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Table of Contents
Virginia Standards of Learning Correlation Chart . . . . . . . . . . . . . . 6
Virginia Standards of Learning
Algebra II
Chapter 1
Expressions and Operations . . . . . . . . . . . . . . . . . . . . 9
Lesson 1
Properties of Exponents . . . . . . . . . . . . . . . . . . . . . . . . 10
AII.1.a
Lesson 2
Use Patterns to Factor Polynomials . . . . . . . . . . . . . . . 14
AII.1.d
Lesson 3
Operations on Rational Expressions. . . . . . . . . . . . . . . 19
AII.1.a
Lesson 4
Expressions and Operations with Radicals . . . . . . . . . 24
AII.1.b
Lesson 5
Convert Between Radical and Exponential
Forms of Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . 29
AII.1.c
Lesson 6
Introduction to Complex Numbers . . . . . . . . . . . . . . . . 33
AII.3
Lesson 7
Add and Subtract Complex Numbers . . . . . . . . . . . . . 38
AII.3
Lesson 8
Multiply Complex Numbers . . . . . . . . . . . . . . . . . . . . . 42
AII.3
Chapter 2
Equations and Inequalities. . . . . . . . . . . . . . . . . . . . . 49
Lesson 9
Solve Absolute Value Equations . . . . . . . . . . . . . . . . . . 50
AII.4.a
Lesson 10
Solve Absolute Value Inequalities. . . . . . . . . . . . . . . . . 56
AII.4.a
Lesson 11
Solve Quadratic Equations Graphically . . . . . . . . . . . . 62
AII.4.b
Lesson 12
Solve Quadratic Equations Algebraically . . . . . . . . . . . 68
AII.4.b
Lesson 13
Solve Rational Equations . . . . . . . . . . . . . . . . . . . . . . . 74
AII.4.c
Lesson 14
Solve Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . 80
AII.4.d
Lesson 15
Use Equations and Inequalities to Solve
Real-World Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 86
AII.4.a, AII.4.b, AII.4.c, AII.4.d
Solve Nonlinear Systems of Equations. . . . . . . . . . . . . 92
AII.5
Lesson 16
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 3
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Lesson 17
Arithmetic Sequences and Series. . . . . . . . . . . . . . . . 104
AII.2
Lesson 18
Graph Transformations of Functions . . . . . . . . . . . . . 111
AII.6
Lesson 19
Characteristics of Functions . . . . . . . . . . . . . . . . . . . . 119
AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e
Lesson 20
Graph Transformations of Quadratic Functions . . . . . 127
AII.6
Lesson 21
Characteristics of Quadratic Functions . . . . . . . . . . . 134
AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.f
Lesson 22
Graph Transformations of Cubic Functions . . . . . . . . 141
AII.6
Lesson 23
Graph Transformations of Polynomial Functions
of Higher Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
AII.6
Duplicating any part of this book is prohibited by law.
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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Virginia Standards of Learning
Algebra II
Lesson 24
Characteristics of Polynomial Functions . . . . . . . . . . 156
AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.f
Lesson 25
Graph Transformations of Exponential Functions . . . 163
AII.6
Lesson 26
Characteristics of Exponential Functions . . . . . . . . . . 169
AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e,
AII.7.f
Lesson 27
Geometric Sequences and Series . . . . . . . . . . . . . . . 175
AII.2
Lesson 28
Graph Transformations of Logarithmic Functions . . . 182
AII.6
Lesson 29
Characteristics of Logarithmic Functions . . . . . . . . . . 189
AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e
Lesson 30
Functions and Their Inverses . . . . . . . . . . . . . . . . . . . 195
AII.7.g
Lesson 31
Characteristics of Functions and Their Inverses . . . . 203
AII.7.a, AII.7.e, AII.7.g
Lesson 32
Composition of Functions. . . . . . . . . . . . . . . . . . . . . . 209
AII.7.g, AII.7.h
Lesson 33
Solutions, Zeros, x-Intercepts, and Factors
of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
AII.8
Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Chapter 4
Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Lesson 34
Curves of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
AII.9
Lesson 35
Direct, Inverse, and Joint Variation. . . . . . . . . . . . . . . 237
AII.10
Lesson 36
Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 242
AII.11
Lesson 37
Permutations and Combinations . . . . . . . . . . . . . . . . 252
AII.12
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Practice Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Duplicating any part of this book is prohibited by law.
Practice Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Algebra II Formula Sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Z-score Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
5
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Chapter 1 • Lesson 1
Properties of Exponents
SOL: AII.1.a
You can apply the properties of exponents to simplify expressions in exponential form. In the
examples below, the variables a and b are real numbers and the variables m and n are integers.
Product of Powers: To multiply powers with the same base, add the
exponents.
m
n
mn
a a a
Power of a Power: To raise a number in exponential form to a power,
multiply the exponents.
m n
mn
(a ) a
Power of a Product: To find a power of a product, find the power of each
factor and multiply.
(ab)m ambm
Quotient of Powers: To divide powers with the same base, subtract the
exponents.
am
__
amn if a 0
an
Power of a Quotient: To raise a quotient to a power, raise both the
numerator and the denominator to that power.
( __ba )
m
m
a
__
m
b
Power of Zero: Any nonzero number raised to the power of zero is 1.
0
a 1 if a 0
n
a
1 if a 0
__
n
a
1 an if a 0
___
a n
Duplicating any part of this book is prohibited by law.
Negative Exponents: Any nonzero number raised to a negative exponent
is equal to the reciprocal of that number with a positive exponent.
10 • Chapter 1: Expressions and Operations
230VA_Alg_II_SE_Final.indd 10
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Example 1
4
73
Simplify and then evaluate the expression: 7_____
5
7
Strategy
Step 1
Use the product of powers and quotient of powers to simplify.
Use the product of powers to rewrite the expression in the numerator.
4
3
43
7
7 7 7 7
4
3
7
7 __
7
So, 7_____
5
5
7
Step 2
7
Use the quotient of powers to rewrite the fraction.
77 775 72
__
75
Step 3
Evaluate the expression.
2
7 49
Solution
4
73 in simplified form is 72 and has a value of 49.
The expression 7_____
5
7
You can also apply properties of exponents to evaluate an algebraic expression. An
algebraic expression is a combination of numbers and variables that are connected by
one or more operations.
Example 2
Simplify the expression: (x3y2z0)3
Strategy
Step 1
Apply the properties of exponents that relate to multiplication.
Use the power of a product to rewrite the expression.
(x3y2z0)3 (x3)3 (y2)3 (z0)3
Step 2
Find the power of each power.
Duplicating any part of this book is prohibited by law.
3 3
33
9
(x ) x x
(y2)3 y 2 3 y6
(z0)3 z0 3 z0 1
Note: You apply the power of zero to determine that z0 1.
Step 3
Rewrite the expression in simplest form.
(x3y2z0)3 x9 y6 1 x9y6
Solution
The simplified form of (x3y2z0)3 is x9y6.
11
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Example 3
3
6
What is the value of the expression ___
1 ?
2
Strategy
Step 1
Apply the properties of negative exponents.
Write an equivalent expression with positive exponents.
6
3
1
__
3
6
1 21
___
1
2
1
6 3
2
__
So, ___
1 3
2
6
Step 2
Evaluate the numerator and the denominator. Then simplify.
1
2 ___
2
__
3
216
6
2 is a common factor in both the numerator and the denominator, so:
1
2 ______
1 2 ___
1
___
216
2 108
108
1
Solution
3
6
1 .
___
The value of the expression ___
1 is
108
2
Coached Example
3
What is the result when a4b
is divided by a 2b2?
3
Write an expression that represents a4b
2 a b
divided by a 2b : ______
2 ____
4
____
a b
Simplify the expression using the quotient of powers property.
b___
Add and subtract the exponents.
a
4 (______)
b___ a___ b___
Rewrite the expression so that all the exponents are positive.
___
a
___
b___
3
When a4b
is divided by a 2b2, the quotient is ____________.
Duplicating any part of this book is prohibited by law.
4 (2)
a
12 • Chapter 1: Expressions and Operations
230VA_Alg_II_SE_Final.indd 12
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Lesson 1: Properties of Exponents
Lesson Practice
Choose the correct answer.
1.
What is d 6 d 4 in simplified form?
A. d
2
B. d
10
5.
25s
A. ___
6
t
25s 2
___
B.
t5
10s 2
C. ___
5
t
2
25s
D. ___
6
t
D. d 64
Which is equivalent to x5 x 8?
1
A. __
40
B.
(t )
2
C. d 24
2.
5s 2
?
Which is equivalent to ___
3
6.
x
1
__
x3
yz
4 2
81x y
A. _____
z
C. x3
z
B. ____
4 2
40
D. x
Evaluate: (80)2
3z
D. ___
4 2
( )
x y
A. 64
Duplicating any part of this book is prohibited by law.
4.
( 3x y )
3x y
( ____
z )
4 2
C.
3.
4
3x ?
Which is equivalent to _____
2 1
B.
8
C.
1
D.
0
7.
( )
Simplify: (2jk)3 _13 j 2k 5
2
A. _89 j 7k 13
B. _23 j 7k 13
3
(5)
Evaluate: _3
C. _23 j 7k 13
15
A. __
9
27
B. ___
125
125
C. ___
27
27
D. ___
125
D. _89 j 12k 30
8.
2
ab
Which is equivalent to ________
3 2 3 ?
(4a b )
A. 64a11b5
11
a
B. ____
5
64b
a11
C. ____
5
64b
11
a
D. 64__
5
b
13
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