Algebra II Workbook For Dummies®, 3rd Edition Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com Copyright © 2019 by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 7486011, fax (201) 748-6008, or online at www.wiley.com/go/permissions Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc and may not be used without written permission All other trademarks are the property of their respective owners John Wiley & Sons, Inc is not associated with any product or vendor mentioned in this book LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ For general information on our other products and services, please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993, or fax 317-572-4002 For technical support, please visit https://hub.wiley.com/community/support/dummies Wiley publishes in a variety of print and electronic formats and by printon-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Library of Congress Control Number: 2018962324 ISBN 978-1-119-54311-4 (pbk); ISBN 978-1-119-54313-8 (ebk); ISBN 978-1-119-54312-1 (ebk) Algebra II Workbook For Dummies® To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Algebra II Workbook For Dummies Cheat Sheet” in the Search box Table of Contents Cover Introduction About This Book Foolish Assumptions Icons Used in This Book Beyond the Book Where to Go from Here Part 1: Getting Started with Algebra II Chapter 1: Going Beyond Beginning Algebra Good Citizenship: Following the Order of Operations and Other Properties Specializing in Products and FOIL Variables on the Side: Solving Linear Equations Dealing with Linear Absolute Value Equations Greater Math Skills: Equalizing Linear Inequalities Answers to Problems on Going Beyond Beginning Algebra Chapter 2: Handling Quadratic (and Quadratic-Like) Equations and Inequalities Finding Reasonable Solutions with Radicals UnFOILed Again! Successfully Factoring for Solutions Your Bag of Tricks: Factoring Multiple Ways Keeping Your Act Together: Factoring by Grouping Resorting to the Quadratic Formula Solving Quadratics by Completing the Square Working with Quadratic-Like Equations Checking Out Quadratic Inequalities Answers to Problems on Quadratic (and Quadratic-Like) Equations and Inequalities Chapter 3: Rooting Out the Rational, the Radical, and the Negative Doing Away with Denominators with an LCD Simplifying and Solving Proportions Wrangling with Radicals Changing Negative Attitudes toward Negative Exponents Divided Powers: Solving Equations with Fractional Exponents Answers to Problems on Rooting Out the Rational, the Radical, and the Negative Chapter 4: Graphing for the Good Life Coordinating Axes, Coordinates of Points, and Quadrants Crossing the Line: Using Intercepts and Symmetry to Graph Graphing Lines Using Slope-Intercept and Standard Forms Graphing Basic Polynomial Curves Grappling with Radical and Absolute Value Functions Enter the Machines: Using a Graphing Calculator Answers to Problems on Graphing for the Good Life Part 2: Functions Chapter 5: Formulating Functions Evaluating Functions Determining the Domain and Range of a Function Recognizing Even, Odd, and One-to-One Functions Composing Functions and Simplifying the Difference Quotient Solving for Inverse Functions Answers to Problems on Formulating Functions Chapter 6: Specializing in Quadratic Functions Finding Intercepts and the Vertex of a Parabola Applying Quadratics to Real-Life Situations Graphing Parabolas Answers to Problems on Quadratic Functions Chapter 7: Plugging in Polynomials Finding Basic Polynomial Intercepts Digging up More-Difficult Polynomial Roots with Factoring Determining Where a Function Is Positive or Negative Graphing Polynomials Possible Roots and Where to Find Them: The Rational Root Theorem and Descartes’s Rule Getting Real Results with Synthetic Division and the Remainder Theorem Connecting the Factor Theorem with a Polynomial’s Roots Answers to Problems on Plugging in Polynomials Chapter 8: Acting Rationally with Functions Determining Domain and Intercepts of Rational Functions Introducing Vertical and Horizontal Asymptotes Getting a New Slant with Oblique Asymptotes Removing Discontinuities Going the Limit: Limits at a Number and Infinity Graphing Rational Functions Answers to Problems on Rational Functions Chapter 9: Exposing Exponential and Logarithmic Functions Evaluating e-Expressions and Powers of e Solving Exponential Equations Making Cents: Applying Compound Interest and Continuous Compounding Checking out the Properties of Logarithms Presto-Chango: Expanding and Contracting Expressions with Log Functions Solving Logarithmic Equations They Ought to Be in Pictures: Graphing Exponential and Logarithmic Functions Answers to Problems on Exponential and Logarithmic Functions Part 3: Conics and Systems of Equations Chapter 10: Any Way You Slice It: Conic Sections Putting Equations of Parabolas in Standard Form Shaping Up: Determining the Focus and Directrix of a Parabola Back to the Drawing Board: Sketching Parabolas Writing the Equations of Circles and Ellipses in Standard Form Determining Foci and Vertices of Ellipses Rounding Out Your Sketches: Circles and Ellipses Hyperbola: Standard Equations and Foci Determining the Asymptotes and Intercepts of Hyperbolas Sketching the Hyperbola Answers to Problems on Conic Sections Chapter 11: Solving Systems of Linear Equations Solving Two Linear Equations Algebraically Using Cramer’s Rule to Defeat Unruly Fractions A Third Variable: Upping the Systems to Three Linear Equations A Line by Any Other Name: Writing Generalized Solution Rules Decomposing Fractions Using Systems Answers to Problems on Systems of Equations Chapter 12: Solving Systems of Nonlinear Equations and Inequalities Finding the Intersections of Lines and Parabolas Crossing Curves: Finding the Intersections of Parabolas and Circles Appealing to a Higher Power: Dealing with Exponential Systems Solving Systems of Inequalities Answers to Problems on Solving Systems of Nonlinear Equations and Inequalities Part 4: Other Good Stuff: Lists, Arrays, and Imaginary Numbers Chapter 13: Getting More Complex with Imaginary Numbers Simplifying Powers of i Not Quite Brain Surgery: Doing Operations on Complex Numbers “Dividing” Complex Numbers with a Conjugate Solving Equations with Complex Solutions Answers to Problems on Imaginary Numbers Chapter 14: Getting Squared Away with Matrices Describing Dimensions and Types of Matrices Adding, Subtracting, and Doing Scalar Multiplication on Matrices Trying Times: Multiplying Matrices by Each Other The Search for Identity: Finding Inverse Matrices Using Matrices to Solve Systems of Equations Answers to Problems on Matrices Chapter 15: Going Out of Sequence with Sequences and Series Writing the Terms of a Sequence Differences and Multipliers: Working with Special Sequences Backtracking: Constructing Recursively Defined Sequences Using Summation Notation Finding Sums with Special Series Answers to Problems on Sequences and Series Chapter 16: Everything You Ever Wanted to Know about Sets and Counting Writing the Elements of a Set from Rules or Patterns Get Together: Combining Sets with Unions, Intersections, and Complements Multiplication Countdowns: Simplifying Factorial Expressions Checking Your Options: Using the Multiplication Property Counting on Permutations When Order Matters problems on, 288 sines of angles, computing, 321 squares, 318 summation notation, 284–285, 291 symmetry answers to problems on, 77–79 axis of, 111–112, 194–195 general discussion, 64–65 polynomial curves, 69 problems on, 66 synthetic division answers to problems on, 140–141 general discussion, 130–131 problems on, 131–132 systems of linear equations Cramer’s rule, 223–225, 231–232 decomposition of fractions, 229–230, 235–236 generalized solution rules, 227–228, 234–235 matrices, solving with, 272–273, 278 overview, 221 systems of three, solving, 225–226, 233–234 systems of two, solving, 221–222, 231 systems of nonlinear equations and inequalities exponential systems, 240–242, 247 intersections of lines and parabolas, 237–239, 245 intersections of parabolas and circles, 239–240, 245–246 overview, 237 systems of inequalities answers to problems on, 247–249 general discussion, 242–243 problems on, 244 T terms of sequences answers to problems on, 289 general discussion, 279–280 problems on, 281 test numbers, 123, 124 test points, 242 three linear equations, solving systems of, 225–226, 233–234 Tip icon, trees, 300 trinomials, factoring, 25 turning points, polynomial curves, 69 two linear equations, solving systems of, 221–222, 231 2, equations with variables with power of See quadratic equations 2, power of, sum formulas for sequences with, 322 U unFOIL method quadratic equations, solving with, 23–25 quadratic-like equations, solving with, 30–32 union symbol (∪), 124 unions, 295–296, 304 universal set, 295 U-shaped curves, 108, 111–112, 191, 196 See also parabolas V vertical asymptotes answers to problems on, 156–157 general discussion, 145–146 graphing rational functions, 152–154 problems on, 146–147 vertices of ellipses, 201–202, 214–215 of parabolas identifying from standard form of equation, 194–195 quadratic functions, 108–109, 113 W Warning icon, X x-axis, 62, 65 x-intercepts hyperbolas, 206–207 lines, 312–313 overview, 64 polynomial curves, 69 polynomial functions, 120, 134 quadratic functions, 108 rational functions, 144, 156 Y y-axis, 62, 65 y-intercepts hyperbolas, 206–207 lines, 312–313 overview, 64 polynomial functions, 120, 134 quadratic functions, 108 rational functions, 144, 156 Z zero, multiplication property of, zero matrices, 264 About the Author Mary Jane Sterling is also the author of Algebra I For Dummies, Algebra II For Dummies, Math Word Problems For Dummies, Business Math For Dummies, and Linear Algebra For Dummies She taught junior high school and high school math for several years before beginning her 30-plus-year tenure at Bradley University in Peoria, Illinois Mary Jane especially enjoys working with future teachers and trying out new technology She and her husband, Ted, enjoy spending their leisure time with their children and grandchildren, traveling, and fishing Dedication I’d like to dedicate Algebra II Workbook For Dummies to my children, Howard Jonathan, James Theodore, and Jane Christine (yes, three J’s: Jon, Jim, and Jane) My husband and I raised our children to be “independent,” and, oh my, we got what we wished for in that quest We’re so proud of the people our children have become Author’s Acknowledgments I warmly thank my project editor, Christopher Morris, for working on this newest endeavor; she is so thorough and professional, yet understanding and patient Thank you to the technical editor, Michael McAsey; without his skills and careful checking, mathematical mistakes may have gone unnoticed and uncorrected And thank you to Lindsay Lefevere for her continued efforts to keep this author happily occupied Publisher’s Acknowledgments Executive Editor: Lindsay Sandman Lefevere Project Editor: Christopher Morris Copy Editor: Christopher Morris Technical Editor: Michael McAsey Production Editor: Mohammed Zafar Ali Cover Image: © jm1366 / Shutterstock Take Dummies with you everywhere you go! Go to our Website Like us on Facebook Follow us on Twitter Watch us on YouTube Join us on LinkedIn Pin us on Pinterest Circle us on google+ Subscribe to our newsletter Create your own Dummies book cover Shop Online WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... ISBN 978-1-119-54312-1 (ebk) Algebra II Workbook For Dummies? ? To view this book's Cheat Sheet, simply go to www .dummies. com and search for ? ?Algebra II Workbook For Dummies Cheat Sheet” in the... Algebra II For Dummies, where I go into more depth on what’s involved with each type of problem (If you need more-basic information, you can try Algebra I For Dummies and Algebra I Workbook For. .. Cheat Sheet for a handy guide that covers tips and tricks for answering Algebra II questions To get this Cheat Sheet, simply go to www .dummies. com and enter ? ?Algebra II Workbook For Dummies? ?? in