Download ExamView - Algebra II H Semester Exam Review.tst

Name: ________________________ Class: ___________________ Date: __________
ID: A
Algebra II Honors--Semester Exam Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. An irrational number can ________ be expressed as a quotient of integers.
a. always
b. sometimes
c. never
2. An inequality ____ has a real number solution.
a. always
b. sometimes
c. never
3. An absolute value equation ____ has an extraneous solution.
a. always
b. sometimes
c. never
4. The theoretical probability of an event is ____ a negative number.
a. always
b. sometimes
c. never
5. An independent system of two linear equations ____ has an infinite number of solutions.
a. always
b. sometimes
c. never
6. Equivalent systems of two linear equations ____ have the same solutions.
a. always
b. sometimes
c. never
7. A system of two linear inequalities ____ has a solution.
a. always
b. sometimes
c. never
8. The maximum value of a linear objective function ____ occurs at exactly one vertex of the feasible region.
a. always
b. sometimes
c. never
9. Which of the following points lies in the plane represented by −x − 7y − 4z = 28?
a. (–5, 3, –5)
b. (2, 10, 3)
c. (8, 0, –9)
d. (–1, 10, 7)
10. The solution to a system of three equations in three variables is ____ one point.
a. always
b. sometimes
c. never
11. The graph models a train’s distance from a river as the train travels at a constant speed. Which equation best
represents the relation?
a.
y = | x | + 60
b.
y = | x + 60 |
c.
y = | 60x |
1
d.
| 1 |
y = |
x|
| 60 |
12. Which is the graph of y = −2(x − 2) 2 − 4?
a.
b.
c.
d.
To which sets of numbers does the number belong?
13. –13
a.
b.
c.
d.
rational numbers, real numbers
whole numbers, integers, rational numbers, real numbers
whole numbers, integers, real numbers
integers, rational numbers, real numbers
Name the property of real numbers illustrated by the equation.
14. −2(x + 2) = −2x − 4
a. Associative Property of Multiplication
b. Commutative Property of Addition
c. Associative Property of Addition
d. Distributive Property
2
Which of the following is the multiplicative inverse of the given matrix?
ÍÈÍ
ÍÍ −1
15. ÍÍÍÍ
ÍÍ
ÍÎ 0
a.
˙˘˙
2 ˙˙˙˙
˙˙
˙
−1 ˙˙˚
ÍÈÍ
ÍÍ −1
ÍÍ
ÍÍ
ÍÍ
ÍÎ 0
˙˘˙
2 ˙˙˙˙
˙˙
˙
−1 ˙˙˚
b.
ÍÈÍ
ÍÍ −1
ÍÍ
ÍÍ
ÍÍ
ÍÎ 0
˙˘˙
−2 ˙˙˙˙
˙˙
˙
−1 ˙˙˚
c.
ÍÈÍ
ÍÍ −1
ÍÍ
ÍÍ
ÍÍ
ÍÎ 0
˙˘˙
−2 ˙˙˙˙
˙˙
˙
1 ˙˙˚
Short Answer
Find the opposite and the reciprocal of the number.
16.
5
7
Evaluate the expression for the given value of the variable(s).
17. |4b + 8| + || −1 − b 2 || + 2b 3 ; b = –2
18. −x 3 + x 2 + x − 2; x = 2
19. Find the perimeter of the figure. Simplify the answer.
Solve the equation.
20. −4y + 10 = −6(y − 6)
3
d.
ÍÈÍ
ÍÍ −1
ÍÍ
ÍÍ
ÍÍ
ÍÎ 0
˙˘˙
2 ˙˙˙˙
˙˙
˙
−1 ˙˙˚
21. 2 |x + 1| − 1 = 19
22. x 2 + 18x + 81 = 25
Solve the equation or formula for the indicated variable.
23. The formula for the time a traffic light remains yellow is t =
1
s + 1, where t is the time in seconds and s is the
8
speed limit in miles per hour.
a.
Solve the equation for s.
b.
What is the speed limit at a traffic light that remains yellow for 6 seconds?
Solve for x. State any restrictions on the variables.
24. ax + bx − 9 = 11
25. A rectangle is 6 times as long as it is wide. The perimeter is 90 cm. Find the dimensions of the rectangle. Round to
the nearest tenth if necessary.
26. Two cars leave Denver at the same time and travel in opposite directions. One car travels 5 mi/h faster than the
other car. The cars are 315 mi apart in 3 h. How fast is each car traveling?
4
Solve the inequality. Graph the solution set.
27. 5(2b + 1) < –5 + 10b
Solve the problem by writing an inequality.
28. A club decides to sell T-shirts for $12 as a fund-raiser. It costs $20 plus $8 per T-shirt to make the T-shirts. Write
and solve an equation to find how many T-shirts the club needs to make and sell in order to profit at least $100.
Solve the compound inequality. Graph the solution set.
29. 6x – 3 < –33 or 6x + 7 > 31
30. The perimeter of a square garden is to be at least 22 feet but not more than 46 feet. Find all possible values for the
length of its sides.
Solve the inequality. Graph the solution.
31. | 2x + 2 | ≥ 26
32. When Spheres-R-Us ships bags of golf balls, the number of balls in each bag must be within 10 balls of 100. Write
an absolute value inequality and a compound inequality for an acceptable number of golf balls b in each bag.
33. Lynn and Dawn tossed a coin 30 times and got heads 14 times. What is the experimental probability of tossing
heads using Lynn and Dawn’s results?
5
34. A bag contains 9 red marbles, 7 white marbles, and 5 blue marbles. Find P(red or blue).
35. If a dart hits the target at random, what is the probability that it will land in the shaded region?
36. Write the ordered pairs for the relation. Find the domain and range.
37. Make a mapping diagram for the relation.
{(–3, –5), (0, 1), (2, –2), (5, –1)}
6
38. Find the domain and range of the relation and determine whether it is a function.
39. Suppose f (x) = 4x − 2 and g (x) = −2x + 1.
f (1)
.
Find the value of
g (4)
40. Graph the equation y =
3
2
x + 1.
7
41. Graph the equation x + 2y = 10 by finding the intercepts.
Find the slope of the line through the pair of points.
42.
Write in standard form an equation of the line passing through the given point with the given slope.
43. slope = 3; (0, –2)
44. Find the point-slope form of the equation of the line passing through the points (–4, 2) and (7, 6).
8
Find the slope of the line.
45.
Find an equation for the line:
3
46. through (–1, 1) and perpendicular to y = x – 2.
2
47. through (7, 0) and parallel to y = 2x – 2.
48. through (4, –6) and horizontal.
Determine whether y varies directly with x. If so, find the constant of variation k and write the equation.
49.
x
y
7
21
35
105
175
525
875
2625
9
Find the value of y for a given value of x, if y varies directly with x.
50. If y = 37 when x = –74, what is y when x = –142?
51. A 6-mi cab ride costs $12.10. A 11-mi cab ride costs $20.10. Find a linear equation that models cost c as a
function of distance d.
52. What is the vertex of the function y = | 3x + 2 | + 4?
53. Write the equation for the translation of y = | x | .
Write an equation for the vertical translation.
54. y =
1
4
x; 21 units down
55. Write the equation that is the translation of y = | x | left 1 unit and down 7 units.
10
56. Graph the function y = | x − 5| + 3.
57. Graph the function y = 3 | x| + 5.
Graph the inequality.
58. 2x + 3y < –9
11
59. Write an inequality for the graph.
Write an inequality for the graph.
60.
61. Is the relation {(–1, –2), (5, 5), (–1, 5), (1, –4), (–2, 4)} a function? Explain.
12
Solve the system by graphing.
ÔÏÔÔ −3x − 4y = −5
62. ÔÔÌ
ÔÔ 4x − y = 13
Ó
Without graphing, classify each system as independent, dependent, or inconsistent.
ÔÏÔÔ 4x − y = −8
63. ÔÔÌ
ÔÔ y = 4x + 6
Ó
Solve the system by the method of your choice.
ÏÔÔ
ÔÔ 3x + y − z = −11
ÔÔÔ
z = −9
64. ÔÔÌ
ÔÔ
ÔÔÔ −x − 3y + 2z = −14
Ó
65. A group of 75 people attended a ball game. There were four times as many children as adults in the group. Set up
a system of equations that represents the numbers of adults and children who attended the game and solve the
system to find the number of children who were in the group.
13
Solve the system by the method of your choice.
ÔÏÔÔ 2x − y = −7
66. ÔÔÌ
ÔÔ 2x − 4y = −10
Ó
Solve the system of inequalities by graphing.
ÏÔÔ
ÔÔ y ≤ −4x + 2
67. ÌÔ
ÔÔ y > 2x − 2
Ó
ÔÏÔÔ y ≥ 3
68. ÔÌÔ
ÔÔ y > |2x − 4|
Ó
14
69. Your club is baking vanilla and chocolate cakes for a bake sale. They need at most 25 cakes. You cannot have
more than 10 chocolate cakes. Write and graph a system of inequalities to model this system.
70. Find the values of x and y that maximize the objective function P = 3x + 2y for the graph. What is the maximum
value?
15
71. Given the system of constraints, name all vertices. Then find the maximum value of the given objective function.
ÔÏÔ x ≥ 0
ÔÔ
ÔÔ
ÔÔÔ y ≥ 0
ÔÌÔÔ
ÔÔ 6x − 2y ≤ 12
ÔÔ
ÔÔÔ
Ó 4y ≤ 4x + 8
Maximum for C = 4x − 3y
72. Describe the location of the point in coordinate space.
(6, –8, 7)
73. Graph 9x + 18y − 12z = 36.
74. State the dimensions of the matrix. Identify the indicated element.
ÍÈÍ −9 7 ˙˘˙
ÍÍ
˙˙
ÍÍ
˙˙
ÍÍ
˙
Í
A = ÍÍ −3 4 ˙˙˙˙ , a 3, 2
ÍÍ
˙
ÍÍ −7 8 ˙˙˙
ÍÍÎ
˙˙˚
Find the values of the variables.
ÍÈÍ
ÍÍ −4 + t
75. ÍÍÍÍ
ÍÍ
ÍÎ 8
˙˘˙
ÍÈÍ
ÍÍ −5
0 ˙˙˙˙
Í
˙˙ = ÍÍÍ
˙˙
ÍÍ
−5 ˙˚
ÍÎ 8
˙˘˙
˙˙
˙˙
˙˙
˙
−3y − 2 ˙˙˚
0
16
Solve the matrix equation.
ÍÈÍ
ÍÍ −6
76. ÍÍÍÍ
ÍÍ
ÍÎ 1
˙˘˙
ÍÈÍ
ÍÍ 1
−7 ˙˙˙˙
Í
˙˙ + X = ÍÍÍ
˙˙
ÍÍ
5 ˙˚
ÍÎ −7
ÍÈÍ
ÍÍ 2
77. X − ÍÍÍÍ
ÍÍ
ÍÎ −4
ÈÍ
ÍÍ
Í9
78. ÍÍÍÍ
ÍÍ
ÍÎ 2
˙˘˙
ÍÈÍ
ÍÍ 4
−8 ˙˙˙˙
Í
˙˙ = ÍÍÍ
˙˙
ÍÍ
2 ˙˚
ÍÎ 2
˙˘˙
4 ˙˙˙˙
˙˙
˙
5 ˙˙˚
˙˘˙
−6 ˙˙˙˙
˙˙
˙
−8 ˙˙˚
˘˙
ÈÍ ˘˙
˙˙
ÍÍ ˙˙
˙
ÍÍ 4 ˙˙
5 ˙˙
ÍÍ ˙˙
X
=
˙˙
ÍÍ ˙˙
˙˙
ÍÍ −4 ˙˙
1 ˙˚
Î ˚
Find the product.
ÈÍ
ÍÍ
Í 1
79. ÍÍÍÍ
ÍÍ
ÎÍ −3
˘˙ ÈÍ
˙Í
7 ˙˙˙˙ ÍÍÍÍ 3
˙˙ ÍÍ
˙Í
−5 ˙˚˙ ÍÎÍ 3
˘˙
˙
−6 ˙˙˙˙
˙˙
˙
2 ˙˙˚
ÏÔÔ
ÔÔ 6a − 9b − 2c = −7
ÔÔÔ
80. Write the system ÔÌÔ −2a + 3b + 2c = −8 as a matrix equation. Then identify the coefficient matrix, the variable
ÔÔ
ÔÔ
ÔÔ a + 6b − 4c = −3
Ó
matrix, and the constant matrix.
Solve the system.
ÏÔÔ
ÔÔ 4x + 3y
= 19
ÔÔÔ
81. ÔÔÌ 4x + 2y − 5z = −2
ÔÔ
ÔÔ
ÔÔ −4x
− 2z = −24
Ó
17
82. A gem store sells beads made of amber and quartz. For 4 amber beads and 2 quartz beads, the cost is $46.50. For 3
amber beads and 2 quartz beads, the cost is $36.00. Find the price of each type of bead.
83. In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again.
a. Find a quadratic model for the data in the table.
b. Use the model to estimate the population of bacteria at 9 hours.
Time (hours)
Population (1000s)
0
1
5.1
3.03
2
3
1.72 1.17
4
5
6
1.38
2.35
4.08
84. Graph y = 2x 2 − 7.
85. Graph y = x 2 + 3x + 2. Identify the vertex and the axis of symmetry.
18
86. Graph y = −3x 2 + 6x + 5. Does the function have a maximum or minimum value? What is this value?
87. A science museum is going to put an outdoor restaurant along one wall of the museum. The restaurant space will
be rectangular. Assume the museum would prefer to maximize the area for the restaurant.
a.
Suppose there is 120 feet of fencing available for the three sides that require fencing.
How long will the longest side of the restaurant be?
b.
What is the maximum area?
19
88. Graph y = (x − 7) 2 + 5.
89. Use the graph of y = (x − 3) 2 + 5.
a.
If you translate the parabola to the right 2 units and down 7 units, what is the equation
of the new parabola in vertex form?
b.
If you translate the original parabola to the left 2 units and up 7 units, what is the
equation of the new parabola in vertex form?
c.
How could you translate the new parabola in part (a) to get the new parabola in part (b)?
Determine whether the function is linear or quadratic. Identify the quadratic, linear, and constant terms.
90. y = (x + 1)(6x − 6) − 6x 2
20
Identify the vertex and the axis of symmetry of the parabola. Identify points corresponding to P and Q.
91.
92. Use vertex form to write the equation of the parabola.
Find a quadratic model for the set of values.
93. (–2, 8), (0, –4), (4, 68)
94. A manufacturer determines that the number of drills it can sell is given by the formula D = −3p 2 + 180p − 285,
where p is the price of the drills in dollars.
a.
At what price will the manufacturer sell the maximum number of drills?
b.
What is the maximum number of drills that can be sold?
21
Write the equation of the parabola in vertex form.
95. vertex (0, 3), point (–4, –45)
96. Identify the vertex and the y-intercept of the graph of the function y = −3(x + 2) 2 + 5.
Factor the expression.
97. −15x 2 − 21x
98. −9x 2 + 15x + 12
99. x 2 + 14x + 48
100. x 2 − 6x + 8
101. 5x 2 − 22x − 15
102. Solve by factoring.
4x 2 + 28x − 32 = 0
103. The function y = −16t 2 + 486 models the height y in feet of a stone t seconds after it is dropped from the edge of
a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a second.
104. A landscaper is designing a flower garden in the shape of a trapezoid. She wants the length of the shorter base to
be 3 yards greater than the height, and the length of the longer base to be 5 yards greater than the height. For what
height will the garden have an area of 360 square yards? Round to the nearest tenth of a yard.
105. A baseball player hits a fly ball that is caught about 4 seconds later by an outfielder. The path of the ball is a
parabola. The ball is at its highest point as it passes the second baseman, who is 127 feet from home plate. About
how far from home plate is the outfielder at the moment he catches the ball? Explain your reasoning.
22
ID: A
Algebra II Honors--Semester Exam Review
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
C
B
B
C
C
A
B
B
C
B
C
A
D
D
B
SHORT ANSWER
5 7
16. − ,
7 5
17. –11
18. –4
19. 10x + 2y
20. 13
21. x = 9 or x = −11
22. –4, –14
23. s = 8t − 8; s = 40 mi/h
20
24. x =
; a ≠ −b
a+b
25. 6.4 cm by 38.6 cm
26. 50 mi/h and 55 mi/h
27. no solutions
28. 12x − (8x + 20) ≥ 100; x ≥ 30
29. x < –5 or x > 4
1
ID: A
30. 5.5 ≤ x ≤ 11.5
31. x ≤ −14 or x ≥ 12
32. |b − 100| ≤ 10; 90 ≤ b ≤ 110
7
33.
15
2
34.
3
1
35.
25
36. {(–2, 5), (–1, 2), (0, 1), (1, 2), (2, 5)}; domain: {–2, –1, 0, 1, 2}; range: {1, 2, 5}
37.
38. Domain: x > 0; range: y > 0; yes, it is a function.
2
39. −
7
40.
2
ID: A
41.
1
3
–3x + y = –2
4
y – 2 = (x + 4)
11
0
2
1
y=− x+
3
3
y = 2x − 14
y = –6
yes; k = 3; y =3x
71
c = 1.60d + 2.50
2
(− , 4)
3
y = |x | + 4
1
y = x − 21
4
y = | x + 1| − 7
42. −
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
3
ID: A
56.
57.
58.
59. –3x + 6y ≥ –18
60. y ≤ |x – 3| + 1
61. No; a domain value corresponds to two or more range values.
4
ID: A
62.
(3, –1)
63. inconsistent
64. (–7, 1, –9)
ÏÔÔ
ÔÔ a + c = 75
65. ÌÔ
; 15 adults, 60 children
ÔÔ c = 4a
Ó
66. (–3, 1)
67.
5
ID: A
68.
69. Let x = the number of vanilla cakes.
Let y = the number of chocolate cakes.
ÏÔ
ÔÔÔ x ≥ 0
ÔÔ
ÔÔÔ y ≥ 0
ÌÔÔ
ÔÔ x + y ≤ 25
ÔÔÔ
ÔÔÔ
Ó y ≤ 10
70. maximum value at (9, 0); 27
71. (0, 2), (2, 0), (4, 6); maximum value of 8
72. From the origin, move 6 units forward, 8 units left, and 7 units up.
73.
6
ID: A
74. 3 × 2, 8
75. t = –1, y = 1
ÍÈÍ
˙˘
ÍÍ 7 11 ˙˙˙
Í
˙˙
76. ÍÍÍ
˙˙
ÍÍ
˙
ÍÎ −8 0 ˙˙˚
ÈÍ
˘˙
ÍÍ
˙
ÍÍ 6 −14 ˙˙˙
Í
˙˙
77. ÍÍ
˙˙
ÍÍ
ÍÎ −2 −6 ˙˙˚
ÈÍ
˘˙
ÍÍ
˙
ÍÍ −24 ˙˙˙
˙˙
78. ÍÍÍ
˙˙
ÍÍ
ÍÎ 44 ˙˙˚
ÈÍ
˘˙
ÍÍ
˙
ÍÍ 24 8 ˙˙˙
Í
˙˙
79. ÍÍ
˙˙
ÍÍ
ÍÎ −24 8 ˙˙˚
ÈÍ
˘È ˘
ÈÍ ˘˙
ÍÍ 6 −9 −2 ˙˙˙ ÍÍÍ a ˙˙˙
ÍÍ −7 ˙˙
ÍÍ
˙˙ ÍÍ ˙˙
ÍÍ ˙˙
ÍÍ
˙˙ ÍÍ ˙˙
ÍÍ ˙˙
Í
˙
Í
˙
80. ÍÍÍ −2 3
2 ˙˙˙ ÍÍÍ b ˙˙˙ = ÍÍÍÍ −8 ˙˙˙˙
ÍÍ
ÍÍ ˙˙
˙˙ ÍÍ ˙˙
ÍÍ 1
ÍÍ −3 ˙˙
6 −4 ˙˙˙˙ ÍÍÍÍ c ˙˙˙˙
ÍÍÎ
ÍÍÎ ˙˙˚
˚Î ˚
ÈÍ
˘
ÍÍ 6 −9 −2 ˙˙˙
ÍÍ
˙˙
ÍÍ
˙˙
Í
coefficient matrix: ÍÍÍ −2 3
2 ˙˙˙˙
ÍÍ
˙˙
ÍÍ 1
6 −4 ˙˙˙˙
ÍÍÎ
˚
ÍÈÍ a ˙˘˙
ÍÍ ˙˙
ÍÍ ˙˙
Í ˙
variable matrix: ÍÍÍÍ b ˙˙˙˙
ÍÍÍ ˙˙˙
ÍÍ c ˙˙
ÍÎ ˙˚
ÈÍ ˘˙
ÍÍ −7 ˙˙
ÍÍ ˙˙
ÍÍ ˙˙
constant matrix: ÍÍÍÍ −8 ˙˙˙˙
ÍÍ ˙˙
ÍÍ −3 ˙˙
ÍÎÍ ˙˚˙
81. (4, 1, 4)
82. amber $10.50, quartz $2.25
83.
a. P = 0.38x 2 − 2.45x + 5.10
b. 13,830 bacteria
7
ID: A
84.
85.
ÁÊÁ 3
1 ˜ˆ˜
3
vertex: ÁÁÁÁ − , − ˜˜˜˜ , axis of symmetry: x = −
ÁË 2
4 ˜¯
2
86.
maximum value; 8
8
ID: A
87.
a.
b.
40 ft
1,600 ft 2
88.
89.
y = (x − 5) 2 − 2
b. y = (x − 1) 2 + 12
c. left 4 units, up 14 units
90. linear function
linear term: 0x
constant term: –6
91. (–1, –2), x = –1
P'(0, –1), Q'(–3, 2)
a.
92. y = 3(x + 2) 2 + 2
y = 4x 2 + 2x − 4
$30; 2,415 drills
y = −3x 2 + 3
vertex: (–2, 5);
y-intercept: –7
97. −3x(5x + 7)
93.
94.
95.
96.
98.
99.
100.
101.
102.
103.
104.
3(−3x 2 + 5x + 4)
(x + 6)(x + 8)
(x − 2)(x − 4)
(5x + 3)(x − 5)
–8, 1
5.51 seconds
17.1 yards
9
ID: A
OTHER
105. The outfielder is about 254 feet from home plate. The parabola is symmetric about its axis, which is a vertical line.
The second baseman is at a point on this axis of symmetry. So the outfielder is the same distance from the second
baseman as the second baseman is from home plate. 127 + 127 = 254, so the distance from home plate to the
outfielder is about 254 feet.
10