Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 1
2 answers: 4 points; #1 and #2: 8 points total each; 20 points
146 points
1. Consider y = 2 | (5/2)x – 10 | – 4
2. Consider 4x – 6y = -3
y = 2 | (5/2)(x – 4) | – 4
6y = 4x – 3 so y = (2/3)x – (1/2)
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
domain
range
x-intercept(s)
y-intercept
slope
vertex
as x → -∞, f(x) →
as x → ∞, f(x) →
0 = 2 | (5/2)x – 10 | – 4
4 = 2 | (5/2)x – 10 |
2 = | (5/2)x – 10 |
all real numbers
y ≥ -4
(16/5, 0), (24/5, 0)
(0, 16)
xxxxxxx
(4, -4)
∞
∞
all real numbers
all real numbers
(-3/4, 0)
(0, 1/2)
2/3
xxxxxxx
-∞
∞
(5/2)x – 10 = 2 or (5/2)x – 10 = -2
(5/2)x = 12 or (5/2)x = 8
x = 24/5 or x = 16/5
3. Solve -2(4 – 5x) + 3x = -1 + (3 – 2x) for x. Present your answer as a simplified fraction.
____________ x = 2/3 (self)
-8 + 10x + 3x = -1 + 3 – 2x
-8 + 13x = 2 – 2x
15x = 10
x = 10/15 = 2/3
4. Solve -5(3 – 5x) – 2x < 1 – 3(5x + 4) for x. Present your answer as a simplified fraction.
____________ x < 2/19 (self)
-15 + 25x – 2x < 1 – 15x – 12
38x < 4
-15 + 23x < -15x – 11
x < 4/38 so x < 2/19
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 2
2 answers: 4 points; #5: 7 points total; #6: 10 points total; 21 points this page; 41 points total
5. Consider y = x3 + 10x2 – 9x – 90
6. Consider y = x2 – 4x – 21
y = (x + 3)(x – 3)(x + 10)
y = (x + 3)(x – 7)
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
domain
range
vertex (exact answer, please)
x-intercept(s)
y-intercept
interval(s) on which y decreases
interval(s) on which y increases
as x → -∞, f(x) →
as x → ∞, f(x) →
all real numbers
all real numbers
xxxxxxxxx
(-3, 0), (3, 0), (-10, 0)
(0, -90)
Xxxxxxxxx
Xxxxxxxxx
-∞
∞
all real numbers
[-25, ∞)
(2, -25)
(-3, 0), (7, 0)
(0, -21)
(-∞, 2)
(2, ∞)
∞
∞
7. Solve 2x2 + 6x + 8 = -1 for x. If necessary, use simplified radicals, and use i for -1. ____________
x = [-3 ± 3i] / 2 (self)
2x2 + 6x + 9 = 0
x = [-6 ± √(36 – 4(18))] / 4
x = [-6 ± √(-36)] / 4
x = [-6 ± 6i] / 4
x = [-3 ± 3i] / 2
8. Solve 12x2 – 11x + 2 > 0 for x. ____________ x > 2/3 or x < 1/4 (self)
(3x – 2)(4x – 1) = 0
x = 2/3 or x = 1/4
pos
|neg |
pos
-----2-----+-----1-----+-----0--|--+-|---1-----+-----2-----|
|
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 3
2 answers: 4 points; #9: 9 points total; #10: 9 points total; 22 points this page; 63 points total
x3 – 2x2 – 9x + 18
x2 – 3
9. Consider y = ------------------------.
10. Consider y = ---------.
x2 – 7x + 10
x+7
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
factored form
domain
hole(s)
x-intercept(s)
y-intercept
horizontal asymptote
vertical asymptote(s)
slant asymptote
(x – 2)(x – 3)(x+3)
-----------------------(x – 2)(x – 5)
(-∞, 2)  (2, 5)  (5, ∞)
(2, 5/3)
(-3, 0), (3, 0)
(0, 9/5)
none
x=5
y=x+5
x2 – 3
-------x+7
(-∞, -7)  (-7, ∞)
none
(-√3, 0), (√3, 0)
(0, -3/7)
none
x = -7
y=x–7
x
14
11. Solve --------- = --------- for x. Present your answer as a simplified fraction. ____________
x+2
x + 11
x = -4 or x = 7
(self)
x(x + 11) = 14(x + 2)
(x + 4)(x – 7) = 0
x2 + 11x = 14x + 28
x = -4 or x = 7
x2 – 3x – 28 = 0
x2 + 10x + 16
12. Solve ------------------- ≥ 0 for x. ____________ [-8, -5)  [-2, ∞) (self: Stewart 5th: 4.RE.41)
x+5
[(x + 8)(x + 2)] / (x + 5) , so critical numbers: {-8, -5, -2}
neg |
pos
|
neg
|
pos
---9---|---7---6---|---4---3---|---1---0--|
|
|
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 4
2 answers: 4 points; #13: 8 points total; #14: 8 points total; 20 points this page; 83 points total
13. Consider y = (-3/2)√[(3/2)x – 3] + 4.
14. Consider y = (1/2){3√[(-5/2)x – 5]} + 1
y = (-3/2)[(3/2)(x – 2)] + 4
y = (1/2){3√[(-5/2)(x + 2)]} + 1
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
domain
range
x-intercept (exact answer, please)
y-intercept (exact answer, please)
initial point
point of symmetry
as x → -∞, f(x) →
as x → ∞, f(x) →
[2, ∞)
(-∞, 4]
(182/27, 0)
none
(2, 4)
xxxxxxxxx
does not apply
-∞
all real numbers
all real numbers
(6/5, 0)
(0, (1/2)[3√(-5)] + 1)
xxxxxxxxx
(-2, 1)
∞
-∞
0 = (-3/2)√[(3/2)x – 3] + 4
0 = (1/2){3√[(-5/2)(x + 2)]} + 1
-4 = (-3/2)√[(3/2)x – 3]
-1 = (1/2){3√[(-5/2)(x + 2)]}
8/3 = √[(3/2)x – 3]
-2 = 3√[(-5/2)(x + 2)]
64/9 = (3/2)x – 3
-8 = (-5/2)(x + 2)
91/9 = (3/2)x
16/5 = x + 2
x = 182/27
x = 6/5
15. Solve √(3x – 2) + 4 = 1 for x. ____________ no solution (self)
√(3x – 2) = -3
no solution; principal square roots are never negative
16. Solve 3√[(-5/2)(x + 5)] – 6 = 9 for x. Present your answer as a simplified fraction. ____________
-1,355 (self)
3
√[(-5/2)(x + 5)] = 15
x + 5 = -1,350
(-5/2)(x + 5) = 3,375
x = -1,355
-5(x + 5) = 6,750
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 5
2 answers: 4 points; #17: 9 points total; #18: 9 points total; 20 points this page; 105 points total
17. Consider y = e(-3/2)x – 4
18. Consider y = -2 log ((3/2)x + (3/2)) + 4
y = -2 log ((3/2)(x + 1)) + 4
For each equation, complete the table below. Then sketch a graph of the equation on the axes below.
domain
range
x-intercept (exact answer, please)
y-intercept (exact answer, please)
horizontal asymptote
vertical asymptote
as x → -∞, f(x) →
as x → ∞, f(x) →
all real numbers
(-4, ∞)
(-2/3)(ln 4) OR (-0.924, 0)
(0, -3)
y = -4
none
∞
-4
(-1, ∞)
all real numbers
(197/3, 0) OR (65.667, 0)
(0, -2 log (3/2) + 4) OR (0, 3.648)
none
x = -1
does not apply
-∞
0 = e(-3/2)x – 4
4 = e(-3/2)x
(-3/2)x = loge 4
x = (-2/3)(ln 4)
0 = -2 log ((3/2)x + (3/2))) + 4
-4 = -2 log ((3/2)x + (3/2)))
2 = log ((3/2)x + (3/2)))
102 = (3/2)x + (3/2)
(3/2)x = 100 – 3/2 = 98.5
x = (2/3)(197/2) = 197/3
19. Solve log3 (2x – 1) – log3 (x – 4) = 2 for x. ____________ 5 (SLO)
log3 [(2x – 1) / (x – 4)] = 2
2x – 1 = 9x – 36
(2x – 1) / (x – 4) = 9
35 = 7x
2x – 1 = 9(x – 4)
x=5
20. Solve (4)22x+8 + 3 ≥ -3 for x. ____________ all real numbers (self)
(4)22x+8 ≥ -6
subtract 3 from both sides
2x+8
2
≥ -3/2
divide both sides by 4
x = all real numbers
22x+8 is always positive
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 6
9 answers: 12 points; 117 points total
21. Suppose you start a business making widgets. After a short period of time, you find that your total
cost of producing 16 widgets is $420 and your total cost of producing 42 widgets is $680. Assume that
the total cost of producing widgets varies linearly with the number of widgets your produce. (self:
1.CT.29)
a. Write a linear function which models the total cost of producing widgets, y, with the number of
widgets you produce, x. ____________ y = 10x + 260
b. Use your answer from part a to find the cost of producing 11 widgets. ____________ $370
c. Use your answer from part a to find the number of widgets you produce if your total cost is $560.
____________ 30
a. (16, 420) and (42, 680)
y – 420 = 10x – 160
c. 560 = 10x + 260
m = (680 – 420) / (42 – 16)
y = 10x + 260
10x = 300
m = 260/26 = 10
b. 10(11) + 260
x = 30
y – 420 = 10(x – 16)
110 + 260 = 370
22. A firefighter holds a hose 3 meters off the ground and directs a stream of water toward a burning
building. The height of the water above the ground can be approximated by h = -0.026x2 + 0.577x + 3,
where h is the height of the water, in meters, at a point x meters horizontally from the firefighter to the
building. (2.1.46)
a. How many meters from the firefighter will the water reach its maximum height? ____________
b. What is the maximum height of the water? ____________ 6.2 meters
c. The water reaches the house after it reaches its maximum point and at a height of 6 meters above
the ground. How far is the firefighter from the house? ____________ 13.9 meters
a. –b/2a
b. -0.026(11.1)2 + 0.577(11.1) + 3
c. 6 = -0.026x2 + 0.577x + 3
-0.577/-0.052 -0.026(123.21) + 0.577(11.1) + 3
-0.026x2 + 0.577x – 3 = 0
11.1 meters
-3.20346 + 6.4047 + 3
x = [-0.577 ± √(0.332929 – 4(-0.026)(-3))] / -0.052
6.20124
x = [-0.577 ± √(0.332929 – 0.312)] / -0.052
x = [-0.577 ± √(0.020929)] / -0.052
x = (-0.577 ± 0.144669) / -0.052
x = 8.314058 or x = 13.87825
23. According to cbsnews.com, the most expensive ticket to Super Bowl I (played in 1967) was $12. The
most expensive ticket to Super Bowl XLIX (played in 2015) was $1,350. When answering parts a and b,
use the discrete model: F = P(1 + i)n). (self)
a. Calculate the rate of return on the most expensive ticket to the Super Bowl over the time interval
from 1967 to 2015. Round your answer to the nearer thousandth. ____________ 10.340%
b. Formulate an equation which you can use to model the price of the most expensive ticket to the
Super Bowl n years after 1967. ____________ F = 12(1.103398)n
c. If most expensive ticket to the Super Bowl has appreciated at the same rate to the present year, how
much should the most expensive ticket to Super Bowl XLVI (played in 2022) cost? Round your answer
to the nearer dollar. ____________ $2,688
1,350 = 12(1 + i)48
F = 12(1.103398)55
48
112.5 = (1 + i)
12(244.011639)
1.103398 = 1 + i
2,688.139667
i = 0.103398
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 7
4 answers: 6 points; #26: 5 points total; 11 points this page; 128 points total
24. The equation y = 3.33 + 14.9 ln x relates the diameter of a sugar maple tree, x (measured in
millimeters) to the age of the tree, y (measured in years). Use this model when answering the following
questions. (3.6 Ex.8, alt.)
a. Estimate the age of a sugar maple tree which has a diameter of 81 millimeters. ____________
68.807
b. Estimate the diameter of a sugar maple tree which is seven years old. ____________ 1.279 mm
a. y = 3.33 + 14.9 ln (81)
b. 7 = 3.33 + 14.9 ln x
y = 3.33 + 14.9 (4.394449)
3.67 = 14.9 ln x
y = 3.33 + 65.477292
3.67/14.9 = ln x
y = 68.507292
x = e3.67/14.9
x = 1.279294
25. An administrative assistant is hired after graduating from high school and learns to type on the job.
The number of words he can type per minute, N, is given by
130x + 260
N = --------------, (Young 2nd: 2.6.65)
x+5
where x is the number of days he has spent working. Use this equation when answering parts a and b.
a. How many words per minute would you expect him to type at the beginning of his first day of
work? ____________ 52
[130(0) + 260] / [(0) + 5] = 52
b. How many words per minute would you expect him to type on his last day of work if he worked at
this job until he retired? ____________ 130
horizontal asymptote at y = 130/1 = 130
26. Solve the system of equations below for x and y. Use any algebraic method you choose. Please
provide exact answers.
(4/3)x – (3/2)y = (3/4)
(-1/4)x + (1/3)y = (2/3)
a. x = ____________ 18
 4/3 -3/2
-1/4 1/3
3/4 
2/3 
 1
-9/8
-1/4 1/3
9/16 multiply row 1 by 3/4
2/3 
b. y = ____________ 31/2
 1 -9/8
9/16  multiply row 1 by 1/4 and add the results to row 2
 0 5/96 155/192 1/4 -9/32 9/64
32/96 – 27/96
128/192 + 27/192
 1 -9/8 9/16
 0
1 31/2 
multiply row 2 by 96/5
155/192 x 96/5 = 31 x 1/2 = 29/2
 1
 0
multiply row 2 by 9/8 and add the results in row 1
0 9/8 279/16
9/16 + 279/16 = 288/16 = 18
0
1
18 
31/2 
Math 161 Final Exam Version 1 (144 points)
Name ___Answers____________________
Precalculus I
2022 May 09
John Scalea
Page 8
6 answers: 12 points; #28: 6 points total; 18 points this page; 146 points total
15x + 54
27. Find the partial fraction decomposition of ----------------. ____________ [8 / (x + 10)] + [7 / (x – 2)]
x2 + 8x – 20
(self: 8.CT.27)
[(15x + 54) / ((x + 10)(x – 2))] = [A / (x + 10)] + [B / (x – 2)]
15x + 54 = A(x – 2) + B(x + 10)
15x + 54 = Ax – 2A + Bx + 10B A = 15 – B
23B = 84
15x + 54 = Ax + Bx – 2A + 10B -2(15 – B) + 10B = 54
B=7
15x + 54 = (A + B)x – 2A + 10B
-30 + 2B + 10B = 54
A = 15 – (7)
so A + B = 15 and -2A + 10B = 54
12B = 54 + 40
A=8
28. Consider f(x) = x2 and g(x) = (-2x + 14)2 – 7. State the transformations present in g(x) when
compared to f(x). Use only the number of answer lines necessary.
g(x) = (-2(x – 7))2 – 7
a. ____________ reflection over the y-axis
a = 1; b = -2; c = -7; d = -7
b. ____________ horizontal compression 1/2
c. ____________ horizontal shift 7 right
d. ____________ vertical shift 7 down
e. ____________
f. ____________
Questions 29 through 33 pertain to the matrices below. (self)
A
-3 -4 9 
-3 -9 -6 
-1 2 -1 
B
 7 9 
-9 -4 
-1 -9 
C
-6
 4
0 
2 
D
 7 9 
 1 -5 
E
-8 -4 
-4 8 
29. Find E + B.
no solution
32. Find CD.
-42 -54 
 30 26 
30. Find A – D.
no solution
33. Find 6E.
-48 -24 
-24 48 
31. Find BE.
-92 44 
 88
4 
 44 -68 
Math 161 Final Exam Version 1 (144 points)
Precalculus I
John Scalea
Name ___Answers____________________
2022 May 09
Page 9
34. Solve the system of equations below for x and y. Use any algebraic method you choose. Please
provide exact answers.
(4/3)x – (3/2)y = (3/4)
(-1/4)x + (1/3)y = (2/3)
a. x = ____________
b. y = ____________
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