Pre-calculus workbook for dummies PDF
Preview Pre-calculus workbook for dummies
P r e - Ca l c u l u s Wo r k b o o k FOR DUMmIES‰ by Michelle Rose Gilman, Christopher Burger, Karina Neal Pre-Calculus Workbook For Dummies® Published by Wiley Publishing, Inc. 111 River St. Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2009 by Wiley Publishing, Inc., Indianapolis, Indiana Published by Wiley Publishing, Inc., Indianapolis, Indiana 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 either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600. 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Library of Congress Control Number: 2009923971 ISBN: 978-0-470-42131-4 Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1 Part I: Foundation (And We Don’t Mean Makeup!).........................5 Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus....................................7 Reviewing Order of Operations: The Fun in Fundamentals..................................................7 Keeping Your Balance While Solving Equalities...................................................................10 A Picture Is Worth a Thousand Words: Graphing Equalities and Inequalities.................12 Graphing using the plug and chug method.................................................................13 Graphing using the slope-intercept form....................................................................13 Using Graphs to Find Information (Distance, Midpoint, Slope).........................................15 Finding the distance.......................................................................................................15 Calculating the midpoint...............................................................................................16 Discovering the slope....................................................................................................16 Answers to Problems on Fundamentals................................................................................20 Chapter 2: Get Real!: Wrestling with Real Numbers.....................................................25 Solving Inequalities..................................................................................................................25 Expressing Inequality Solutions in Interval Notations........................................................28 Don’t Get Drastic with Radicals and Exponents — Just Simplify Them!..........................30 Getting Out of a Sticky Situation or Rationalizing................................................................33 Answers to Problems on Real Numbers................................................................................36 Chapter 3: Understanding the Function of Functions.....................................................41 Battling Out Even versus Odd................................................................................................41 Leaving the Nest: Transforming Parent Graphs...................................................................43 Quadratic functions.......................................................................................................43 Square root functions....................................................................................................44 Absolute value functions...............................................................................................44 Cubic functions...............................................................................................................44 Cube root functions.......................................................................................................45 Vertical transformations................................................................................................45 Horizontal transformations...........................................................................................46 Translations....................................................................................................................46 Reflections.......................................................................................................................47 Combinations of transformations................................................................................47 Lucid Thinking? Graphing Rational Functions.....................................................................50 Picking Up the Pieces: Graphing Piece-Wise Functions......................................................53 Operating on Functions: No Scalpel Necessary...................................................................54 Evaluating Composition of Functions....................................................................................56 Working Together: Domain and Range..................................................................................58 Finding the Inverse of a Function (Who Knew It Was Lost?)..............................................60 Answers to Questions on Functions......................................................................................62 Chapter 4: Go Back to Your Roots to Get Your Degree...................................................73 Reason Through It: Factoring a Factorable Polynomial......................................................73 Get Your Roots Done while Solving a Quadratic Polynomial.............................................76 Completing the square..................................................................................................76 Quadratic formula..........................................................................................................76 Climb the Mountains by Solving High Order Polynomials.................................................78 Determining positive and negative roots: Descartes’ Rule of Signs........................78 Counting on imaginary roots........................................................................................78 Getting the rational roots..............................................................................................79 Synthetic division finds some roots.............................................................................79 Strike That! Reverse It! Using Roots to Find an Equation...................................................81 Graphing Polynomials...................................................................................................82 Answers to Questions on Finding Roots...............................................................................86 Chapter 5: Exponential and Logarithmic Functions.......................................................91 Things Get Bigger (Or Smaller) All the Time – Solving Exponential Functions...............91 The Only Logs You Won’t Cut: Solving Logarithms.............................................................93 Putting Them Together: Solving Equations Using Exponents and Logs...........................96 Solving Exponential Word Problems . . . Argh!.....................................................................98 Answers to Problems on Exponential and Logarithmic Functions.................................100 Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs........................................................105 Chapter 6: Basic Trigonometry and the Unit Circle......................................................107 It’s All Right-Triangle Trig — Finding the Six Trigonometric Ratios................................107 Solving Word Problems with Right Triangles.....................................................................110 Unit Circle and the Coordinate Plane: Finding Points and Angles...................................112 Isn’t That Special? Finding Right Triangle Ratios on the Unit Circle...............................115 Solving Trig Equations...........................................................................................................117 Making and Measuring Areas...............................................................................................119 Answers to Problems.............................................................................................................121 Chapter 7: Graphing and Transforming Trig Functions................................................127 Getting a Grip on Period Graphs..........................................................................................127 Sine and Cosine: Parent Graphs and Transformations.....................................................128 Tangent and Cotangent: Mom, Pops, and Children...........................................................131 Secant and Cosecant: Generations......................................................................................134 Answers to Problems on Graphing and Transforming Trig Functions............................137 Part III: Advanced Trig: Identities, Theorems, and Applications ....143 Chapter 8: Basic Trig Identities........................................................................................145 Using Reciprocal Identities to Simplify Trig Expressions.................................................145 Simplifying with Pythagorean Identities.............................................................................147 Discovering Even-Odd Identities..........................................................................................148 Solving with Co-Function Identities.....................................................................................149 Moving with Periodicity Identities.......................................................................................150 Tackling Trig Proofs...............................................................................................................151 Answers to Problems on Basic Trig Identities...................................................................153 Chapter 9: Advanced Identities........................................................................................161 Simplifying with Sum and Difference Identities..................................................................161 Using Double Angle Identities..............................................................................................163 Reducing with Half-Angle Identities.....................................................................................165 Changing Products to Sums..................................................................................................166 Expressing Sums as Products...............................................................................................168 Powering Down: Power-Reducing Formulas.......................................................................169 Answers to Problems on Advanced Identities...................................................................170 Chapter 10: Solving Oblique Triangles...........................................................................177 Solving a Triangle with the Law of Sines: ASA and AAS....................................................177 Tackling Triangles in the Ambiguous Case: SSA................................................................179 Conquering a Triangle with the Law of Cosines: SAS and SSS.........................................180 Using Oblique Triangles to Solve Word Problems.............................................................182 Figuring Flatness (Area)........................................................................................................184 Answers to Problems on Solving Triangles........................................................................186 Part IV: And the Rest . . ............................................................193 Chapter 11: Complex Numbers and Polar Coordinates...............................................195 Performing Operations with and Graphing Complex Numbers.......................................195 Round a Pole: Graphing Polar Coordinates........................................................................199 Changing to and from Polar..................................................................................................202 Graphing Polar Equations.....................................................................................................204 Archimedean spiral......................................................................................................204 Cardioid.........................................................................................................................204 Rose................................................................................................................................204 Circle..............................................................................................................................204 Lemniscate....................................................................................................................204 Limaçon.........................................................................................................................205 Answers to Problems on Complex Numbers and Polar Coordinates..............................207 Chapter 12: Conquering Conic Sections.........................................................................211 A Quick Conic Review............................................................................................................211 Going Round and Round with Circles..................................................................................212 Graphing Parabolas: The Ups and Downs..........................................................................213 Standing tall: Vertical parabolas.................................................................................214 Lying down on the job: Horizontal parabolas..........................................................216 Graphing Ellipses: The Fat and the Skinny.........................................................................218 Short and fat: The horizontal ellipse.........................................................................219 Tall and skinny: The vertical ellipse..........................................................................220 Graphing Hyperbolas: No Caffeine Required......................................................................222 Hip horizontal hyperbolas (alliteration!)..................................................................223 Vexing vertical vyperbolas (er, hyperbolas).............................................................225 Identifying Conic Sections....................................................................................................227 Converting from Parametric Form to Polar Coordinates and Back.................................230 Parametric form for conic sections...........................................................................230 Changing from parametric form to rectangular form..............................................232 Conic sections on the polar coordinate plane..........................................................233 Answers to Problems on Conic Sections............................................................................235 Chapter 13: Finding Solutions for Systems of Equations.............................................243 A Quick-and-Dirty Technique Overview..............................................................................243 Solving Two Linear Equations with Two Variables............................................................244 The substitution method.............................................................................................244 The elimination method..............................................................................................245 Not-So-Straight: Solving Nonlinear Systems.......................................................................247 One linear equation and one not................................................................................247 Two equations that are nonlinear..............................................................................248 Systems of equations disguised as rational equations...........................................248 Solving More than Two (Equations and/or Variables)......................................................249 Graphing Systems of Inequalities.........................................................................................251 We’re Partial to . . . Decomposing Partial Fractions!.........................................................253 There Is No Spoon: Working with a Matrix.........................................................................255 Getting It in the Right Form: Simplifying Matrices.............................................................257 Solving Systems of Equations Using Matrices....................................................................259 Gaussian elimination....................................................................................................260 Inverse matrices...........................................................................................................261 Cramer’s Rule................................................................................................................262 Answers to Problems on Systems of Equations.................................................................264 Chapter 14: Sequences, Series, and Binomials — Oh My!........................................275 Major General Sequences and Series: Calculating Terms.................................................275 Working Out the Common Difference: Arithmetic Sequences and Series......................277 Be Fruitful and Multiply: Simplifying Geometric Sequences and Series.........................279 Expanding Polynomials Using the Binomial Theorem......................................................281 Answers to Problems on Sequences, Series, and Binomials............................................283 Chapter 15: The Next Step Is Calculus...........................................................................287 Finding Limits: Graphically, Analytically, and Algebraically............................................287 Graphically....................................................................................................................288 Analytically....................................................................................................................289 Algebraically.................................................................................................................290 Knowing Your Limits..............................................................................................................292 Determining Continuity.........................................................................................................293 Answers to Problems on Calculus.......................................................................................296 Part V: The Part of Tens.............................................................299 Chapter 16: Ten Uses for Parent Graphs.........................................................................301 Squaring Up with Quadratics...............................................................................................301 Cueing Up for Cubics.............................................................................................................302 Rooting for Square Roots and Cube Roots.........................................................................302 Graphing Absolutely Fabulous Absolute Value Functions................................................303 Flipping over Rational Functions.........................................................................................304 Exploring Exponential Graphs and Logarithmic Graphs..................................................304 Seeing the Sine and Cosine...................................................................................................305 Covering Cosecant and Secant.............................................................................................306 Tripping over Tangent and Cotangent................................................................................306 Chapter 17: Ten Pitfalls to Pass Up in Pre-Calc............................................................309 Going Out of Order (Of Operations)....................................................................................309 FOILed Again! FOILing Binomials Incorrectly.....................................................................309 Gettin’ Down with Denominators.........................................................................................310 Combining Terms That Can’t Be Combined.......................................................................310 Forgetting to Flip the Fraction..............................................................................................310 Losing the Negative (Sign)....................................................................................................311 Oversimplifying Roots...........................................................................................................311 Avoiding Exponent Errors.....................................................................................................311 Canceling Too Quickly...........................................................................................................312 Dealing with Distribution......................................................................................................312 Index........................................................................................313 Part I Foundation (And We Don’t Mean Makeup!) In this part . . . P re-calculus is really just another stop on the road to calculus. You started with the village of Algebra I, moved on to the small town of Geometry, made your way to Algebra II city, and now find yourself in the mega- metropolis known as Pre-Calculus. The skills, for the most part, are the same. This part takes those skills and reviews them (and, in some cases, expands on them). The chapters here begin with a review of the basics: using the order of operations, solving and graphing equations and inequalities, and using the distance and midpoint for- mulas. Some new material pops up in the form of interval notation, so be sure and check that out. Then we move on to real numbers, including radicals. Everything you ever wanted to know about functions is covered in one of the chapters: graphing and transforming parent graphs, ratio- nal functions, and piece-wise functions. We also go over performing operations on functions and how to find the inverse. We then move on to solving higher degree poly- nomials using techniques like factoring, completing the square, and the quadratic formula. You also learn how to graph these complicated polynomials. Lastly, you discover exponential and logarithmic functions and what you’re expected to know about them. Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter (cid:1)Brushing up on order of operations (cid:1)Solving equalities (cid:1)Graphing equalities and inequalities (cid:1)Finding distance, midpoint, and slope P re-calculus is the stepping stone for Calculus. It’s the final hurdle after all those years of math: Pre-algebra, Algebra, Geometry, and Algebra II. Now all you need is Pre-calculus to get to that ultimate goal — Calculus. And as you may recall from your Algebra II class, you were subjected to much of the same material you saw in Algebra and even Pre-algebra (just acouple steps up in terms of complexity — but really the same stuff). As the stepping stone, pre-calculus begins with certain concepts that you’re expected to solidly understand. Therefore, we’re starting here, at the very beginning, reviewing those concepts. If you feel you’re already an expert at everything algebra, feel free to skip past this chapter and get the full swing of pre-calc going. If, however, you need to review, then read on. If you don’t remember some of the concepts we discuss in this chapter, or even in this book, you can pick up another For Dummiesmath book for review. The fundamentals are impor- tant. That’s why they’re called fundamentals. Take the time now to review — it will save you countless hours of frustration in the future! Reviewing Order of Operations: The Fun in Fundamentals You can’t put on your sock after you put on your shoe, can you? The same concept applies tomathematical operations. There’s a specific order to which operation you perform first, second, third, and so on. At this point, it should be second nature. However, because the concept is so important as we continue into more complex calculations, we review it here.
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