Table of Contents
Georgia Performance Standards Correlation Chart . . . 4
Georgia Performance
Standards
Common Core
State Standards
MM2N1.a, MM2N1.b, MM2N1.c,
MM2N1.d, MM2P2.d
N-CN.1
Chapter 1
Number and Operations . . . . . . . . . . . . . . . 7
Lesson 1
Introduction to Complex Numbers . . . . . . . . 8
Lesson 2
Adding and Subtracting Complex
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
MM2N1.c, MM2N1.d
N-CN.2
Multiplying and Dividing Complex
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
MM2N1.c, MM2N1.d
N-CN.2, N-CN.3
Properties of Exponents . . . . . . . . . . . . . . . .21
MM2A2.a
Lesson 3
Lesson 4
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
Piecewise and Exponential Functions . . . 29
Lesson 5
Introduction to Piecewise Functions. . . . . . 30
MM2A1.a, MM2P5.a, MM2P5.b,
MM2P5.c
A-CED.3, F-IF.7.b
Lesson 6
Characteristics of Piecewise Functions . . . .37
MM2A1.b, MM2P3.a, MM2P3.d
F-IF.4
Lesson 7
Solving Absolute Value Equations . . . . . . . .44
MM2A1.c, MM2P1.a, MM2P1.c
A-REI.1, A-REI.11, F-IF.2
Lesson 8
Solving Absolute Value Inequalities . . . . . . .49
MM2A1.c, MM2P1.a, MM2P1.c
A-REI.1, F-IF.2
Lesson 9
Basic Exponential Functions. . . . . . . . . . . . 55
MM2A2.e, MM2P1.b, MM2P4.c,
MM2P5.c
A-SSE.1a, F-IF.7.e, F-LE.1.c
Lesson 10
Characteristics of Exponential
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .61
MM2A2.b, MM2P3.a, MM2P3.d
F-IF.4, F-IF.8.b
Lesson 11
Solving Exponential Equations. . . . . . . . . . .67
MM2A2.d, MM2P1.a, MM2P1.c
A-SSE.3.c, A-REI.1, A-REI.11,
F-IF.2
Lesson 12
Solving Exponential Inequalities . . . . . . . . . 72
MM2A2.d, MM2P1.a, MM2P1.c,
MM2P1.d
A-SSE.3.c, A-REI.1, F-IF.2
Lesson 13
Transformations of Exponential
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .77
MM2A2.c
F-BF.3
MM2A2.f, MM2A2.g, MM2P4.a,
MM2P4.b
A-SSE.4, F-IF.3, F-BF.2,
F-LE.2
Lesson 14
Geometric Sequences. . . . . . . . . . . . . . . . . 84
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
Duplicating any part of this book is prohibited by law.
Chapter 2
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Chapter 1 • Lesson 1
Introduction to Complex Numbers
Georgia Performance Standard(s):
MM2N1.a, MM2N1.b, MM2N1.c,
MM2N1.d, MM2P2.d
Squaring a number means raising it to the power of 2. Taking the square root of a number is
the inverse of squaring it.
__
__
32 3 3 9, so √ 9 √ 32 3
Squaring a negative number always results in a positive number.
(3)2 (3) (3) 9
___
Now consider √ 9 . That number is not a real number, because there is no real number you
___
can multiply by itself to get 9. However, we can represent √ 9 as an imaginary number.
___
Imaginary numbers are written with i, which is equal to √ 1 . By using the property that
___
__
__
√ ab √ a √ b , you can write the square root of any negative number as the product of a real
number and i. That is, you can write it in the form bi.
___
______
__
___
√ 9 √ 9 1 √ 9 √ 1 3i
Example 1
_____
Write √ 20 in imaginary form.
Strategy
Rewrite the square root in the form bi.
_____
Step 1
Write √ 20 as a product of b and i.
_____
___
___
√ 20 √ 1 √ 20
___
Step 2
Simplify √ 20 .
___
__
__
__
√ 20 √ 4 √ 5 2√ 5
Step 3
Substitute and multiply.
___
___
__
__
Solution
__
In imaginary form, the number is 2i√ 5 .
Duplicating any part of this book is prohibited by law.
√ 1 √ 20 i 2√ 5 2i√ 5 .
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You should also know how to find different powers of i using the properties of exponents.
1 i
兹苶
The square of i is equal to 1.
1)2 1
i 2 (兹苶
i i i 1 i i
3
2
1
i 4 i 2 i 2 1 1 1
i5 i4 i1 1 i i
i 6 i 5 i 1 i i i 2 1
i 7 i 6 i 1 1 i i
When the exponent is a multiple
of 4, the power of i is equal to 1.
i 8 i 4 i 4 1 1 1
Notice that the pattern repeats: i, 1, i, 1. You can use this pattern to simplify large powers of i.
Example 2
Simplify i 35.
Strategy
Step 1
Use the pattern of powers of i to find i 35.
Find the closest multiple of 4 that is not greater than the exponent, 35.
32 is the closest multiple of 4. So, i
Step 2
1.
Write i 35 as the product of i to a power of a multiple of 4 and another power of i.
Simplify.
i
Solution
32
i
35
35
i 32 i 3 1 i i
i
A complex number has two parts: a real part and an imaginary part. It is written in the standard
form a bi, in which a and b are real numbers, a is the real part, and bi is the imaginary part.
Duplicating any part of this book is prohibited by law.
Example 3
___
Write 3 √ 7 as a complex number in standard form.
Strategy
Step 1
Identify the real part and the imaginary part. Then rewrite the imaginary part
in the form bi.
Identify the real part and the imaginary part.
3 7
real part
imaginary part
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Step 2
Rewrite the imaginary part in imaginary form.
___
___
__
__
√ 7 √ 1 √ 7 i√ 7
Rewrite the number in the form a bi.
Step 3
___
__
3 √ 7 3 i√ 7
__
In standard form, the complex number is 3 i√ 7 .
Solution
_______
The absolute value of a complex number in standard form a bi is equal to √ a2 b2 .
Example 4
Find the absolute value of 5 12i.
_______
Identify a and b. Then find √ a2 b2 .
Strategy
a_______
5 and b___________
12
_________
____
√a2 b2 √(5)2 122 √25 144 √169 13
The absolute value of 5 12i is 13.
Solution
Coached Example
_____
Write 18 √ 25 as a complex number in standard form.
Identify the real part and the imaginary part of the complex number.
The real part is
. The imaginary part is
.
Rewrite the imaginary part so it is in imaginary form.
_____
18 √ 25 The standard form is
.
Duplicating any part of this book is prohibited by law.
Rewrite in the form a bi.
10 • Chapter 1: Number and Operations
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Lesson 1: Introduction to Complex Numbers
Lesson Practice
Choose the correct answer.
1.
____
Which is equivalent to √ 16 ?
6.
What is the absolute value of 3 4i ?
__
__
A. √ 5
__
B. √ 7
A. i√ 8
__
B. 2i√ 2
C. 5
D. 7
C. 4i
D. 8i
2.
7.
____
Which shows the standard form of
___
70 √ 3 ?
___
Which is equivalent to √ 27 ?
A. 3 i√ 70
__
A. 3i√ 3
B. 3 70i
B. 3i√ 3
C. 70 i√ 3
C. 3i√ 9
D. 70 3i
__
__
__
D. 9i
3.
_____
8.
Which is equivalent to √ 200 ?
__
A. 10i√ 2
A. 9i
B. 20i
B. 1 8i
C. 100i
C. 1 8i
__
D. 9i
D. 100i√ 2
4.
Which shows
the standard form
____
of 1 √ 64 ?
Which is equivalent to i 13?
9.
What is the absolute value of 14 7i?
__
Duplicating any part of this book is prohibited by law.
A. 1
A. 7√ 5
B. i
B. 5√ 7
__
C. √ 7
__
D. √ 5
__
C. i
D. 1
5.
96
Which is equivalent to i ?
A. 1
B. i
C. i
10. Which shows the standard form
____
of 12 √ 50 ?
__
A. 10 2i√ 3
__
B. 12 2i√ 5
C. 12 5i
__
D. 12 5i√ 2
D. 1
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