removable discontinuities by factoring

Evaluating the removals

Looking at Limits of Rational Functions

Determining limits at function discontinuities

Finding infinity

Looking at infinity

Chapter 9: Examining Exponential and Logarithmic Functions

Computing Exponentially

Getting to the Base of Exponential Functions

Classifying bases

Introducing the more frequently used bases: 10 and e

Exponential Equation Solutions

Creating matching bases

Quelling quadratic patterns

Looking into Logarithmic Functions

Presenting the properties of logarithms

Doing more with logs than sawing

Solving Equations Containing Logs

Seeing all logs created equal

Solving log equations by changing to exponentials

Chapter 10: Getting Creative with Conics

Posing with Parabolas

Generalizing the form of a parabola’s equation

Making short work of a parabola’s sketch

Changing a parabola’s equation to the standard form

Circling Around a Conic

Getting Eclipsed by Ellipses

Determining the shape

Finding the foci

Getting Hyped for Hyperbolas

Including the asymptotes

Graphing hyperbolas

Chapter 11: Solving Systems of Equations

Looking at Solutions Using the Standard Linear-Systems Form

Solving Linear Systems by Graphing

Interpreting an intersection

Tackling the same line

Putting up with parallel lines

Using Elimination (Addition) to Solve Systems of Equations

Finding Substitution to Be a Satisfactory Substitute

Variable substituting made easy

Writing solutions for coexisting lines

Taking on Systems of Three Linear Equations

Finding the solution of a system of three linear equations

Generalizing with a system solution

Increasing the Number of Equations

Intersecting Parabolas and Lines

Determining if and where lines and parabolas cross paths

Determining that there’s no solution

Crossing Parabolas with Circles

Finding multiple intersections

Sifting through the possibilities for solutions

Chapter 12: Taking the Complexity Out of Complex Numbers

Simplifying Powers of i

Getting More Complex with Complex Numbers

Performing complex operations

Performing complex division by multiplying by the conjugate

Simplifying reluctant radicals

Unraveling Complex Solutions in Quadratic Equations

Investigating Polynomials with Complex Roots

Classifying conjugate pairs

Making use of complex zeros

Chapter 13: Ten (Or So) Special Formulas

Using Multiplication to Add

Factoring in Factorial

Picking Out Permutations

Collecting Combinations

Adding n Integers

Adding n Squared Integers

Adding Odd Numbers

Going for the Geometric

Calculating Compound Interest

Algebra II Essentials For Dummies®

by Mary Jane Sterling

Algebra II Essentials For Dummies®

Published byWiley Publishing, Inc.111 River St.Hoboken, NJ 07030-5774www.wiley.com

Copyright © 2010 by Wiley Publishing, Inc., Indianapolis, Indiana

Published simultaneously in Canada

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Library of Congress Control Number: 2010925242

ISBN: 978-0-470-61840-0

Manufactured in the United States of America

10 9 8 7 6 5 4 3 2 1

About the Author

Mary Jane Sterling has been an educator since graduating from college. Teaching at the junior high, high school, and college levels, she has had the full span of experiences and opportunities to determine how best to explain how mathematics works. She has been teaching at Bradley University in Peoria, Illinois, for the past 30 years. She is also the author of Algebra II For Dummies, Trigonometry For Dummies, Math Word Problems For Dummies, Business Math For Dummies, and Linear Algebra For Dummies.

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Introduction

Here you are, perusing a book on the essentials of Algebra II. You’ll find here, as Joe Friday (star of the old Dragnet series) said, “The facts, ma’am, just the facts.” For those of you too young to remember Dragnet, just think of this essentials book as being the Twitter version — not too detailed but with all the necessary information. In this book, you find the information you need with enough examples to show you the processes, but not a bunch of nitty-gritty details that tend to get in the way.

About This Book

A book on Algebra II isn’t a romance novel (although I do love math), and it isn’t science fiction. You could think of this book as a cross between a travel guide and a mathematical laboratory manual. How do travel and math go together? Let me try some situations that may fit:

You just finished working through Algebra I and feel eager to embark on a new adventure.

You haven’t worked with algebra in a while, but math has always been your strength, so you think that a little prepping with some basic concepts will bring you up to speed.

You’re helping a friend or family member with Algebra II and want just the most necessary information — no frills or extra side-trips.

Even though I’ve pared the material in this book down to the basics, I haven’t lost sight of the fact that other math areas are what drive Algebra II. Algebra is the passport to studying calculus, trigonometry, number theory, geometry, and all sorts of good mathematics. Algebra is basic, and the algebra you find here will help you grow your skills and knowledge so you can do well in math courses and possibly pursue other math topics.

Conventions Used in This Book

To help you navigate this book, I use the following conventions:

I italicize special mathematical terms and define them right then and there so you don’t have to search around.

I use boldface text to indicate keywords in bulleted lists or the action parts of numbered steps. I describe many algebraic procedures in a step-by-step format and then use those steps in an example or two.

Foolish Assumptions

Algebra II is essentially a continuation of Algebra I, so I need to make some assumptions about readers of this book.

I assume that a person taking on Algebra II has a grasp of working with operations on signed numbers, simplifying radical expressions, and manipulating with rational terms. Another assumption I make is that your order of operations is in order. You should be able to work your way through algebraic equations and expressions using the ordering rules. I also assume that you know how to solve basic linear and quadratic equations and can make quick sketches of basic graphs. Even though I lightly cover these topics in this book, I assume that you have a general knowledge of the necessary procedures.

If you feel a bit over your head after reading through some chapters, you may want to refer to Algebra I For Dummies, 2nd Edition (Wiley), or Algebra II For Dummies (Wiley) for a more complete explanation of the basics. My feelings won’t be hurt; I wrote those, too!

Icons Used in This Book

The icons that appear in this book are great for calling attention to the hot topics when doing algebra.

This icon provides you the rule or law or instruction on how to proceed whenever encountering the particular mathematical situation. The algebra rule given is “the law” — it always applies and always must be followed.

When you see the Example icon, you know that you’ll find the result of an attempt at working out an equation or concept. An example often has a hidden agenda — it shows you more of a process than a basic rule can get across by itself.

This icon is like the sign alerting you to the presence of something special to watch out for on your adventure. It can save you time and energy. Use this information to cut to the chase and avoid unnecessary detours.

This icon helps you bring back information that you may have misplaced along the way. The information is needed to get you from here to the goal.

This icon alerts you to common hazards and stumbling blocks that could trip you up — cause accidents or get you into trouble with the math police. Those who have gone before you have found that these items can cause a big problem — so pay heed.

Where to Go from Here

You can use the table of contents at the beginning of the book and the index in the back to navigate your way to the topic that you’re most interested in. You may want to start with some problem solving — in the form of equations or inequalities. If that’s the case, then look at Chapter 2 for linear equations and inequalities or Chapters 3 and 4 for quadratic and other degree equations. Chapter 5 is your destination if you want to see what constitutes a function and its characteristics. And specific functions such as linear and quadratics are found in Chapter 6; polynomials are found in Chapter 7, rationals in Chapter 8, and exponentials and logs in Chapter 9. I saved the imaginary for last, in Chapter 12. But you could stop off and look at conics in Chapter 10, if those curves are of interest. Also, systems of equations incorporate several types of functions, and you find them in Chapter 11.

And, if you’re more of a freewheeling type of guy or gal, take your finger, flip open the book, and mark a spot. No matter your motivation or what technique you use to jump into this book, you won’t get lost because you can go in any direction from there.

Enjoy!

Chapter 1

Making Advances inAlgebra

In This Chapter

Making algebra orderly with the order of operations and other properties

Enlisting rules of exponents

Focusing on factoring

Algebra is a branch of mathematics that people study before they move on to other areas or branches in mathematics and science. Algebra all by itself is esthetically pleasing, but it springs to life when used in other applications.

Any study of science or mathematics involves rules and patterns. You approach the subject with the rules and patterns you already know, and you build on those rules with further study. In this chapter, I recap for you the basic rules from Algebra I so that you work from the correct structure. I present these basics so you can further your study of algebra and feel confident in your algebraic ability.