Download Algebra II semester review

Name: ________________________ Class: ___________________ Date: __________
ID: A
Algebra II semester review 2016
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. What quadratic function does the graph represent?
a.
b.
f(x)  x 2  8x  14
f(x)  x 2  8x  14
c.
d.
f(x)  x 2  8x  14
f(x)  x 2  8x  14
Numeric Response
1. What is the x-coordinate of the vertex of the graph of f(x)  7(x  9) 2  3?
2. Find the positive root of x 2  2x  35  0 .
3. Evaluate log 4 4096 .
Short Answer
1. Translate the point (1, 1) right 2 units and down 2 units. Give the coordinates of the translated point.
1
Name: ________________________
ID: A
2. Use a table to translate the graph 3 units to the left. Use the same coordinate plane as the original function.
3. The graph shows Carmen’s savings each week. She decides to save 2.5 times as much money each week.
Sketch a graph that represents the new savings and identify the transformation of the original graph that it
represents.
4. Identify the parent function for g x   x  3  and describe what transformation of the parent function it
represents.
3
5. Let g(x) be the transformation, vertical translation 3 units down, of f(x)  4x  8 . Write the rule for g(x) .
6. Let g(x) be a horizontal compression of f(x)  3x  5 by a factor of 2 . Write the rule for g(x) and graph the
1
function.
7. Using the graph of f(x)  x 2 as a guide, describe the transformations, and then graph the function
g(x)  (x  6) 2  2 .
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Name: ________________________
ID: A
8. Using the graph of f(x)  x 2 as a guide, describe the transformations, and then graph the function
g(x)  8x 2 .
9. Consider the function f(x)  4x 2  8x  10. Determine whether the graph opens upward or downward. Find
the axis of symmetry, the vertex and the y-intercept. Graph the function.
10. Let g x  be a vertical shift of f x   x up 4 units followed by a vertical stretch by a factor of 3. Write the
rule for g x  .
11. Find the zeros of the function h x   x 2  23x  60 by factoring.
12. Simplify
2  2i
.
5  3i
13. Graph the complex number 4  2i.
14. Find the minimum or maximum value of f(x)  x 2  2x  6. Then state the domain and range of the function.
15. Identify the axis of symmetry for the graph of f(x)  x 2  2x  3.
16. The parent function f(x)  x 2 is reflected across the x-axis, vertically stretched by a factor of 10, and
translated right 10 units to create g. Use the description to write the quadratic function in vertex form.
17. Find the roots of the equation 30x  45  5x 2 by factoring.
18. Write the function f(x)  5x 2  60x  181 in vertex form, and identify its vertex.
19. Complete the square for the expression x 2  16x  ____. Write the resulting expression as a binomial
squared.
20. Graph the complex number 4i.
21. Graph the complex number –2.
22. Divide by using synthetic division.
(x 2  9x  10)  x  2 
Graph each function. How is each graph a translation of f(x)  x 2 ?
23. y  x 2  2
24. y  (x  3) 2  4
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Name: ________________________
ID: A
25. Solve the equation x 2  3  2x by completing the square.
26. Find the complex conjugate of 3i  4.
27. Write a quadratic function in standard form with zeros 6 and –8.
28. Solve the equation 2x 2  18  0.
29. Find the zeros of the function f(x)  x 2  6x  18.
30. Find the zeros of f(x)  x 2  7x  9 by using the Quadratic Formula.
31. Find the zeros of g x   4x 2  x  5 by using the Quadratic Formula.
32. Express 8 84 in terms of i.
33. Solve the equation x 2  10x  25  54.
34. The daily profit P for a cake bakery can be modeled by the function P(x)  15x 2  330x  815 , where x is
the price of a cake. What should the price of a cake be to provide a daily profit of at least $600? Round your
answer(s) to the nearest dollar.
35. Subtract. Write the result in the form a  bi.
(5 – 2i) – (6 + 8i)
36. Multiply 6i 4  6i . Write the result in the form a  bi.
37. Solve the inequality 8x 2  14x  4  11.
38. Find the absolute value 7  9i .
39. Simplify 8i 20 .
40. Rewrite the polynomial 12x2 + 6 – 7x5 + 3x3 + 7x4 – 5x in standard form. Then, identify the leading
coefficient, degree, and number of terms. Name the polynomial.
41. What expression is equivalent to (3  2i) 2 ?
42. Identify the degree of the monomial 5r 3 s 5 .
43. Add. Write your answer in standard form.
(5a 5  a 4 )  (a 5  7a 4  2)
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Name: ________________________
ID: A
44. Factor x 3  5x 2  9x  45.
45. Find the product 5x  3(x 3  5x  2).
46. For h(x)  2x 2  6x  9 and k(x)  3x 2  8x  8, find h(x)  2k(x).
47. Find the product 2c d 4 (4c 6 d 5  c 3 d ).
48. Determine whether the binomial (x  4 ) is a factor of the polynomial P x   5x 3  20x 2  5x  20 .
49. Divide by using long division: (5x  6x 3  8)  (x  2).
50. Find the product (x  2y) 3 .
51. Use Pascal’s Triangle to expand the expression 4x  3 .
4
52. Use synthetic substitution to evaluate the polynomial P x   x 3  4x 2  4x  5 for x  4 .
53. Solve the polynomial equation 3x 5  6x 4  72x 3  0 by factoring.
54. Factor the expression 81x 6  24x 3 y 3 .
55. Identify the leading coefficient, degree, and end behavior of the function P(x)  –5x 4 – 6x 2 + 6.
56. Identify whether the function graphed has an odd or even degree and a positive or negative leading
coefficient.
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Name: ________________________
ID: A
57. A bacteria population starts at 2,032 and decreases at about 15% per day. Write a function representing the
number of bacteria present each day. Graph the function. After how many days will there be fewer than 321
bacteria?
58. What quartic function does the graph represent (in factored form)?
59. A initial investment of $10,000 grows at 11% per year. What function represents the value of the investment
after t years?
60. Mira bought $300 of Freerange Wireless stock in January of 1998. The value of the stock is expected to
increase by 7.5% per year. Use a graph to predict the year the value of Mira’s stock will reach $700.
61. Use inverse operations to write the inverse of f x   x  3 .
2
62. Tell whether the function y  2 5 shows growth or decay. Then graph the function.
x
63. Use inverse operations to write the inverse of f(x) 
x
4
– 5.
64. Graph f(x)  5x  1 . Then, write and graph the inverse.
65. Write the exponential equation 2 3  8 in logarithmic form.
66. Write the logarithmic equation log 4 16  2 in exponential from.
67. Evaluate log 4
1
16
by using mental math.
68. Express log 3 6  log 3 4.5 as a single logarithm. Simplify, if possible.
69. Express log 3 27 3 as a product. Simplify, if possible.
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Name: ________________________
ID: A
70. Simplify the expression log 4 64 .
71. Simplify log 7 x 3  log 7 x .
72. Nadav invests $6,000 in an account that earns 5% interest compounded continuously. What is the total
amount of her investment after 8 years? Round your answer to the nearest cent.
73. Simplify lne 5x .
74. Identify the maximum or minimum value and the domain and range of the graph of the function
y  2(x  2) 2  3 .
75. Identify the vertex and the axis of symmetry of the graph of the function y  2(x  2) 2  4 .
76. Use the vertex form to write the equation of the parabola.
What are the vertex and the axis of symmetry of the equation?
77. y  2x 2  4x  10
78. y  2x 2  16x  16
What is the maximum or minimum value of the function? What is the range?
79. y  2x 2  20x  2
What is the vertex form of the equation?
80. y  x 2  8x  6
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Name: ________________________
ID: A
81. 3x 2  26x  35
What is the expression in factored form?
82. 16x 2  8x
What are the solutions of the quadratic equation?
83. 3x 2  25x  42 = 0
84. x 2  11x  28
What is the solution of each equation?
85. 3x 2  21
What value completes the square for the expression?
86. x 2  18x
Rewrite the equation in vertex form. Name the vertex and y-intercept.
87.
y  x 2  12x  34
88. Write –2x2(–5x2 + 4x3) in standard form.
89. Classify –6x5 + 4x3 + 3x2 + 11 by degree.
Consider the leading term of each polynomial function. What is the end behavior of the graph?
90. 2x 3  5x
91. 5x 8  2x 7  8x 6  1
Write the polynomial in factored form.
92. 4x3 + 8x2 – 96x
What are the zeros of the function? Graph the function.
93. y  x(x  2)(x  5)
94. What is a cubic polynomial function in standard form with zeros 5, 2, and –5?
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Name: ________________________
ID: A
What is the relative maximum and minimum of the function?
95. f(x)  x 3  6x 2  36x
What are the zeros of the function? What are their multiplicities?
96. f(x)  x 4  4x 3  3x 2
What are the real or imaginary solutions of each polynomial equation?
97. x 4  40x 2  144  0
98. x 4  20x 2  64
Use Pascal’s Triangle to expand the binomial.
99. (s  2v) 5
100. Divide 4x 3  2x 2  3x  4 by x + 4.
Find the roots of the polynomial equation.
101. x 3  3x 2  5x  15  0
102. x 3  2x 2  10x  136  0
Write the expression as a single logarithm.
103. 4 log x  6 log (x  2)
104. 3 log b q  6 log b v
Expand the logarithmic expression.
105. log 3
d
12
106. log 3 11p 3
107. Find the annual percent increase or decrease that y  0.35(2.3) x models.
108. An initial population of 820 quail increases at an annual rate of 23%. Write an exponential function to model
the quail population. What will the approximate population be after 3 years?
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Name: ________________________
ID: A
109. The half-life of a certain radioactive material is 71 hours. An initial amount of the material has a mass of 722
kg. Write an exponential function that models the decay of this material. Find how much radioactive material
remains after 17 hours. Round your answer to the nearest thousandth.
Write the equation in logarithmic form.
110. 2 5  32
Evaluate the logarithm.
111. log 0.01
112. log 5
1
625
10
ID: A
Algebra II semester review 2016
Answer Section
MULTIPLE CHOICE
1. B
NUMERIC RESPONSE
1. 9
2. 5
3. 6
SHORT ANSWER
1.
2.
1
ID: A
3.
The graph represents a vertical stretch by a factor of 2.5.
4. The parent function is the cubic function, f x   x 3 .
g x   x  3 represents a horizontal translation of the parent function 3 units to the left.
5. g(x)  4x  5
6. g(x)  6x  5
3
7. g(x) is f(x) translated 6 units left and 2 units down.
2
ID: A
8. A reflection across the x-axis and a vertical stretch by a factor of 8.
9. The parabola opens downward.
The axis of symmetry is the line x  1.
The vertex is the point (1,14).
The y-intercept is 10.
10. g x   3x  12
11. x  20 or x  3
12.  17 +
2
8
17
i
3
ID: A
13.
14. The minimum value is –7. D: {all real numbers}; R: {y | y  –7}
15. x  1
16. g(x)  10(x  10) 2
17. x  3
18. f(x)  5(x  6) 2  1 ;
vertex: (–6, –1)
19. x  8
20.
2
4
ID: A
21.
22. x  7 
23.
4
x2
24.
25. x = 1 or x = –3
26. 4  3i
f(x) translated up 2 unit(s)
f(x) translated up 4 unit(s) and translated to the left 3 unit(s).
5
ID: A
27. f(x)  x 2  2x  48
28. x  3i
29. x = –3 + 3i or –3 – 3i
30. x 
31. x 
32. 16i
33.
34.
35.
36.
37.
7  13
2
1
8

21
79
8
i
x  5 3 6
6  x  16
–1 – 10i
36  24i
2.5  x  0.75
38.
130
39. –8
40. 7x 5  7x 4  3x 3  12x 2  5x  6
leading coefficient: –7; degree: 5; number of terms: 6; name: quintic polynomial
41. 5  12i
42. 8
43. 6a 5  6a 4  2
44. (x  5)(x  3)(x  3)
45. 5x 4  3x 3  25x 2  25x  6
46. 4x 2  22x  25
47. 8c 7 d 9  2c 4 d 5
48. (x  4 ) is a factor of the polynomial P x   5x 3  20x 2  5x  20 .
49. 6x 2  12x  29 
50
(x  2)
50. x 3  6x 2 y  12xy 2  8y 3
51. 256x 4  768x 3  864x 2  432x  81
52. P 4  11
53. The roots are 0, –6, and 4.
54. 3x 3 (3x  2y)(9x 2  6xy  4y 2 )
55. The leading coefficient is –5. The degree is 4.
As x  , P(x)  – and as x  +, P(x)  –
56. The degree is odd, and the leading coefficient is positive.
6
ID: A
57. f(x)  2032(0.85) t
After about 11.3 days, there will be fewer than 321 bacteria.
58. f(x)  x  2  2x  1 2x  1 2x  3
59. f(t)  10000(1.11) t
60. 2009
61. f 1 x   x 
2
3
62. This is an exponential growth function.
63. f 1 (x)  4(x + 5)
7
ID: A
64.
(x  1)
5
65. log 2 8  3
f 1 (x) 
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
4 2  16
–2
3
–9
3
2log 7 x
$8950.95
–5x
minimum value: –3
domain: all real numbers
range: all real numbers  3
vertex: (–2, –4);
axis of symmetry: x  2
y  3(x  2) 2  2
vertex: ( –1, – 6)
axis of symmetry: x  1
vertex: ( 4, 8)
axis of symmetry: x  4
maximum: 24
range: y  24
80. y  (x  4) 2  22
81. (3x  5)(x  7)
82. 4x(4x  2)
7
83. –6, 
3
84. –4, –7
85.
7, –
7
8
ID: A
86. 81
87. y  (x  6) 2  2
vertex: (6, – 2)
y-intercept: (0, 34)
88. –8x5 + 10x4
89. quintic
90. The leading term is 2x 3 . Since n is odd and a is positive, the end behavior is down and up.
91. The leading term is 5x 8 . Since n is even and a is positive, the end behavior is up and up.
92. 4x(x – 4)(x + 6)
93. 0, 2, –5
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
f(x)  x 3  2x 2  25x  50
The relative maximum is at (–6, 216) and the relative minimum is at (2, –40).
the number 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeros of multiplicity 1
6, –6, 2, –2
4, –4, 2, –2
s 5  10s 4 v  40s 3 v 2  80s 2 v 3  80sv 4  32v 5
4x 2  14x  59, R –232
i 5 , i 5 , –3
3 ± 5i, –4
none of these
log b (q 3 v 6 )
105. log 3 d  log 3 12
106. log 3 11  3 log 3 p
107. 130% increase
108. f(x)  820(1.23) x ; 1526
1
 1  71 x
109. y  722  
; 611.589 kg
 2
110. log 2 32  5
111. –2
9
ID: A
112. –4
10