Algebra 2
2nd Semester Final Review
1. Use inverse operations to write the
1
7
inverse of
.
17. Solve
2. Use inverse operations to write the
5
8
inverse of
.
3. Write the exponential equation
in logarithmic form.
4. Write the logarithmic equation
in exponential form.
5. Write the logarithmic equation
in exponential form.
6. Evaluate
1
16
by using mental
math.
7. Evaluate
math.
0.0001 by using mental
8. Express
as a single
logarithm. Simplify, if possible.
9. Express
as a
single logarithm. Simplify, if
possible.
10. Express
as a single
logarithm. Simplify, if possible.
11. Express
as a product.
Simplify, if possible.
12. Express
Simplify, if possible.
as a product.
13. Simplify the expression
.
14. Simplify the expression
.
18. Solve
16. Solve
.
.
.
19. Simplify
.
20. Simplify
.
21. Distance varies directly as time
because as time increases, the
distance traveled increases
proportionally. The speed of sound
in air is about 335 feet per
second. How long would it take for
sound to travel 11,725 feet?
22. The volume V of a cylinder varies
jointly with the height h and the
radius squared r2, and
cm3 when
cm and
cm2. Find V when
cm and
cm2. Round your answer
to the nearest hundredth.
23. 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?
24. Multiply
.
Assume that all expressions are
defined.
25. Divide
15. Simplify
.
. Assume that
all expressions are defined.
26. The area of a rectangle is equal to
square units. If the
length of the rectangle is equal to
units, what expression
represents its width?
37. Given
find
.
.
,
.
38. Given
27. Simplify the expression
Assume that all variables are
positive.
28. Simplify the expression
Assume that all variables are
positive.
and
and
, write the
composite function
state its domain.
39. Given
and
and
, write the
composite function
state its domain.
29. Write the expression
in radical
form, and simplify. Round to the
nearest whole number if necessary.
30. Write the expression
using rational exponents.
by
and
40. Find the center and radius of a
circle that has a diameter with
endpoints
and
.
41. Graph the equation
.
31. Simplify the expression
42. Write the equation of a circle with
center
and radius
.
.
43. Write the equation of a circle with
center
and radius
.
32. Simplify the expression
.
33. Evaluate the piecewise function
for
and
44. Write the equation of the circle
with center
and containing
the point
.
45. Write the equation of the circle
with center
and
containing the point
.
46. Write the equation of the line that
is tangent to the circle
.
34. Graph the piecewise function
.
35. Given
, find
36. Given
, find
and
.
and
.
at the point
47. Write an equation in standard
form for the ellipse with center
(0, 0), a Focus at (0, 6), and a
vertex at (0, -10).
.
48. Graph the ellipse
56. Write the equation in standard
form for the parabola with vertex
and the directrix
.
.
57. Write the equation in standard
form for the parabola with vertex
and the directrix
.
49. Graph the ellipse
.
50. Find the constant difference for a
hyperbola with foci
and
and the point on the
hyperbola
the parabola
.
Then, graph the parabola.
.
51. Find the constant difference for a
hyperbola with foci
and
and the point on the
hyperbola
58. Find the vertex, value of p, axis of
symmetry, focus, and directrix of
.
59. Identify the conic section the
equation
represents.
52. Write an equation in standard form
for the hyperbola with center
,
vertex
, and focus
.
53. Find the vertices and asymptotes of
60. Identify the conic section the
equation
represents.
61. Identify the conic section that the
equation
the hyperbola
, and then graph.
54. Find the vertices, co-vertices, and
asymptotes of the hyperbola
, and then
represents.
62. Find the standard form of the
equation
by
graph.
55. Use the Distance Formula to find the
equation of a parabola with focus
and directrix
.
y
completing the square. Then,
identify and graph the conic
section.
63. Find the standard form of the
equation
by
5
–1
–5
1
2
3
4
5
6
7
8
9
x
completing the square. Then,
identify and graph the conic
section.
64. Find the standard form of the
equation
by
completing the square. Then,
identify and graph the conic section.
65. Louise wears an outfit everyday that
consists of one top (shirt, T-shirt, or
blouse), one bottom (pants or skirt)
and one scarf. Her wardrobe consists
of a tan skirt, a pair of black pants, 2
T-shirts, one silk blouse, 1 buttondown shirt, and a set of 3 scarves.
How many different outfits can
Louise put together?
66. There are 7 singers competing at a
talent show. In how many different
ways can the singers appear?
67. Joel owns 12 shirts and is selecting
the ones he will wear to school next
week. How many different ways can
Joel choose a group of 5 shirts?
(Note that he will not wear the same
shirt more than once during the
week.)
68. An experiment consists of rolling a
number cube. What is the probability
of rolling a number greater than 4?
Express your answer as a fraction in
simplest form.
69. An experiment consists of spinning a
spinner. The table shows the results.
Find the experimental probability
that the spinner does not land on
red. Express your answer as a
fraction in simplest form.
Outcome Frequency
red
10
purple
11
yellow
13
70. A grab bag contains 3 football
cards and 7 basketball cards. An
experiment consists of taking one
card out of the bag, replacing it,
and then selecting another card.
What is the probability of selecting
a football card and then a
basketball card? Express your
answer as a decimal.
71. A grab bag contains 7 football
cards and 3 basketball cards. An
experiment consists of taking one
card out of the bag, replacing it,
and then selecting another card.
What is the probability of selecting
a football card and then a
basketball card? Express your
answer as a decimal.
72. Find the first 5 terms of the
sequence with
and
for
.
73. Find the first 5 terms of the
sequence with
and
for
.
74. Find the first 5 terms of the
sequence
.
75. Find the first 5 terms of the
sequence
.
76. Write a possible explicit rule for
nth term of the sequence 23.1,
20.2, 17.3, 14.4, 11.5, 8.6, ...
77. Write a possible explicit rule for
nth term of the sequence 23.1,
20.2, 17.3, 14.4, 11.5, 8.6, ...
78. Write the series
in
summation notation.
79. Write the series
in
summation notation.
92. Find the sum of the geometric
series: 3  6 12  ...  384
80. Expand the series
93. Find the sum of the geometric
and evaluate.
series with a1  2 , r 
81. Evaluate the series
.
94.
3
, and n  5
2
Write an equation of the line that
is tangent to the circle x 2  y 2  100
at the point  6,8 .
82. Evaluate the series
.
83. Find the missing terms in the
arithmetic sequence 18, ____, ____,
____, 42.
84. Find the missing terms in the
arithmetic sequence 16, ____, ____,
____, 52.
85. Find the missing terms in the
arithmetic sequence 18, ____, ____,
____, –14.
86. Find the 5th term of the arithmetic
sequence with
and
.
87. Find the sum for the arithmetic
series
.
88. Find the sum for the arithmetic
95.
Write an equation of the line that
is tangent to the circle x 2  y 2  169
at the point  5,12 .
96.
Find the sum for the infinite
geometric series: 12  4 
97.
4 4
  ...
3 9
A piece of machinery costs
$50,000. The value of the
machine depreciates 9% per year.
What is the machine worth after 1
year? After 2 years? After 5
years? In how many years will
the machine be worth $10,000?
98. A ball thrown into the air from a
roof 15 feet above the ground
with an initial vertical velocity of
30 ft/sec can be modeled by the
equation: h(t )  16t 2  30t  15 .
How long will the ball be in the
air? What is it’s maximum height?
89. Determine whether the sequence 12,
40, 68, 96 could be geometric or
arithmetic. If possible, find the
common ratio or difference.
99. A circle is inscribed in a square
with a side measuring 24 inches.
If you randomly throw darts at the
board, what is the probability of
hitting the board but missing the
circle?
100. Solve for x:
log2 (𝑥 + 2) + log2 (𝑥) = 3
90. Find the 7th term of the geometric
sequence –4, 12, –36, 108, –324, ...
101. Solve for x:
log(𝑥2 + 5𝑥 + 6) − log(𝑥 + 3) = 2
91. Find the 7th term of the geometric
sequence with
and
.
102. Solve for x:
ln(𝑥2 − 𝑥) − ln(𝑥) = 4
series
.
Algebra 2 Final Exam
Answer Section
SHORT ANSWER
1
7
1.
5
8
2.
23. 16 lawns
24.
25.
26.
3.
27.
4.
28.
5.
29.
; 32
6. –2
7. –4
8.
3
9.
3
10.
4
30.
31. 27
32. 16
33.
11. –9
;
34.
y
12. –4
10
13. 3
8
6
14. 4
4
2
15.
16.
–10 –8
x = 12
–6
–4
–2
–2
2
–4
–6
17.
x=4
–8
–10
18.
19. –5x
35.
20. –6x
36.
21. 35 sec
22.
37.
cm3
=
=
4
6
8
10
x
38.
,
48.
,
y
18
12
39.
,
,
6
–18
40. center
–12
–6
6
; radius 5
–6
41.
12
18
x
(6, –5)
–12
y
–18
10
8
49.
6
y
4
18
2
12
–10 –8
–6
–4
–2
–2
2
4
6
8
10
x
6
–4
–6
–8
–18
–12
–6
6
–6
–10
(1, –4)
12
18
x
–12
42.
43.
–18
50.
2
51. 8
44.
45.
3
25
46. y =  x 
4
4
52.
53.
Vertices:
Asymptotes:
47.
and
and
y
y
5
10
–10 –8
–6
–4
8
4
6
3
4
2
2
1
–2
–2
2
4
6
8
10
x
–5
–4
–3
–2
–1
–1
–4
–2
–6
–3
–8
–4
–10
1
2
3
4
–5
63. Same as #62
54.
64. ellipse
Same as #53
65. 24 outfits
55.
66.
56.
66. 5,040 ways
67. 792 ways
57.
58. Vertex
axis of symmetry
, focus
,
, and directrix
,
.
68.
y
5
69.
4
12
17
3
2
70. 0.21
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
71. 0.21
–2
–3
–4
72. 6, 11, 21, 41, 81
–5
59. ellipse
60. ellipse
61. hyperbola
62. ellipse
73. 7, 23, 87, 343, 1367
74. –3, –1, 3, 11, 27
75. –4, 2, 20, 74, 236
5
x
76.
90. –2,916
77.
91.
78.
256
92. 765
93.
79.
80.
6
81. 253
82. 276
211
3
 26  26.375
8
8
94. y  8 
3
x  x  6
4
95. y  12 
5
x  x  5
12
96. 18
83. 24, 30, 36
97. $45,500; $41,405; $31,201.61; 17.1years
84. 25, 34, 43
98. 2.3 seconds in the air; max height of 29.1 ft
85. 10, 2, –6
4 
 .2146  21.5%
4
86. 15
99.
87. 1313
100. x = 2
88. 630
101. x = 98
89. It could be arithmetic with d = 28.
102. 𝑥 = 𝑒4 + 1