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Review Sheets for Final Exam
Chapter 6
1. Divide using synthetic division: 2x 4  6x 3  24 x  28  x  4 
3. One of the zeros of the function f (x)  x 3  5x 2  9x  45 is x = –5, find the other zeros of
the function.

5. Find all rational zeros of the polynomial: y = 3x 3  3x 2  75x  75

7. Find all the zeros of the function: f (x)  6x 4  5x 3  12x 2  5x  6

9. Write a polynomial function that has the zeros 2, –3, and 1 and has a leading coefficient of 1.
Then graph the function to show that 2, –3, and 1 are solutions.

10. Write a polynomial function of least degree that has real coefficients, the given zeros, and a
leading coefficient of 1.
(–3, 0), (0, 0), (5, 0)
12. Write the polynomial as a product of linear factors. x 3  3x 2  4 x  12
13. Solve for x: x 4  12x 2  11 = 0

14. Find the four real zeros (including multiplicity) of the polynomial
f (x)  x 4  5x 3  6x 2  32x  32 . (Hint: The real zeros are integers that lie between –4 and 4,

inclusive.)
Chapter 7

17. Evaluate. 16 5/4
18. Evaluate. (27) 2/3

19. Evaluate. 16 5/4

21. Rewrite 181/6 using radical notation.

Simplify:
251/6

22. 2/3
25
23.



3
40  4 3 5
24. 54 5  54 5 
10
49 5/6
25.
491/3
Simplify:
26. x1 3  x1 4

5
27.

9  5 81
5
3
28. (73/4 )2/9

30. Write the expression in simplest form.

16xy 2
27z 5
31. Let f (x) = 1 x 2 and g(x) = 1 x. Find f (x)  g(x).

32. Let f (x)  1 x 2 and g(x)  1 x . Find f (x)  g(x) .



33. Let f (x)  16  x 2 and g(x)  4  x . Find f (g(x)) .

 of the relation f= {(–1, –5), (–5, –1), (9, –9)}.
35. Find the inverse

 for the inverse ofthe relation y  5x  3.
36. Find an equation
39. Sketch the graph of the function. Is the inverse of f(x) a function? f (x)  2  x 2

40. Describe how to obtain the graph of y 

Graph:
41. f(x) =
43. f(x) =
x  3  3 from the graph of y  x.
x  5
3

x3  5

Refer to the function g(x)  2  x 1 .
45. Whatis the domain of g(x)?

46. What is the range of g(x)?

Solve the equation. Check for extraneous solutions.
47. x + 2 = x
49.



3
x  5  4
51. 3x 3 4  192

52.
3
y2 5

Chapter 8
59. Graph: f (x)  3x
60. The amount of money, A, accrued at the end of n years when a certain amount, P, is invested
at a compound annual rate, r, is given by A  P(1  r)n . If a person invests $270 in an account
 pays 10% interest compounded annually, find the balance after 15 years.
that
62. Find the value of $1000 deposited for 10 years in an account paying 7% annual interest

compounded yearly.
64. How much money must be deposited now in an account paying 8% annual interest,
compounded quarterly, to have a balance of $1000 after 10 years?
68. A piece of equipment costs $85,000 new but depreciates 15% per year in each succeeding
year. Find its value after 10 years.
70. Graph f (x) =  3  e x .
Simplify the expression.
72. e x  6e 3x 1

5e 2x (e 2x ) 2
73. 4 x 
e
10e x

74. Is f (x)  13.7e 0.04 t an example of exponential growth or decay?


1
75. Sketch the graph of the function. f (x)  e x  2
3
76. Use the formula A  Pe rt . If $3900 is deposited in an account at the bank and earns 9%
annual interest, compounded continuously, what is the amount in the account, rounded to the

nearest dollar, after 7 years?

3
78. Write the equation log 243 27  in exponential form.
5
Find the inverse of the function.
79. y  log 8 x


Find the inverse of the function.
80. y  ln 3x
81. Evaluate without using a calculator. log 7
1
49
83. Graph the function. State the domain and range. y  log 2 (x 1)

84. Use the change-of-base formula to evaluate the expression. log 4 7
85. Express as a single logarithm: log a11  log a 35
87. Expand using the properties of logarithms: log 4

  3)
x(x
x5
88. Expand the expression. log 3 (x 2 y 3 )

1
89. Condense the expression. log 516  3log 5 x  4 log 5 y
2

90. Solve for x correct to four decimal places: e 4 x = 5.8

1
1
91. Solve for x. log 7 3  log 7 x  log 7
2
 2
1
92. Solve for x. log 10 8  log 10 x  log 10 2
3

95. Solve the equation. 35log 6 x  18

96. Solve the equation. 75log 7 x  61

97. Write an exponential
function of the form y  ab x whose graph passes through the given
points.
(5, 3), (7, 9) 

Chapter 9
98. The price per person of renting a bus varies inversely with the number of people renting the
bus. It costs $22 per person if 47 people rent the bus. How much will it cost per person if 74
people rent the bus?
99. The variables x and y vary inversely. x = 7 when y = –4. Find an equation that relates the
variables.
100. The variable z varies jointly with the product of x and y. z = 2.4 when x = 3 and y = 2. Find
an equation that relates the variables.
103. Sketch the graph of the function. f (x) 
x2
x 2
3
104. Locate the asymptotes and graph the rational function f (x) =  2
.
x 4

x2
106. Identify all horizontal and vertical asymptotes of the graph of the function. f (x)  2
x 4

5x
107. Identify all horizontal and vertical asymptotes of the graph of the function. f (x)  2
x 1

Simplify the rational expression.
n 2  8n  15
108.
n 2  25


Divide:
x6
x 2  9x  18
110.

2
x6
x 9
x 2  2x  3
112. Simplify the expression.
x2 1

2x  3
x2  4 x  3
114. Multiply and simplify.

(x  3) 2
4 x2  9

x 2  8x  20
x 2  9x

115. Divide and simplify.
5x 3  50x 2
x 2  7x  18

116. Simplify.
(2x 2 y 3 ) 2
(4 x) 2 y 3

3 2 3
(x
(xy) 4
 y )
3x  4
2

118. Perform the operations and simplify. 2
x  16
x4

4x
2
2


119. Perform the operations and simplify. 2
x 9
x3
x3

Solve:
x
x

= 7
120.
3
10
121.


4
3

= 0
f 3
f 4


122.
r
r

1
r 1 r  9
Solve:

124. Is x = –2 a solution of
4x
8
 1
?
x 2
x 2

Chapter 13
127. Find x to the nearest hundredth.
10
32°
x
129. Evaluate without using a calculator. sec 30
130. Find the missing side lengths for x and y.

30°
8
x
y
3
45°
x
y
131. Find the missing side lengths for x and y.
132. Solve ABC using the diagram and the given measurements. (The triangle is not drawn to
scale.)
B = 34  , a = 14
A

c
b
C
a
B
Find one positive angle and one negative angle that are coterminal with the given angle.
134. 59 
2
135.
3

137. Convert 48 to radian measure.

138. Find the arc length of a sector with a radius of 6 feet and a central angle of 12  .

139. Find the reference angle for 228  .
140. The point (–3, 3) is on the terminal side of an angle  . Find cos .


141. Triangle ABC is a right triangle. Find sin A and the measure of  A to the nearest degree.
A
21

C


20
29
B
142. Use a calculator to evaluate the expression. Round your answer to three significant digits.
tan 1 8
Solve the equation for  . Round your answer to three significant digits.
143. sin   0.62; 180   270

144. tan   3.2; 180   270



145. You are flying a kite and want to know its angle of elevation. The string on the kite is 41
meters long and the kite is level with the top of a building that you know is 24 meters high. Use
an inverse trigonometric function to find the angle of elevation of the kite.
146. Given triangle ABC with a = 7, b = 10, and A = 23  , find c. Round your answer to two
decimal places.
148. Solve triangle ABC given that A = 48  , B= 54  , and b = 67.
150. Find the area of ABC . The figure is not drawn to scale.



B
3 cm
60°
A
10 cm
C
152. Given triangle ABC with b = 2, c = 4, and A = 82°, find a. Round the answer to two decimal
places.
154. Solve triangle ABC given that a = 15, b = 11, and c = 21.
155. Find the area of ABC using Heron’s formula.
B

28
A
21
22
C
Chapter 14
156. Sketch one cycle of the graph of the function.
x
f x   4 cos
3


157. Write two x-values at which the function has a minimum.
f (x)  4sin 2x
 

158. Graph y = 3 cos x 
on the interval   x   .

4 
x
159. Write the equation of the resultinggraph when y  6sin
is translated up two units.
3

161. Simplify the expression.

sin


 x sec x
2

162. Simplify the expression.
sin 2  x 
tan 2 x

163. Simplify the expression.
cosx
 sin 2 x 
sec x


164. Verify the identity.
sin x
1 cos x

1  cos x
sin x

165. Verify the identity.
tan 2 x csc 2 x  1 1
166. Solve 6cos x  3  0 in the interval 0   x  360  .

167. Solve 4 sin 2 x  1  0 in the interval 0  x  2 .


 Write a function for the sinusoidal
168.
graph below. (The period is 2  .)


y
5






2

2

–5


x
169. Evaluate the following exactly
b. tan( 53  4 )
a. cos 512
170. Simplify the following sin 2 x  cos2x  cos2 x
 Verify thefollowing:
171.

sin 
tan 

sin   cos  1  tan 
172. Given cos A 

a. cos2B

3
5
with 0  A 
b. sin B2


2
and sin B   12
13 with   B 
c. tan2A

3
2
find the following:
d. sin(A – B)

173. Simplify the following (sec x  tan( x))(1  sin(  x))

174. Given that sinu 
a. cos v2
3
with 0  u  2 and cosv   12
13 with   v 
5
b. sin( u  v)
c. tan 2v


175. Solve the following:


a. sin 2x  sin x  0





b. sin 2x sec x  2cos x

3
2
, find the following
d. cos2u
Reference: [6.5.1.69]
[1] 2x 3  2x 2  8x  8 +

4
x4
Reference: [6.5.1.79]
[3] 3, -3
Reference: [6.6.1.81]
[5] x = 5,  5, and 1

Reference: [6.7.1.87]
2 3
[7] –1, 1,  ,
3 2
Reference: [6.7.1.89]


y
10
10 x
–10
[9]
–10
Reference: [6.7.1.92]
[10] f (x)  x 3  2x 2  15x

Reference: [6.7.1.95]
[12] (x  2)(x  2)(x  3)

Reference: [6.7.2.97]
[13]  1,  11
Reference: [6.7.2.98]
[14]
 
 4, 4, –2, –1
Reference: [7.1.1.6]
[17] 32
Reference: [7.1.1.7]
[18] 9
y = x 3  7x  6
Reference: [7.1.1.8]
1
[19]
32





Reference: [7.1.1.15]
[21] 6 18
Reference: [7.2.1.25]
1
[22]
5
Reference: [7.2.1.26]
[23] 83 25
Reference: [7.2.1.27]
1
[24] 16
5
Reference: [7.2.1.28]
[25] 7
Reference: [7.2.1.30]
[26] x

7
12
Reference: [7.2.1.32]
[27] 3
Reference: [7.2.1.33]
1
[28] 1/6
7

Reference: [7.2.1.36]
4 y 3xz
[30]
9z 3

Reference: [7.3.1.40]
[31]  x 2  x  2

Reference: [7.3.1.39]
[32]  x 2  x

Reference: [7.3.1.41]
[33] 8x  x 2
Reference: [7.4.1.45]
[35] {(–5, –1), (–1, –5), (–9, 9)}
Reference: [7.4.1.46]
x3
[36] y =
5

Reference: [7.4.2.63]
[39] No
f(x )
3
1
–3
1
3
x
–2
–3
Reference: [7.5.1.65]
[40] Shift the graph of y 
x left 3 units, and down 3 units.
Reference: [7.5.1.66]

y
10
10 x
–10
[41]
–10
Reference: [7.5.1.68]
y
10
10 x
–10
–10
[43]
Reference: [7.5.1.70]
[45] x  1

Reference: [7.5.1.71]
[46] g(x)  2

Reference: [7.6.1.81]
[47] 2
Reference: [7.6.1.83]
[49] –59
Reference: [7.6.1.86]
[51] x  256

Reference: [7.6.1.87]
[52] y = 127
Reference: [8.1.1.1]
y
10
10 x
–10
[59]
–10
Reference: [8.1.2.2]
[60] $1128
Reference: [8.1.2.4]
[62] $1967.15
Reference: [8.1.2.7]
[64] $452.89
Reference: [8.2.2.22]
[68] $16,734.32
Reference: [8.3.1.27]
y
10
10 x
–10
[70]
–10
Reference: [8.3.1.29]
[72] 6e 4 x 1


Reference: [8.3.1.31]
1
[73] x
2e
Reference: [8.3.1.37]
[74] Decay
Reference: [8.3.1.38]
f (x)
4
3

2
1
–3 –2 –1
–1
1
2
3
x
–2
[75]
Reference: [8.3.2.41]
[76] $7323
Reference: [8.4.1.43]
[78] 2433 5 = 27



Reference: [8.4.1.45]
[79] y  8 x
Reference: [8.4.1.47]
ex
[80] y 
3
Reference: [8.4.1.49]
[81] –2
Reference: [8.4.2.56]
y
10
10 x
–10
–10
[83]
Domain: x | x  1; Range: all real numbers
Reference: [8.5.1.59]
log 7

1.4
[84]
log 4
Reference: [8.5.1.62]
[85] log a 385

Reference: [8.5.1.64]
[87] log 4 x  log 4 (x  3)  5log 4 x

Reference: [8.5.1.66]
[88] 2log 3 x  3log 3 y


Reference: [8.5.1.69]
4 y4
[89] log 5 3
x
Reference: [8.6.1.76]
[90] –0.4395
Reference: [8.6.2.77]
[91] x = 36
Reference: [8.6.2.78]
[92] x = 64
Reference: [8.6.2.82]
[95] x = 2.513
Reference: [8.6.2.83]
[96] 4.868
Reference: [8.7.1.86]
x
[97] y 0.1921.73

Reference: [9.1.1.1]
[98] $13.97
Reference: [9.1.1.6]
28
[99] y 
or xy  28
x

Reference: [9.1.2.9]
[100] z 0.4 xy
Reference: [9.3.1.19]

y
10
10 x
–10
[104]
–10
Reference: [9.3.1.21]
[106] x = 2, x = –2, y = 1
Reference: [9.3.1.22]
[107] x = 1, x = –1, y = 0
Reference: [9.4.1.27]
n3
[108]
n5
Reference: [9.4.1.30]
 [110] x  6
x3
Reference: [9.4.1.32]
 [112] x  3
x 1
Reference: [9.4.1.36]
x 1
 [114]
(x  3)(2x  3)
Reference: [9.4.1.38]
2
 [115] (x  2)
5x 3
Reference: [9.4.1.41]
y
 [116]
4x3
Reference: [9.5.1.49]
 [118] 1
x4
Reference: [9.5.1.51]
4
[119]
x3
Reference: [9.6.1.61]
 [120] 30
Reference: [9.6.1.64]
[121] 25
Reference: [9.6.1.65]
 [122] 3,  3
Reference: [9.6.1.68]
 [124] It is not.
Reference: [13.1.1.3]
[127] 8.48
Reference: [13.1.1.12]
2
2 3
[129]

3
3
Reference: [13.1.1.31]
 [130] x = 4 3 , y = 4
Reference: [13.1.1.32]
[131] x = 3,

 y = 3 2
Reference: [13.1.1.33]
A  56, b  9.44, c  16.89
[132] 
Reference: [13.2.1.37]
 [134] Positive coterminal angle: 419 
Negative coterminal angle: –301 
Reference: [13.2.1.39]

 angle: 8
[135] Positive coterminal
3
4
Negative coterminal angle: –
3
Reference: [13.2.1.48]


[137]
4
radians
15
Reference: [13.2.2.54]
 [138] arc length: 2  feet
5
Reference: [13.3.1.57]
 [139] 48 
Reference: [13.3.1.65]
[140]  1
2
Reference: [13.3.1.68]
20
 [141] sin A =
, mA = 44
29
Reference: [13.4.1.75]
 [142] 82.9 
Reference: [13.4.1.76]
[143]
 218 
Reference: [13.4.1.77]
[144]
253 

Reference: [13.4.2.79]
35.83
[145]

Reference: [13.5.1.80]
 [146] c = 15.01 or 3.40
Reference: [13.5.1.84]
[148] C = 78, a = 61.54, c = 81.01
Reference: [13.5.2.89]
2
 [150] 12.99 cm
Reference: [13.6.1.95]
[152]
 4.22
Reference: [13.6.1.97]
[154] A = 43.2°, B = 30.1°, C = 106.7°
Reference: [13.6.2.102]
[155] 228.29
Reference: [14.1.1.9]
f(x )
6
2
1
–2
3
5
6
x
–4
–6
[156]
Reference: [14.1.1.16]
3 7
,
[157]
(There are other correct values.)
4
4
Reference: [14.2.1.22]

y
5


[158]



2

2

–5
Reference: [14.2.1.25]
x
[159] y  2  6sin
3
Reference: [14.3.1.38]
 [161] 1
Reference: [14.3.1.42]
[162] cos2 x
Reference: [14.3.1.43]
 [163] 1
Reference: [14.3.1.44]


x
[164]
sin x
sin x1 cos x 
sin x
1 cos xsin x 1 cos x


1 cos x 

2
1  cos x 1  cos x 1 cos x  1 cos x
sin 2 x
sin x
Reference: [14.3.1.45]
2
2
2
2
2
 [165] tan x csc x  tan x  sec x  tan x  1
Reference: [14.4.1.49]
 [166] 120.00  , 240.00 
Reference: [14.4.1.53]
7 11
  , 5 
,
,
[167]
6 6
6
6
Reference: [14.5.1.61]
 [168] y = 4  sin(x +  )
2

2 3
2
169. a.
2
170. 2cos x

172. a.
7
25
a.
 3 1
1 3
c.
24
7

b.
3 13
13
173. cos
x
174.
b.
 26
26

b.
d.

56
65


175. a. 90, 210, 330
c.
120
119

3 7
b.
,
4
4

16
65

d.

7
25