Name: ________________________ Class: ___________________ Date: __________
ID: A
Algebra II H semester review 2015
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Fabric that regularly sells for $4.90 per square foot is on sale for 10% off. Write an equation that represents
the cost of s square feet of fabric during the sale. Write a transformation that shows the change in the cost of
fabric.
a. 4.90s − 10; (x, y) → (x, 0.1y)
c. 4.90s − 10; (x, y) → (x, 0.9y)
b. (4.90 − 0.49)s; (x, y) → (x, 0.1y)
d. (4.90 − 0.49)s; (x, y) → (x, 0.9y)
____
2. For which function is 3 NOT an element of the range?
a. y = −2x + 4
c. y = −(−x) 2
b.
____
d.
y=3
3. What quadratic function does the graph represent?
a.
b.
____
y = x5
f(x) = x 2 + 8x − 14
f(x) = −x 2 + 8x − 14
4. What expression is equal to log50?
a. ln(50 ÷ 10)
b. ln50 ÷ ln10
d.
f(x) = −x 2 + 8x + 14
f(x) = −x 2 − 8x − 14
c.
d.
ln50 − ln10
ln50
c.
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 .
1
Name: ________________________
ID: A
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.
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.
3
4. Identify the parent function for g (x ) = (x + 3 ) and describe what transformation of the parent function it
represents.
2
Name: ________________________
ID: A
5. Graph the data from the table. Describe the parent function and the transformation that best approximates the
data set.
x
y
–3
0
–2
1
1
2
6
3
13
4
6. Let g(x) be the transformation, vertical translation 3 units down, of f(x) = −4x + 8 . Write the rule for g(x) .
1
7. 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
function.
8. 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 .
9. Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph the function
g(x) = −8x 2 .
10. 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.
11. 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 ) .
12. Find the zeros of the function h (x ) = x 2 + 23x + 60 by factoring.
13. Simplify
−2 + 2i
.
5 + 3i
14. Graph the complex number 4 + 2i.
15. Find the minimum or maximum value of f(x) = x 2 − 2x − 6. Then state the domain and range of the function.
16. Identify the axis of symmetry for the graph of f(x) = x 2 + 2x − 3.
17. 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.
18. Find the roots of the equation 30x − 45 = 5x 2 by factoring.
19. Write the function f(x) = −5x 2 − 60x − 181 in vertex form, and identify its vertex.
3
Name: ________________________
ID: A
20. Complete the square for the expression x 2 − 16x + ____. Write the resulting expression as a binomial
squared.
21. Graph the complex number 4i.
22. Graph the complex number –2.
23. Graph y ≤ −x 2 − 5x + 4 .
24. 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 ?
25. y = x 2 + 2
26. y = (x + 3) 2 + 4
27. Solve the equation x 2 = 3 − 2x by completing the square.
28. Find the complex conjugate of 3i + 4.
29. Write a quadratic function in standard form with zeros 6 and –8.
30. Solve the equation 2x 2 + 18 = 0.
31. Find the zeros of the function f(x) = x 2 + 6x + 18.
32. Find the zeros of f(x) = x 2 + 7x + 9 by using the Quadratic Formula.
33. Find the zeros of g (x ) = 4x 2 − x + 5 by using the Quadratic Formula.
34. Express 8 −84 in terms of i.
35. Solve the equation x 2 − 10x + 25 = 54.
36. Find the number and type of solutions for x 2 − 9x = −8 .
37. Solve the inequality x 2 − 14x + 45 ≤ −3 by using algebra.
4
Name: ________________________
ID: A
38. 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.
39. Subtract. Write the result in the form a + bi.
(5 – 2i) – (6 + 8i)
40. Multiply 6i (4 − 6i) . Write the result in the form a + bi.
41. Solve the inequality −8x 2 − 14x + 4 > −11.
42. Find the absolute value |−7 − 9i| .
43. Simplify −8i 20 .
44. 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.
45. Graph the function f(x) = x 3 + 3x 2 − 6x − 8 .
46. iGraph f(x) = e x + 2.
47. Given: y varies directly as x, and y = −5 when x = 2.5. Write and graph the direct variation function.
48. What expression is equivalent to (3 − 2i) 2 ?
49. Identify the degree of the monomial −5r 3 s 5 .
50. Add. Write your answer in standard form.
(5a 5 − a 4 ) + (a 5 + 7a 4 − 2)
51. Factor x 3 + 5x 2 − 9x − 45.
52. Find the product (5x − 3) (x 3 − 5x + 2).
53. For h(x) = 2x 2 + 6x − 9 and k(x) = 3x 2 − 8x + 8, find h(x) − 2k(x).
54. Find the product 2c d 4 (−4c 6 d 5 − c 3 d ).
55. Determine whether the binomial ( x − 4 ) is a factor of the polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 .
56. Divide by using long division: (5x + 6x 3 − 8) ÷ (x − 2).
5
Name: ________________________
ID: A
57. Graph g (x ) = 4x 3 − 24x + 9 on a calculator, and estimate the local maxima and minima.
58. Find the product (x − 2y) 3 .
4
59. Use Pascal’s Triangle to expand the expression (4x + 3) .
60. Use synthetic substitution to evaluate the polynomial P (x ) = x 3 − 4x 2 + 4x − 5 for x = 4 .
61. Solve the polynomial equation 3x 5 + 6x 4 − 72x 3 = 0 by factoring.
62. Factor the expression 81x 6 + 24x 3 y 3 .
63. Identify the leading coefficient, degree, and end behavior of the function P(x) = –5x 4 – 6x 2 + 6.
64. Identify whether the function graphed has an odd or even degree and a positive or negative leading
coefficient.
65. Write a function that transforms f(x) = 2x 3 + 4 in the following way:
stretch vertically by a factor of 6 and shift 5 units left.
66. 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?
6
Name: ________________________
ID: A
67. What quartic function does the graph represent (in factored form)?
68. A initial investment of $10,000 grows at 11% per year. What function represents the value of the investment
after t years?
69. 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.
2
70. Use inverse operations to write the inverse of f (x ) = x + 3 .
x
71. Tell whether the function y = 2 (5) shows growth or decay. Then graph the function.
72. Use inverse operations to write the inverse of f(x) =
x
4
– 5.
73. Graph f(x) = 5x − 1 . Then, write and graph the inverse.
74. Write the exponential equation 2 3 = 8 in logarithmic form.
75. Write the logarithmic equation log 4 16 = 2 in exponential from.
76. Evaluate log 4
1
16
by using mental math.
77. Express log 3 6 + log 3 4.5 as a single logarithm. Simplify, if possible.
78. Express log 3 27 −3 as a product. Simplify, if possible.
79. Simplify the expression log 4 64 .
7
Name: ________________________
ID: A
80. Simplify log 7 x 3 − log 7 x .
81. Solve 8 x + 8 = 32 x .
82. Use a table and graph to solve 3 2x = 6561.
83. Solve log 5 x 10 − log 5 x 6 = 21.
84. 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.
85. Distance that sound travels through air d varies directly as time t, and d = 1,675 ft when t = 5 s. Find t when
d = 5,025 ft.
86. Given: y varies inversely as x, and y = 4 when x = –4. Write and graph the inverse variation function.
87. The volume V of a cylinder varies jointly with the height h and the radius squared r2, and V = 157.00 cm3
when h = 2 cm and r 2 = 25 cm2. Find V when h = 3 cm and r 2 = 36 cm2. Round your answer to the nearest
hundredth.
88. Simplify lne −5x .
89. The pressure P of a gas varies inversely with the volume V of its container and directly with the temperature
T. A certain gas has a pressure of 1.6 atmospheres with a volume of 14 liters and a temperature of 280
kelvins. If the gas is cooled to a temperature of 250 kelvins and the container is expanded to 16 liters, what
will be the new pressure?
90. Identify the maximum or minimum value and the domain and range of the graph of the function
y = 2(x + 2) 2 − 3 .
91. The number of lawns l that a volunteer can mow in a day varies inversely with the number of shrubs s that
need to be pruned that day. If the volunteer can prune 6 shrubs and mow 8 lawns in one day, then how many
lawns can be mowed if there are only 3 shrubs to be pruned?
92. Determine whether the data set represents a direct variation, an inverse variation, or neither.
x
2
3
4
y
420
280
210
93. Identify the vertex and the axis of symmetry of the graph of the function y = 2(x + 2) 2 − 4 .
8
Name: ________________________
ID: A
94. Use the vertex form to write the equation of the parabola.
What are the vertex and the axis of symmetry of the equation?
95. y = 2x 2 + 4x − 10
96. y = −2x 2 + 16x − 16
What is the maximum or minimum value of the function? What is the range?
97. y = −2x 2 + 20x − 2
What is the vertex form of the equation?
98. y = x 2 + 8x − 6
99. 3x 2 + 26x + 35
What is the expression in factored form?
100. 16x 2 + 8x
What are the solutions of the quadratic equation?
101. 3x 2 + 25x + 42 = 0
102. x 2 + 11x = −28
What is the solution of each equation?
103. 3x 2 = 21
9
Name: ________________________
ID: A
What value completes the square for the expression?
104. x 2 − 18x
Rewrite the equation in vertex form. Name the vertex and y-intercept.
105.
y = x 2 − 12x + 34
106. Write –2x2(–5x2 + 4x3) in standard form.
107. Classify –6x5 + 4x3 + 3x2 + 11 by degree.
Consider the leading term of each polynomial function. What is the end behavior of the graph?
108. 2x 3 + 5x
109. 5x 8 − 2x 7 − 8x 6 + 1
Write the polynomial in factored form.
110. 4x3 + 8x2 – 96x
What are the zeros of the function? Graph the function.
111. y = x(x − 2)(x + 5)
112. What is a cubic polynomial function in standard form with zeros 5, 2, and –5?
What is the relative maximum and minimum of the function?
113. f(x) = x 3 + 6x 2 − 36x
What are the zeros of the function? What are their multiplicities?
114. f(x) = x 4 − 4x 3 + 3x 2
What are the real or imaginary solutions of each polynomial equation?
115. x 4 − 40x 2 + 144 = 0
116. x 4 − 20x 2 = −64
10
Name: ________________________
ID: A
Graph the logarithmic equation.
117. y = log 3 x
Use Pascal’s Triangle to expand the binomial.
118. (s + 2v) 5
119. Divide 4x 3 + 2x 2 + 3x + 4 by x + 4.
Find the roots of the polynomial equation.
120. x 3 − 3x 2 − 5x − 15 = 0
121. x 3 − 2x 2 + 10x + 136 = 0
Write the expression as a single logarithm.
122. 4 log x − 6 log (x + 2)
123. 3 log b q + 6 log b v
Expand the logarithmic expression.
124. log 3
d
12
125. log 3 11p 3
126. Find the annual percent increase or decrease that y = 0.35(2.3) x models.
127. 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?
Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary.
128. 3 log 2x = 4
129. Solve log(4x + 10) = 3.
130. log(x + 9) − log x = 3
131. 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.
11
Name: ________________________
ID: A
132. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will you
have in the account after 4 years?
Write the equation in logarithmic form.
133. 2 5 = 32
Evaluate the logarithm.
134. log 0.01
135. log 5
1
625
136. Use the Change of Base Formula to evaluate log 4 20 .
137. Use the Change of Base Formula to evaluate log 7 28 .
Solve the exponential equation.
138.
1
= 64 4x − 3
16
139. 4 4 x = 8
140. Solve 15 2x = 36. Round to the nearest ten-thousandth.
141. Solve log 3x + log 9 = 0. Round to the nearest hundredth if necessary.
Write the expression as a single natural logarithm.
142. 3 ln x − 2 lnc
143. Simplify ln e 3 .
144. Solve ln(2x − 1) = 8. Round to the nearest thousandth.
12
ID: A
Algebra II H semester review 2015
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
D
C
B
B
NUMERIC RESPONSE
1. 9
2. 5
3. 6
SHORT ANSWER
1.
1
ID: A
2.
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 .
3
g (x ) = (x + 3 ) represents a horizontal translation of the parent function 3 units to the left.
5.
Square root function translated 3 units to the left.
6. g(x) = −4x + 5
2
ID: A
7. g(x) = 6x + 5
8. g(x) is f(x) translated 6 units left and 2 units down.
9. A reflection across the x-axis and a vertical stretch by a factor of 8.
3
ID: A
10. 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.
11. g (x ) = −3x + 12
12. x = −20 or x = −3
2
13. − 17 +
8
17
i
14.
15. The minimum value is –7. D: {all real numbers}; R: {y | y ≥ –7}
16. x = −1
17. g(x) = −10(x − 10) 2
18. x = 3
19. f(x) = −5(x + 6) 2 − 1 ;
vertex: (–6, –1)
20. (x − 8 )
2
4
ID: A
21.
22.
23.
24. x − 7 +
−4
x−2
5
ID: A
25.
f(x) translated up 2 unit(s)
26.
f(x) translated up 4 unit(s) and translated to the left 3 unit(s).
27. x = 1 or x = –3
28. 4 − 3i
29. f(x) = x 2 + 2x − 48
30. x = ±3i
31. x = –3 + 3i or –3 – 3i
32. x =
−7 ± 13
2
33. x =
1
8
34. 16i
35.
36.
37.
38.
39.
40.
±
79
8
i
21
x = 5 ±3 6
The equation has two real solutions.
6≤x≤8
6 ≤ x ≤ 16
–1 – 10i
36 + 24i
6
ID: A
41. −2.5 < x < 0.75
42.
130
43. –8
44. −7x 5 + 7x 4 + 3x 3 + 12x 2 − 5x + 6
leading coefficient: –7; degree: 5; number of terms: 6; name: quintic polynomial
45.
46.
7
ID: A
47. y = –2x
48. 5 − 12i
49. 8
50. 6a 5 + 6a 4 − 2
51. (x + 5)(x − 3)(x + 3)
52. 5x 4 − 3x 3 − 25x 2 + 25x − 6
53. −4x 2 + 22x − 25
54. −8c 7 d 9 − 2c 4 d 5
55. (x − 4 ) is a factor of the polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 .
56. 6x 2 + 12x + 29 +
50
(x − 2)
57. The local maximum is about 31.627417. The local minimum is about –13.627417.
58. x 3 − 6x 2 y + 12xy 2 − 8y 3
59. 256x 4 + 768x 3 + 864x 2 + 432x + 81
60. P (4) = 11
61. The roots are 0, –6, and 4.
62. 3x 3 (3x + 2y)(9x 2 − 6xy + 4y 2 )
63. The leading coefficient is –5. The degree is 4.
As x → −∞, P(x) → –∞ and as x → +∞, P(x) → –∞
64. The degree is odd, and the leading coefficient is positive.
65. g(x) = 12(x + 5) 3 + 24
8
ID: A
66. f(x) = 2032(0.85) t
After about 11.3 days, there will be fewer than 321 bacteria.
67. f(x) = (x + 2 ) (2x + 1) (2x − 1) (2x − 3)
68. f(t) = 10000(1.11) t
69. 2009
70. f −1 (x ) = x −
2
3
71. This is an exponential growth function.
72. f −1 (x) = 4(x + 5)
9
ID: A
73.
(x + 1)
5
74. log 2 8 = 3
f −1 (x) =
4 2 = 16
–2
3
–9
3
2log 7 x
81. x = 12
82. x = 4
75.
76.
77.
78.
79.
80.
21
4
83. x = 5
84. $8950.95
85. 15 sec
16
86. y = − x
87. 339.12 cm3
88. –5x
10
ID: A
89. 1.25 atmospheres
90. minimum value: –3
domain: all real numbers
range: all real numbers ≥ −3
91. 16 lawns
92. Inverse variation
93. vertex: (–2, –4);
axis of symmetry: x = −2
94. y = 3(x + 2) 2 + 2
95. vertex: ( –1, – 6)
axis of symmetry: x = −1
96. vertex: ( 4, 8)
axis of symmetry: x = 4
97. maximum: 24
range: y ≤ 24
98. y = (x + 4) 2 − 22
99. (3x + 5)(x + 7)
100. 4x(4x + 2)
7
101. –6, −
3
102. –4, –7
103.
7, –
104. 81
7
105. y = (x − 6) 2 − 2
vertex: (6, – 2)
y-intercept: (0, 34)
106. –8x5 + 10x4
107. quintic
108. The leading term is 2x 3 . Since n is odd and a is positive, the end behavior is down and up.
109. The leading term is 5x 8 . Since n is even and a is positive, the end behavior is up and up.
110. 4x(x – 4)(x + 6)
11
ID: A
111. 0, 2, –5
112.
113.
114.
115.
116.
117.
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
118. s 5 + 10s 4 v + 40s 3 v 2 + 80s 2 v 3 + 80sv 4 + 32v 5
119. 4x 2 − 14x + 59, R –232
120.
121.
122.
123.
124.
125.
126.
127.
128.
i 5 , −i 5 , –3
3 ± 5i, –4
none of these
log b (q 3 v 6 )
log 3 d − log 3 12
log 3 11 + 3 log 3 p
130% increase
f(x) = 820(1.23) x ; 1526
10.7722
12
ID: A
495
2
130. 0.0090
129.
1
x
ÁÊÁ 1 ˜ˆ˜ 71
; 611.589 kg
131. y = 722 ÁÁÁ ˜˜˜
Ë 2¯
132. $1,923.23
133. log 2 32 = 5
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
–2
–4
2.161
1.712
7
12
3
8
0.6616
0.04
x3
ln 2
c
3
1,490.979
13