Table Of Content

MathSkills Algebra 2 By Michael Buckley Development and Production: Frishco Ltd. and Pearl Production ISBN-13: 978-1-61651-424-2 ISBN-10: 1-61651-424-8 eBook: 978-1-60291-801-6 Copyright ©2006, 2011 by Saddleback Educational Publishing. All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, scanning, or by any information storage and retrieval system, without the written permission of the publisher, with the exception below. Pages labeled with the statement ©Saddleback Educational Publishing are intended for reproduction. Saddleback Publishing, Inc. grants to individual purchasers of this book the right to make sufficient copies of reproducible pages for use by all students of a single teacher. This permission is limited to a single teacher, and does not apply to entire schools or school systems. Printed in the United States of America 15 14 13 12 11 1 2 3 4 5 SBM4_i-v,1-50.indd 1 1/18/11 3:54 PM SBM4_i-v,1-50.indd 2 1/18/11 3:54 PM Table of Contents Properties of Real Numbers ....................................................... 1 Absolute Value .................................................................. 2 Absolute Value Equations ......................................................... 3 Compound Inequalities........................................................... 4 Absolute Value Inequalities........................................................ 5 Graphing Absolute Value Inequalities ............................................... 6 Introduction to Matrices.......................................................... 7 Matrix Addition ................................................................. 8 Matrix Subtraction............................................................... 9 Scalar Multiplication ............................................................ 10 Matrix Multiplication ........................................................... 11 Writing the Inverse of a Matrix ................................................... 12 Solving a Matrix Equation ....................................................... 13 Relations and Functions ......................................................... 14 Types of Functions.............................................................. 15 Direct Variation ................................................................ 16 Slope-Intercept Form............................................................ 17 Point-Slope Form I ............................................................. 18 Point-Slope Form II............................................................. 19 Linear Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Exponential Growth and Decay ................................................... 21 Writing an Exponential Function.................................................. 22 Solving an Exponential Equation.................................................. 23 Negative Exponents ............................................................. 24 Rational Exponents ............................................................. 25 Properties of Rational Exponents.................................................. 26 Doubling Time................................................................. 27 The Number e ................................................................. 28 Logarithmic Functions .......................................................... 29 Properties of Logarithms: Product Property......................................... 30 Properties of Logarithms: Quotient Property........................................ 31 Properties of Logarithms: Power Property .......................................... 32 Properties of Logarithms: Summary ............................................... 33 Solving Logarithmic Functions ................................................... 34 Solving a Natural Logarithmic Function............................................ 35 Graphing Quadratic Functions.................................................... 36 Properties of a Graph of a Quadratic Function ...................................... 37 Writing a Quadratic Function from Its Graph ....................................... 38 Algebra 2 iii SBM4_i-v,1-50.indd 3 1/18/11 3:54 PM Quadratic Functions in Intercept Form............................................. 39 Solving Quadratic Equations Using Square Roots .................................... 40 Solving a Quadratic Equation by Completing the Square.............................. 41 Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Solving a Quadratic Equation by Factoring ......................................... 43 Using the Discriminant.......................................................... 44 Methods for Solving Quadratic Functions .......................................... 45 Writing an Equation of an Ellipse ................................................. 46 Foci of an Ellipse ............................................................... 47 Standard Deviation ............................................................. 48 Margin of Error ................................................................ 49 Binomial Theorem.............................................................. 50 Writing a System of Equations as a Matrix.......................................... 51 Using Matrices to Solve a System of Two Equations .................................. 52 Cramer’s Rule.................................................................. 53 Simplifying Radical Expressions by Removing Perfect Squares ......................... 54 Simplifying Radical Expressions with Variables ...................................... 55 Adding Radical Expressions ...................................................... 56 Subtracting Radical Expressions .................................................. 57 Multiplying Radical Expressions .................................................. 58 Dividing Radical Expressions ..................................................... 59 Rationalizing the Denominator of a Radical Expression............................... 60 Square Root of a Negative Real Number: Imaginary Numbers ......................... 61 Complex Numbers.............................................................. 62 Adding Complex Numbers....................................................... 63 Subtracting Complex Numbers ................................................... 64 Multiplying Complex Numbers ................................................... 65 Dividing Complex Numbers...................................................... 66 Absolute Value and Complex Numbers............................................. 67 Finding a Complex Solution to a Simple Quadratic Equation .......................... 68 Finding a Complex Solution to a Quadratic Equation................................. 69 Solving Cubic Equations: Finding x-Intercepts ...................................... 70 Factoring Cubic Equations: Sum of Cubes .......................................... 71 Factoring Cubic Equations: Difference of Cubes ..................................... 72 Writing Cubic Equations in Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Evaluating Polynomials Using Synthetic Substitution ................................. 74 Multiplying Polynomials......................................................... 75 Dividing Polynomials Using Synthetic Division...................................... 76 Algebra 2 iv SBM4_i-v,1-50.indd 4 1/18/11 3:54 PM End Behavior of a Polynomial Function ............................................ 77 Inverse Variation ............................................................... 78 Simplifying Rational Expressions.................................................. 79 Multiplying Rational Expressions ................................................. 80 Dividing Rational Expressions .................................................... 81 Rational Functions—Finding Vertical Asymptotes.................................... 82 End Behavior of Rational Functions: m < n ......................................... 83 End Behavior of Rational Functions: m = n ......................................... 84 End Behavior of Rational Functions: m > n ......................................... 85 End Behavior of Rational Functions: Using All Three Conditions....................... 86 Recursive Formulas: Arithmetic Sequences.......................................... 87 Recursive Formulas: Geometric Sequences.......................................... 88 Finding the Sum of a Finite Arithmetic Series ....................................... 89 Finding the Sum of a Finite Geometric Series ....................................... 90 Finding the Sum of an Infinite Geometric Series..................................... 91 Function Operations—Adding Functions .......................................... 92 Function Operations—Subtracting Functions ....................................... 93 Function Operations—Multiplying Functions....................................... 94 Function Operations—Dividing Functions ......................................... 95 Composition of Functions ....................................................... 96 Inverse of a Function............................................................ 97 Finding Trigonometric Ratios .................................................... 98 Law of Sines ................................................................... 99 Law of Cosines ................................................................ 100 Answer Key...............................................................101–107 Algebra 2 v SBM4_i-v,1-50.indd 5 1/18/11 3:54 PM SBM4_i-v,1-50.indd 6 1/18/11 3:54 PM Name Date Properties of Real Numbers Real numbers include all rational and irrational numbers. You will use the properties of real numbers listed below throughout your study of algebra. Complete the table by filling in the last column. Summary of Properties of Real Numbers Property Using Symbols Example Summary Commutative Addition: a + b = b + a –2 + 3 = 3 + (–2) Multiplication: ab = ba –2(3) = 3(–2) Associative Addition: (a + b) + c = a + (b + c) (–2 + 3) + 4 = –2 + (3 + 4) Multiplication: (ab) c = a (bc) (–2 × 3) × 4 = –2 × (3 × 4) Distributive a(b + c) = ab + ac –2(3 + 4) = (–2 × 3) + (–2 × 4) Example Identify the property being used. (5x)(8y) = (5x × 8)y Step 1 Describe the change from the left side Numbers being multiplied have been to the right side of the equal sign. regrouped. Step 2 Match the description of the change associative property with one of the properties. Practice Identify the property being used. 1. (2a × 7)b = (7 × 2a)b Describe the change from the left side to the right side of the equal sign. Match the description of the change with one of the properties. 2. 5x + (2 − 4x) = 5x + (−4x + 2) 3. (3 × 6x)y = (3 × 6)(xy) Complete the right side of each equation by using the identified property. 4. Associative: (3x + 2y) + 5 = 5. Distributive: x(2 + 5) = Algebra 2 ©Saddleback Educational Publishing • www.sdlback.com 1 SBM4_i-v,1-50.indd 1 1/18/11 3:54 PM Name Date Absolute Value The absolute value of a number is the distance between the origin of a number line and the point representing that number. Look at the number line below. Both 7 and −7 are 7 units from the origin. 7 units 7 units –7 0 7 The notation for absolute value is |a| and is read as “the absolute value of a.” Rules for the Absolute Value of a Number 1. If a is a positive number, then |a| = a (e.g. |4| = 4). 2. If a is zero, then |a| = 0 (e.g. |0| = 0). 3. If a is a negative number, then |−a| = a (e.g. |−4| = 4). Example Solve the equation. |x| = 15 Step 1 Which positive number is 15 units 15 from the origin? Step 2 Which negative number is 15 units −15 from the origin? Step 3 Check that both solutions are true. |15| = 15 true |−15| = 15 true Practice Solve. 1. −|x| = –10 Which positive number is 10 units from the origin? Which negative number is 10 units from the origin? Check that both solutions are true. −| | = −10 −| | = 2. |x| = 3 5. |−x| = 13 3. |x| = 0 6. −|5| = x 4. |−7| = x 7. −|−6| = x Algebra 2 2 ©Saddleback Educational Publishing • www.sdlback.com SBM4_i-v,1-50.indd 2 1/18/11 3:54 PM Name Date Absolute Value Equations The absolute value of a number is the distance between the origin of a number line and the point representing that number. To solve an absolute value equation you need to account for the value inside the absolute value symbol being positive, and the value inside the absolute value symbol being negative. Rules for Solving an Absolute Value Equation 1. Set the expression inside the absolute value bars equal to the original value on the other side of the equality. Then solve. 2. Set the expression inside the absolute value bars equal to the opposite of the original value on the other side of the equality. Then solve. Example Solve. |x + 2| = 10 Step 1 Set the expression inside the absolute x + 2 = 10 value bars equal to the value on the x + 2 − 2 = 10 − 2 other side of the equality. Then solve. x = 8 Step 2 Set the expression inside the absolute x + 2 = −10 value bars equal to the opposite of the x + 2 − 2 = −10 − 2 value on the other side of the equality. x = −12 Then solve. The solutions are −12 and 8. Practice Solve. 1. |2x − 1| = 9 Set the expression inside the absolute 2x − 1 = 9 value bars equal to the value on the 2x − 1 + = 9 + other side of the equality. Then solve. 2x ÷ = ÷ x = Set the expression inside the absolute 2x − 1 = −9 value bars equal to the opposite of the 2x − 1 + = −9 + value on the other side of the equality. Then solve. 2x ÷ = ÷ x = 2. |x − 4| = 8 4. |3x| = 21 3. |x + 3|= 15 5. |3x + 3| = 30 Algebra 2 ©Saddleback Educational Publishing • www.sdlback.com 3 SBM4_i-v,1-50.indd 3 1/18/11 3:54 PM