Table Of ContentMathSkills
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
