Lesson 2.1 Skills Practice
Name
Date
Shape and Structure Forms of Quadratic Functions
2
Vocabulary Write an example for each form of quadratic function and tell whether the form helps determine the x-intercepts, the y-intercept, or the vertex of the graph. Then describe how to determine the concavity of a parabola. 1. Standard form:
2. Factored form:
© Carnegie Learning
3. Vertex form:
4. Concavity of a parabola:
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Lesson 2.1 Skills Practice
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Problem Set Circle the function that matches each graph. Explain your reasoning. 1.
2.
y
2
y
4
6
2
4
0
28 26 24 22
2
x
22 24 22 24
f(x) 5 6(x 2 2)(x 2 8) 1 f(x) 5 2 __ (x 1 2)(x 1 8) 2 f(x) 5 __ 1 (x 1 2)(x 1 8) 2 __ f(x) 5 1 (x 2 2)(x 2 8) 2
0
2
x
4
22
f(x) 5 2x2 2 x 1 7
f(x) 5 22x2 2 x 1 7
f(x) 5 2x2 2 2x 1 7
f(x) 5 22x2 2 x 2 2
The a–value is positive so the parabola opens up. Also, the roots are at 22 and 28.
4.
y
y
8
2
6
0 x
25 24 23 22 21 4
22
2
24
0
160
2
4
6
8
26
x
f(x) 5 0.25(x 2 4)2 1 2
f(x) 5 23(x 1 2)(x 2 5)
f(x) 5 4(x 2 2)2 2 2
f(x) 5 3(x 1 2)(x 1 5)
f(x) 5 20.25(x 1 4)2 1 2
f(x) 5 3(x 2 2)(x 2 5)
f(x) 5 0.25(x 2 2)2 1 4
f(x) 5 23(x 2 2)(x 2 5)
© Carnegie Learning
3.
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Lesson 2.1 Skills Practice
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Name
Date
5.
16
6.
y
y 8
12
4
4 22
0
2
6
8
2 2
4
x 22
f(x) 5 x2 1 5x 2 4
f(x) 5 2x2 1 5x 1 10
f(x) 5 x2 1 5x 1 4
f(x) 5 2x2 1 5x 1 4
0
2
4
6
x
1 f(x) 5 2 __ (x 2 2)2 2 f(x) 5 __ 1 (x 2 2)2 1 2 2 f(x) 5 __ 1 (x 2 2)2 2 __ f(x) 5 1 (x 1 2)2 2
Use the given information to determine the most efficient form you could use to write the quadratic function. Write standard form, factored form, or vertex form. 7. vertex (3, 7) and point (1, 10) vertex form 8. points (1, 0), (4, 23), and (7, 0)
© Carnegie Learning
3 9. y-intercept (0, 3) and axis of symmetry 2 __ 8
10. points (21, 12), (5, 12), and (22, 22)
11. roots (25, 0), (13, 0) and point (27, 40)
12. maximum point (24, 28) and point (23, 215)
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Lesson 2.1 Skills Practice
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Convert each quadratic function in factored form to standard form. 13. f(x) 5 (x 1 5)(x 2 7)
14. f(x) 5 (x 1 2)(x 1 9)
f(x) 5 x 2 7x 1 5x 2 35 2
5 x2 2 2x 2 35
2
15. f(x) 5 2(x 2 4)(x 1 1)
16. f(x) 5 23(x 2 1)(x 2 3)
17. f(x) 5 __ 1 (x 1 6)(x 1 3) 3
5 (x 2 6)(x 1 2) 18. f(x) 5 2 __ 8
Convert each quadratic function in vertex form to standard form. 19. f(x) 5 3(x 2 4)2 1 7
20. f(x) 5 22(x 1 1)2 2 5
f(x) 5 3(x2 2 8x 1 16) 1 7 5 3x2 2 24x 1 55
(
)
2 3 21. f(x) 5 2x 1 __ 7 2 __ 2 2
22. f(x) 5 2(x 2 6)2 1 4
1 23. f(x) 5 2 __ (x 2 10)2 2 12 2
24. f(x) 5 ___ 1 (x 1 100)2 1 60 20
© Carnegie Learning
162
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Lesson 2.1 Skills Practice
page 5
Name
Date
Write a quadratic function to represent each situation using the given information.
2
25. Cory is training his dog, Cocoa, for an agility competition. Cocoa must jump through a hoop in the middle of a course. The center of the hoop is 8 feet from the starting pole. The dog runs from the starting pole for 5 feet, jumps through the hoop, and lands 4 feet from the hoop. When Cocoa is 1 foot from landing, Cory measures that she is 3 feet off the ground. Write a function to represent Cocoa’s height in terms of her distance from the starting pole. h(d) 5 a(d 2 r1) (d 2 r2) 3 5 a(11 2 5) (11 2 12) 3 5 a(6)(21) 3 5 26a
___ 3 5 a
26 20.5 5 a
h(d) 5 20.5(d 2 5) (d 2 12)
© Carnegie Learning
26. Sasha is training her dog, Bingo, to run across an arched ramp, which is in the shape of a parabola. To help Bingo get across the ramp, Sasha places a treat on the ground where the arched ramp begins and one at the top of the ramp. The treat at the top of the ramp is a horizontal distance of 2 feet from the first treat, and Bingo is 6 feet above the ground when he reaches the top of the ramp. Write a function to represent Bingo’s height above the ground as he walks across the ramp in terms of his distance from the beginning of the ramp.
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Lesson 2.1 Skills Practice
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27. Ella’s dog, Doug, is performing in a special tricks show. Doug can fling a ball off his nose into a bucket 20 feet away. Ella places the ball on Doug’s nose, which is 2 feet off the ground. Doug flings the ball through the air into a bucket sitting on a 4-foot platform. Halfway to the bucket, the ball is 10 feet in the air. Write a function to represent the height of the ball in terms of its distance from Doug.
2
© Carnegie Learning
28. A spectator in the crowd throws a treat to one of the dogs in a competition. The spectator throws the treat from the bleachers 19 feet above ground. The treat amazingly flies 30 feet and just barely crosses over a hoop which is 7.5 feet tall. The dog catches the treat 6 feet beyond the hoop when his mouth is 1 foot from the ground. Write a function to represent the height of the treat in terms of its distance.
164
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Lesson 2.1 Skills Practice
Name
page 7
Date
29. Hector’s dog, Ginger, competes in a waterfowl jump. She jumps from the edge of the water, catches a toy duck at a horizontal distance of 10 feet and a height of 2 feet above the water, and lands in the water at a horizontal distance of 15 feet. Write a function to represent the height of Ginger’s jump in terms of her horizontal distance.
2
© Carnegie Learning
30. Ping is training her dog, TinTin, to jump across a row of logs. He takes off from a platform that is 7 feet high with a speed of 18 feet per second. Write a function to represent TinTin’s height in terms of time as he jumps across the logs.
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© Carnegie Learning
2
166
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Lesson 2.2 Skills Practice
Name
Date
Function Sense Translating Functions
2
Vocabulary Complete each sentence with the correct term from the word bank.
transformation
reference point
translation
argument of a function
1. A(n)
is one of a set of key points that help identify the basic function.
2. The mapping, or movement, of all the points of a figure in a plane according to a common operation is called a(n) . 3. The
is the variable, term, or expression on which the function operates.
4. A(n) same distance and direction.
is a type of transformation that shifts an entire figure or graph the
Problem Set Given f(x) 5 x2 , complete the table and graph h(x). 1. h(x) 5 (x 2 1)2 1 3 y
S
Corresponding Points on h(x)
8
(0, 0)
S
(1, 3)
6
(1, 1)
S
(2, 4)
(2, 4)
S
(3, 7)
© Carnegie Learning
Reference Points on f(x)
h(x)
4 2 28 26 24 22
0
2
4
6
8
x
22 24 26 28
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Lesson 2.2 Skills Practice
page 2
2. h(x) 5 (x 1 2)2 2 1
2
Reference Points of f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
y
Corresponding Points on h(x)
8 6 4 2 28 26 24
0
2
4
6
8
x
2
4
6
x
22 24 26 28
3. h(x) 5 (x 1 7)2 Reference Points of f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
y
Corresponding Points on h(x)
8 6 4 2 210 28 26 24 22
0 22 24
28
168
© Carnegie Learning
26
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Lesson 2.2 Skills Practice
page 3
Name
Date
4. h(x) 5 (x 2 3)2 1 4 Reference Points of f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
2
y
Corresponding Points on h(x)
8 6 4 2 28 26 24 22
0
2
4
6
8
x
2
4
6
8
x
22 24 26 28
5. h(x) 5 x2 2 9 Reference Points of f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
y
Corresponding Points on h(x)
6 4 2 28 26 24 22
0 22 24
© Carnegie Learning
26 28 210
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Lesson 2.2 Skills Practice
page 4
6. h(x) 5 (x 1 4)2 2 4
2
Reference Points of f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
y
Corresponding Points on h(x)
8 6 4 2 28 26 24 22
0
2
4
6
8
x
22 24 26 28
Each given function is in transformational function form g(x) 5 Af(B(x 2 C)) 1 D, where f(x) 5 x 2. Identify the values of C and D for the given function. Then, describe how the vertex of the given function compares to the vertex of f(x). 7. g(x) 5 f(x 2 4) 1 12 The C–value is 4 and the D–value is 12, so the vertex will be shifted 4 units to the right and 12 units up to (4, 12).
9. g(x) 5 f(x 2 5) 211
© Carnegie Learning
8. g(x) 5 f(x 1 8) 2 9
10. g(x) 5 f(x 2 6) 1 10
170
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Lesson 2.2 Skills Practice
page 5
Name
Date
11. g(x) 5 f(x 1 2) 1 3
2 12. g(x) 5 f(x 1 4) 2 2
Analyze the graphs of b(x), c(x), d(x), and f(x). Write each function in terms of the indicated function. y b(x)
d(x)
8 6 4
f(x)
2 28 26 24 22
0 22
2
4
6
8
x
c(x)
24 26 28
13. Write b(x) in terms of f(x).
14. Write c(x) in terms of f(x).
© Carnegie Learning
b(x) 5 f(x 1 5) 2 2
15. Write d(x) in terms of f(x).
16. Write d(x) in terms of b(x).
17. Write c(x) in terms of b(x).
18. Write b(x) in terms of c(x).
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© Carnegie Learning
2
172
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Lesson 2.3 Skills Practice
Name
Date
Up and Down Vertical Dilations of Quadratic Functions
2
Vocabulary 1. Label the graph to identify the vertical dilations (vertical compression and vertical stretching) and the reflection of the function f(x) 5 x2 . Also label the line of reflection. y f(x) 5 x2
0
x
Problem Set Graph each vertical dilation of f(x) 5 x2 and tell whether the transformation is a vertical stretch or a vertical compression and if the graph includes a reflection. 2. p(x) 5 __ 1 x2 1. g(x) 5 4x2 8
© Carnegie Learning
y
24 23 22 21
y
f(x)
8
8
6
6
4
4
2
2
0
1
2
3
4
x
28 26 24 22
0
22
22
24
24
26
26
28
28
f(x)
2
4
6
8
x
vertical stretch Chapter 2 Skills Practice
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Lesson 2.3 Skills Practice 4. m(x) 5 2.5x2
3. h(x) 5 25x2 4
y
10 f(x)
3
24 23 22 21
8
2
6
1
4
0
1
2
3
4
24 23 22 21
0
22
22
23
24
24
26
25
28
1
2
3
4
x
8
x
1 6. g(x) 5 2 __ x2 2 3 2
5. d(x) 5 __ 2 x2 5
y
y 10
f(x)
8
12
6
8
4
4 28 26 24 22
0
2
f(x)
16
2
174
f(x)
2
x
21
28 26 24 22
y
4
6
8
x
0
2
4
6
24
22
28
24
212
26
216
© Carnegie Learning
2
page 2
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Lesson 2.3 Skills Practice
Name
page 3
Date
Each given function is in transformational function form g(x) 5 Af(B(x 2 C)) 1 D, where f(x) 5 x 2. Describe how g(x) compares to f(x). Then, use coordinate notation to represent how the A-, C-, and D-values transform f(x) to generate g(x).
2
7. g(x) 5 23(f(x)) 2 1 The A–value is 23, so the graph will have a vertical stretch by a factor of 3 and will be reflected about the line y 5 21. The C–value is 0 and the D–value is 21 so the vertex will be shifted 1 unit down to (0, 21). (x, y) → (x, 23y 2 1) 8. g(x) 5 __ 1 (f(x)) 1 8 4
9. g(x) 5 24(f(x 1 3))
10. g(x) 5 __ 1 f(x 2 6) 2 3 3
© Carnegie Learning
11. g(x) 5 20.75f(x 1 4) 2 2
(
)
12. g(x) 5 __ 4 fx 2 __ 1 1 __ 2 3 3 3
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Lesson 2.3 Skills Practice
page 4
Write the function that represents each graph. 13.
2
14.
y 8
8
4
4
0
28 24
y
4
8
x
28 24
0
24
24
28
28
f(x) 5 3(x 1 2)2 2 4
15.
16.
y
4
8
x
6
8
x
y 8
16
4
12
0
8
2
4
24
4
28 24 22 0
2
4
x
17.
y
8
8
4
4
0
28 24
176
18.
y
4
8
x
216 212 28 24
0
24
24
28
28
x
© Carnegie Learning
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Lesson 2.4 Skills Practice
Name
Date
Side to Side Horizontal Dilations of Quadratic Functions
2
Vocabulary 1. Explain the differences and similarities between horizontal dilation, horizontal stretching, and horizontal compression of a quadratic function.
Problem Set Complete the table and graph m(x). Then, describe how the graph of m(x) compares to the graph of f(x). 1 x 1. f(x) 5 x2 ; m(x) 5 f __ 5
( )
Reference Points on f(x)
S
8 6
(0, 0)
S
(0, 0)
(5, 25)
S
(5, 1)
S
(10, 4)
(10, 100)
y
Corresponding Points on m(x)
4 2 216 212 28 24
© Carnegie Learning
(15, 225)
S
m(x)
(15, 9)
0
4
8
12
16
x
22 24
The function m(x) is a horizontal stretch of f(x) by a factor of 5.
26 28
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Lesson 2.4 Skills Practice
page 2
2. f(x) 5 x2 ; m(x) 5 f(1.5x)
2
Reference Points on f(x)
S
(0, 0)
S
(1, 1)
S
(2, 4)
S
(4, 16)
S
y
Corresponding Points on m(x)
8 6 4 2 28 26 24 22
0
2
4
6
8
x
0.5
1
1.5
2
x
2
4
6
8
x
22 24 26 28
3. f(x) 5 x2 ; m(x) 5 f(4x) Reference Points on f(x)
S
(0, 0)
S
(0.5, 0.25)
S
(1, 1) (2, 4)
y
Corresponding Points on m(x)
16 12 8 4
S
22 21.5 21 20.5 0 24
S
28 212 216
4. f(x) 5 x2 ; m(x) 5 f(0.25x) S
(0, 0)
S
6
(4, 16)
S
4
(8, 64)
S
2
(12, 144)
S
8
28 26 24 22
0
© Carnegie Learning
y
Corresponding Points on m(x)
Reference Points on f(x)
22 24 26 28
178
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Lesson 2.4 Skills Practice
page 3
Name
Date
( )
5. f(x) 5 x2 ; m(x) 5 f __ 2 x 3
2
y
Reference Points on f(x)
S
Corresponding Points on m(x)
(0, 0)
S
14
(3, 9)
S
12
(6, 36)
S
(9, 81)
S
16
10 8 6 4 2 28 26 24 22
0
2
4
6
8
x
2
4
6
8
x
6. f(x) 5 x2 ; m(x) 5 f (2x) y
Corresponding Points on m(x)
S
(0, 0)
S
14
(1, 1)
S
12
(2, 4)
S
(3, 9)
S
© Carnegie Learning
Reference Points on f(x)
16
10 8 6 4 2 28 26 24 22
0
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Lesson 2.4 Skills Practice
page 4
The graph of f(x) is shown. Sketch the graph of the given transformed function. 8. t(x) 5 2f(x 2 4)
7. d(x) 5 f(2x)
d(x)
28 26 24 22
y
8
8
6
6
f(x)
4
4
2
2
0
2
4
6
8
x
28 26 24 22
22
22
24
24
26
26
28
28
9. m(x) 5 22f(x 1 3) 1 5 y
y
8
8 f(x) f(x)
4
4
6
8
x
2
4
6
8
x
6 4
2
2 0
28 26 24 22
2
10. g(x) 5 (2x 1 1) 2 4
6
180
0
f(x)
2
4
6
8
x
28 26 24 22
0
22
22
24
24
26
26
28
28
© Carnegie Learning
2
y
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Lesson 2.4 Skills Practice
page 5
Name
Date
(
)
11. r(x) 5 f__ 1 x 2 1 1 2 2
210 28 26 24 22
f(x)
12. p(x) 5 2f(x 1 1) 2 3 y
y
8
8
6
6
4
4
2
2
0
2
4
6
x
28 26 24 22
0
22
22
24
24
26
26
28
28
2 f(x) 2
4
6
8
x
Write an equation for w(x) in terms of v(x). 13.
14.
y 8
v(x)
6
6
4
4 w(x)
2
© Carnegie Learning
28 26 24 22
( __ )
w(x) 5 v 1 x 4
y 8
0
2
4
6
8
w(x)
v(x)
2 x
28 26 24 22
0
22
22
24
24
26
26
28
28
2
4
6
8
x
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Lesson 2.4 Skills Practice 15.
page 6 16.
y
y 8
8 w(x)
2
6
v(x)
6
v(x)
4
4
2
2 0
28 26 24 22
2
4
6
8
x
28 26 24 22
22
24
24
26
26
28
28
2
4
6
8
x
4
x
w(x)
18.
y
y 4
8
3
6 v(x)
w(x)
4
2 1
2 28 26 24 22
0
2
4
6
8
x
0
24 23 22 21
22
21
24
22 w(x)
26
2
3
v(x)
23 24
28
1
© Carnegie Learning
17.
182
0
22
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Lesson 2.5 Skills Practice
Name
Date
What’s the Point? Deriving Quadratic Functions
2
Problem Set Use your knowledge of reference points to write an equation for the quadratic function that satisfies the given information. Use the graph to help solve each problem. 1. Given: vertex (3, 5) and point (5, 23)
y 8
f(x) 5 22(x 2 3)2 1 5
6
Point (5, 23) is point B because it is 2 units from the axis of symmetry. The range between the vertex and point B on the basic function is 4. The range between the vertex and point B is 4 3 (22), therefore the a-value must be 22.
(3, 5)
4 2 28 26 24 22
0
28
2
22 24
6 8 (5, 23) B´ 2 units
x
6
x
4
26 28
2. Given: vertex (22, 29) and one of two x-intercepts (1, 0)
y 8 6 4 2 (1, 0)
© Carnegie Learning
28 26 24 22
0
2
4
8
22 24 26 (22, 29) 28
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Lesson 2.5 Skills Practice
page 2
3. Given: two x-intercepts (27, 0) and (5, 0) and one point (24, 29)
y 8 6 4
2
2
(27, 0)
(5, 0)
28 26 24 22
0
2
4
6
8
x
4
6
8
x
22 24 26 (24, 29)
4. Given: vertex (24, 3) and y-intercept (0, 11)
28
y 12 10
(0, 11)
8 6 4 (24, 3)
28 26 24 22
2 0
2
24
184
© Carnegie Learning
22
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Lesson 2.5 Skills Practice
page 3
Name
Date
5. Given: exactly one x-intercept (2, 0) and y-intercept (0, 212)
y 4
2
2 (2, 0) 28 26 24 22
0
2
4
6
8
x
2
4
6
8
x
22 24 26 28 210 (0, 212)
6. Given: vertex (26, 21) and point (23, 35)
(23, 35)
y
32 24 16 8 (26, 21)
24 22
0
© Carnegie Learning
28 216 224 232
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Lesson 2.5 Skills Practice
page 4
Use a graphing calculator to determine the quadratic equation for each set of three points that lie on a parabola. 7. (24, 12), (22, 214), (2, 6) f(x) = 3x2 1 5x 2 16
2 8. (5, 256), (1, 24), (210, 226)
9. (28, 8 ), (24, 6), (4, 38)
10. (22, 3), (2, 29), (5, 260)
12. (22, 13), (1, 217), (7, 31)
186
© Carnegie Learning
11. (0, 3), (25, 22.4 ), (15, 27.8)
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Lesson 2.5 Skills Practice
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Create a system of equations and use algebra to write a quadratic equation for each set of three points that lie on a parabola.
2
13. (23, 12), (0, 9), (3, 24) Equation 1: 12 5 9a 2 3b 1 c Equation 2: 9 5 c Equation 3: 24 5 9a 1 3b 1 c Substitute equation 2 into equation 1 and solve for a. 12 5 9a 2 3b 1 9 3 5 9a 2 3b 3 1 3b 5 9a a 5 1 1 1 b 3 3
__ __
Substitute the value for a in terms of b and the value for c into equation 3 and solve for b.
( __ __ )
24 5 9 1 1 1 b 1 3b 1 9 3 3 24 5 3 1 3b 1 3b 1 9 15 5 3 1 6b 12 5 6b b52 Substitute the values for b and c into equation 1 and solve for a. 12 5 9a 2 3(2) 1 9 15 5 9a 1 3 9 5 9a a51
© Carnegie Learning
Substitute the values for a, b, and c into a quadratic equation in standard form. f(x) 5 x2 1 2x 1 9
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14. (22, 22), (1, 25), (2, 218)
© Carnegie Learning
2
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15. (2, 9), (0, 25), (210, 215)
© Carnegie Learning
2
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Lesson 2.5 Skills Practice
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16. (21, 2), (4, 27), (23, 20)
© Carnegie Learning
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17. (5, 26), (22, 8), (3, 4)
© Carnegie Learning
2
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18. (1, 17), (21, 29), (2, 105)
© Carnegie Learning
2
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Lesson 2.6 Skills Practice
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Now It’s Getting Complex . . . But It’s Really Not Difficult! Complex Number Operations
2
Vocabulary
© Carnegie Learning
Match each term to its corresponding definition. 1. the number i
A. a number in the form a 1 bi where a and b are real numbers and b is not equal to 0
2. imaginary number
B. term a of a number written in the form a 1 bi
3. pure imaginary number
C. a polynomial with two terms
4. complex number
D. pairs of numbers of the form a 1 bi and a 2 bi
5. real part of a complex number
E. a number such that its square equals 21
6. imaginary part of a complex number
F. a number in the form a 1 bi where a and b are real numbers
7. complex conjugates
G. a polynomial with three terms
8. monomial
H. a number of the form bi where b is not equal to 0
9. binomial
I. term bi of a number written in the form a 1 bi
10. trinomial
J. a polynomial with one term
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Problem Set Calculate each power of i. 2. i361
1. i48 i48 5 (i4)12
2
5 112 51
4. i1000
5. i222
6. i27
© Carnegie Learning
3. i55
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Rewrite each expression using i. _____
7. √ 272 _____
_________
_____
_____
8. √ 249 1 √ 223
2
√ 272 5 √ 36(2)(21)
5
__ 6√2 i
______
____
9. 38 2 √2200 1√ 121
_____
__
√ 248 2 12 11. ___________
3
___ ___ _____ √ √ 21 12 ____ ____ 2√ 228 1 2
6
_____
___
√ 80 √ 275 1 14. ____________
10
© Carnegie Learning
3
_____
√ √ 215 12. _______________ 1 1 4 2
4
13.
_____
10. √ 245 1 21
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Simplify each expression. 15. (2 1 5i ) 2 (7 2 9i )
16. 26 1 8i 2 1 2 11i 1 13
(2 1 5i ) 2 (7 2 9i ) 5 2 1 5i 2 7 1 9i 5 (2 2 7) 1 (5i 1 9i )
2
5 25 1 14i
17. 2(4i 2 1 1 3i ) 1 (6i 2 10 1 17)
18. 22i 1 13 2 (7i 1 3 1 12i ) 1 16i 2 25
19. 9 1 3i(7 2 2i )
20. (4 2 5i )(8 1 i )
21. 20.5(14i 2 6) 2 4i(0.75 2 3i )
3 1 __ 22. __ 1 i 2 __ 1 2 __ 3 i 4 8 4 2
) (
)
© Carnegie Learning
(
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Determine each product. 24. (4i 2 5)(4i 1 5)
23. (3 1 i )(3 2 i )
2
(3 1 i )(3 2 i ) 5 9 2 3i 1 3i 2 i
2
5 9 2 (21) 5 10
(
) (
)
25. (7 2 2i )(7 1 2i )
26. __ 1 1 3i __ 1 2 3i 3 3
27. (0.1 1 0.6i )(0.1 2 0.6i )
28. 22[(2i 2 8)(2i 1 8)]
Identify each expression as a monomial, binomial, or trinomial. Explain your reasoning. 29. 4xi 1 7x © Carnegie Learning
The expression is a monomial because it can be rewritten as (4i 1 7)x, which shows one x term.
30. 23x 1 5 2 8xi 1 1
31. 6x2i 1 3x2
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32. 8i 2 x3 1 7x2i
33. xi 2 x 1 i 1 2 2 4i
2
34. 23x3i 2 x2 1 6x3 1 9i 2 1
Simplify each expression, if possible. 35. (x 2 6i )2 (x 2 6i )2 5 x2 2 6xi 2 6xi 1 36i2 5 x2 2 12xi 1 36(21) 5 x2 2 12xi 2 36
36. (2 1 5xi )(7 2 xi )
38. (2xi 2 9)(3x 1 5i )
198
© Carnegie Learning
37. 3xi 2 4yi
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39. (x 1 4i )(x 2 4i )(x 1 4i )
2
40. (3i 2 2xi )(3i 2 2xi ) 1 (2i 2 3xi )(2 2 3xi )
For each complex number, write its conjugate. 41. 7 1 2i
42. 3 1 5i
© Carnegie Learning
7 2 2i
43. 8i
44. 27i
45. 2 2 11i
46. 9 2 4i
47. 213 2 6i
48. 221 1 4i
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Calculate each quotient. 49. ______ 3 1 4i 5 1 6i 2 24i2 3 1 4i 5 3 1 4i ? 5 2 6i 5 15 2 18i 1 20i 5 1 6i 5 1 6i 5 2 6i 25 2 30i 1 30i 2 36i2
______ ______ ______ ____________________ 2i 24 5 _______ 39 1 39 1 ___ 2 i 5 ____________ 15 1 2i 1 5 ___ 25 1 36 61 61 61
2
8 1 7i 50. ______ 21i
26 1 2i 51. ________ 2 2 3i
21 1 5i 52. ________ 1 2 4i
4 2 2i 54. ________ 21 1 2i
200
© Carnegie Learning
6 2 3i 53. ______ 22i
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You Can’t Spell “Fundamental Theorem of Algebra” without F-U-N!
2
Quadratics and Complex Numbers Vocabulary Write a definition for each term in your own words. 1. imaginary roots
2. discriminant
3. imaginary zeros
© Carnegie Learning
4. degree of a polynomial equation
5. Fundamental Theorem of Algebra
6. double root
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Problem Set Use the Quadratic Formula to solve an equation of the form f(x) 5 0 for each function. 2. f(x) 5 x2 1 4x 1 4
1. f(x) 5 x2 2 2x 2 3 x2 2 2x 2 3 5 0
2
a 5 1, b 5 22, c 5 23
2b 6 √ b 2 4ac ________________ x 5 2a 2(22) 6 √(22) 2 4(1)(23) _________________________ x 5 2(1) 26√ 16 ________ x 5 2 2 6 4 ______ x 5 _________ 2
_______________ 2
___
2
x 5 3, x 5 21
3. f(x) 5 4x2 2 9
4. f(x) 5 2x2 2 5x 2 6
© Carnegie Learning
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5. f(x) 5 x2 1 2x 1 10
6. f(x) 5 23x2 2 6x 2 11
2
Use the discriminant to determine whether each function has real or imaginary zeros. 7. f(x) 5 x2 1 12x 1 35 b2 2 4ac 5 122 2 4(1)(35) 5 144 2 140 54
© Carnegie Learning
The discriminant is positive, so the function has real zeros.
8. f(x) 5 23x 1 x 2 9
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9. f(x) 5 x2 2 4x 1 7
2
10. f(x) 5 9x2 2 12x 1 4
12. f(x) 5 x2 1 6x 1 9
204
© Carnegie Learning
1 11. f(x) 5 2 __ x2 1 3x 2 8 4
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Use the vertex form of a quadratic equation to determine whether the zeros of each function are real or imaginary. Explain how you know.
2
13. f(x) 5 (x 2 4)2 2 2 Because the vertex (4, 22) is below the x-axis and the parabola is concave up (a . 0), it intersects the x-axis. So, the zeros are real.
14. f(x) 5 22(x 2 1)2 2 5
15. f(x) 5 __ 1 (x 2 2)2 1 7 3
16. f(x) 5 23(x 2 1)2 1 5
© Carnegie Learning
17. f(x) 5 2(x 2 6)2
18. f(x) 5 __ 3 (x 1 4)2 2 6 4
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Factor each function over the set of real or imaginary numbers. Then, identify the type of zeros. 19. k(x) 5 x2 2 25 k(x) 5 (x 1 5)(x 2 5) x 5 25, x 5 5
2
The function k(x) has two real zeros.
20. n(x) 5 x2 2 5x 2 14
21. p(x) 5 2x2 2 8x 2 17
22. g(x) 5 x2 1 6x 1 10
24. m(x) 5 __ 1 x2 1 8 2
206
© Carnegie Learning
23. h(x) 5 2x2 1 8x 2 7
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