Download Chapter 2 Skills Practice

Name: ________________________ Class: ___________________ Date: __________
ID: A
Chapter 2 Skills Practice
Shape and Structure
Forms of Quadratic Functions
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:
3. Vertex form:
4. Concavity of a parabola:
Problem Set
Circle the function that matches each graph. Explain your reasoning.
5.
f(x) = 6(x − 2)(x − 8)
1
f(x) = − (x + 2)(x + 8)
2
f(x) =
1
(x + 2)(x + 8)
2
f(x) =
1
(x − 2)(x − 8)
2
1
Name: ________________________
ID: A
6.
f(x) = −3(x + 2)(x − 5)
f(x) = 3(x + 2)(x + 5)
f(x) = 3(x − 2)(x − 5)
f(x) = −3(x − 2)(x − 5)
7.
f(x) = x 2 + 5x − 4
f(x) = −x 2 + 5x + 10
f(x) = x 2 + 5x + 4
f(x) = −x 2 + 5x + 4
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.
8. vertex (3, 7) and point (1, 10)
9. points (1, 0), (4,− 3), and (7, 0)
10. y-intercept (0, 3) and axis of symmetry −
3
8
11. roots (−5, 0), (13, 0) and point (−7, 40)
2
Name: ________________________
ID: A
12. maximum point (−4, − 8) and point (−3, − 15)
Convert each quadratic function in factored form to standard form.
13. f(x) = (x + 5)(x − 7)
14. f(x) = (x + 2)(x + 9)
15. f(x) = 2(x − 4)(x + 1)
16. f(x) = −3(x − 1)(x − 3)
17. f(x) =
1
(x + 6)(x + 3)
3
5
18. f(x) = − (x − 6)(x + 2)
8
Convert each quadratic function in vertex form to standard form.
19. f(x) = 3(x − 4) 2 + 7
20. f(x) = −2(x + 1) 2 − 5
2
ÊÁ
7 ˆ˜˜˜
3
Á
Á
21. f(x) = 2 ÁÁ x + ˜˜ −
2
2
Ë
¯
22. f(x) = −(x − 6) 2 + 4
1
23. f(x) = − (x − 10) 2 − 12
2
24. f(x) =
1
(x + 100) 2 + 60
20
3
Name: ________________________
ID: A
25. h(x) = (x + 2) 2 − 1
Reference
Points of f(x)
→
(0, 0)
→
(1, 1)
→
(2, 4)
→
Corresponding
Points on h(x)
4
Name: ________________________
ID: A
26. h(x) = (x + 7) 2
Reference
Points of f(x)
→
(0, 0)
→
(1, 1)
→
(2, 4)
→
Corresponding
Points on h(x)
5
Name: ________________________
ID: A
27. h(x) = x 2 − 9
Reference
Points of f(x)
→
(0, 0)
→
(1, 1)
→
(2, 4)
→
Corresponding
Points on h(x)
Write the function that represents each graph.
28.
6
Name: ________________________
ID: A
29.
30.
31.
32.
7
Name: ________________________
ID: A
33.
What’s the Point?
Deriving Quadratic Functions
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.
34. Given: vertex (3, 5) and point (5, − 3)
8
Name: ________________________
ID: A
35. Given: vertex (−2, − 9) and one of two x-intercepts (1, 0)
36. Given: two x-intercepts (−7, 0) and (5, 0) and one point (−4, − 9)
9
Name: ________________________
ID: A
37. Given: vertex (−6, − 1) and point (−3,35)
Use a graphing calculator to determine the quadratic equation for each set of three points that lie on a
parabola.
38. (−4, 12), (−2, − 14), (2, 6)
39. (5, − 56), (1, − 4), (−10, − 26)
40. (−8, 8),(−4, 6),(4, 38)
41. (−2, 3), (2, − 9), (5, − 60)
42. (0, 3), (−5, − 2.4), (15, − 7.8)
43. (−2, 13), (1, − 17), (7, 31)
10
Name: ________________________
ID: A
Now It’s Getting Complex … But It’s Really Not Difficult!
Complex Number Operations
Vocabulary
Match each term to its corresponding definition.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
a number in the form a + bi where a and b are real numbers and b is not equal to 0
term a of a number written in the form a + bi
a polynomial with two terms
pairs of numbers of the form a + bi and a − bi
a number such that its square equals −1
a number in the form a + bi where a and b are real numbers
a polynomial with three terms
a number of the form bi where b is not equal to 0
term bi of a number written in the form a + bi
a polynomial with one term
____ 44. the number i
____ 45. imaginary number
____ 46. pure imaginary number
____ 47. complex number
____ 48. real part of a complex number
____ 49. imaginary part of a complex number
____ 50. complex conjugates
____ 51. monomial
____ 52. binomial
____ 53. trinomial
Problem Set
Calculate each power of i.
54. i 48
55. i 361
56. i 55
57. i 1000
11
Name: ________________________
ID: A
58. i −22
59. i −7
Rewrite each expression using i.
60.
−72
61.
−49 +
62. 38 −
−23
−200 +
63.
−45 + 21
64.
−48 − 12
4
65.
1+
4−
3
−15
21
−
3
66. − −28 +
67.
−75 +
10
121
12
6
80
Simplify each expression.
68. (2 + 5i) − (7 − 9i)
69. −6 + 8i − 1 − 11i + 13
70. −(4i − 1 + 3i) + (6i − 10 + 17)
71. 22i + 13 − (7i + 3 + 12i) + 16i − 25
72. 9 + 3i(7 − 2i)
73. (4 − 5i)(8 + i)
74. −0.5(14i − 6) − 4i(0.75 − 3i)
12
Name: ________________________
ÁÊ 1
75. ÁÁÁÁ i −
Ë2
ID: A
3 ˜ˆ˜˜ ÁÊÁÁ 1 3 ˜ˆ˜˜
+
− i
4 ˜˜¯ ÁÁË 8 4 ˜˜¯
Determine each product.
76. (3 + i)(3 − i)
77. (4i − 5)(4i + 5)
78. (7 − 2i)(7 + 2i)
ÊÁ 1
ˆ˜ ÊÁ 1
ˆ˜
79. ÁÁÁÁ + 3i ˜˜˜˜ ÁÁÁÁ − 3i ˜˜˜
˜¯
Ë3
¯Ë 3
80. (0.1 + 0.6i)(0.1 − 0.6i)
È
˘
81. −2 ÍÍÎ (−i − 8)(−i + 8) ˙˙˚
Identify each expression as a monomial, binomial, or trinomial. Explain your reasoning.
82. 4xi + 7x
83. −3x + 5 − 8xi + 1
84. 6x 2 i + 3x 2
85. 8i − x 3 + 7x 2 i
86. xi − x + i + 2 − 4i
87. −3x 3 i + x 2 + 6x 3 + 9i − 1
Simplify each expression, if possible.
88. (x − 6i) 2
89. (2 + 5xi)(7 − xi)
90. 3xi − 4yi
91. (2xi − 9)(3x + 5i)
92. (x + 4i)(x − 4i)(x + 4i)
13
Name: ________________________
ID: A
93. (3i − 2xi)(3i − 2xi) + (2i − 3xi)(2 − 3xi)
For each complex number, write its conjugate.
94. 7 + 2i
95. 3 + 5i
96. 8i
97. −7i
98. 2 − 11i
99. 9 − 4i
100. −13 − 6i
101. −21 + 4i
Calculate each quotient.
102.
3 + 4i
5 + 6i
103.
8 + 7i
2+i
104.
−6 + 2i
2 − 3i
105.
−1 + 5i
1 − 4i
106.
6 − 3i
2−i
107.
4 − 2i
−1 + 2i
14
Name: ________________________
ID: A
You Can’t Spell “Fundamental Theorem of Algebra” without F-U-N!
Quadratics and Complex Numbers
Vocabulary
Write a definition for each term in your own words.
108. imaginary roots
109. discriminant
110. imaginary zeros
111. degree of a polynomial equation
112. Fundamental Theorem of Algebra
113. double root
Problem Set
Use the Quadratic Formula to solve an equation of the form f (x) = 0 for each function.
114. f(x) = x 2 − 2x − 3
115. f(x) = x 2 + 4x + 4
116. f(x) = 4x 2 − 9
117. f(x) = −x 2 − 5x − 6
118. f(x) = x 2 + 2x + 10
119. f(x) = −3x 2 − 6x − 11
Use the discriminant to determine whether each function has real or imaginary zeros.
120. f(x) = x 2 + 12x + 35
121. f(x) = −3x + x − 9
122. f(x) = x 2 − 4x + 7
123. f(x) = 9x 2 − 12x + 4
15
Name: ________________________
ID: A
1
124. f(x) = − x 2 + 3x − 8
4
125. f(x) = x 2 + 6x + 9
Use the vertex form of a quadratic equation to determine whether the zeros of each function are real or
imaginary. Explain how you know.
126. f(x) = (x − 4) 2 − 2
127. f(x) = −2(x − 1) 2 − 5
128. f(x) =
1
(x − 2) 2 + 7
3
129. f(x) = −3(x − 1) 2 + 5
130. f(x) = −(x − 6) 2
131. f(x) =
3
(x + 4) 2 − 6
4
Factor each function over the set of real or imaginary numbers. Then, identify the type of zeros.
132. k(x) = x 2 − 25
133. n(x) = x 2 − 5x − 14
134. p(x) = −x 2 − 8x − 17
135. g(x) = x 2 + 6x + 10
136. h(x) = −x 2 + 8x − 7
137. m(x) =
1 2
x +8
2
16