MATH107 College Algebra Standard Mathematical Conventions Final Exam

Please see attachment for the problems (30 questions)

Math 107 Final Examination
Spring, 2019
1
Math 107 College Algebra
Name______________________________
Final Examination: Spring, 2019
Instructor __________________________
Answer Sheet
Instructions:
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator.
Record your answers and work in this document.
There are 30 problems.
Problems #1-12 are multiple choice. Record your choice for each problem.
Problems #13-21 are short answer. Record your answer for each problem.
Problems #22-30 are short answer with work required. When requested, show all work and write
all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either
case, show work neatly and correctly, following standard mathematical conventions. Each step
should follow clearly and completely from the previous step. If necessary, you may attach extra
pages.
You must complete the exam individually. Neither collaboration nor consultation with
others is allowed. Your exam will receive a zero grade unless you complete the following
honor statement.
Please sign (or type) your name below the following honor statement:
I have completed this final examination myself, working independently and not consulting
anyone except the instructor. I have neither given nor received help on this final examination.
Name _____________________
Date___________________
Math 107 Final Examination
Spring, 2019
MULTIPLE CHOICE. Record your answer choices.
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
SHORT ANSWER. Record your answers below.
13.
14.
15.
16.
17.
18.
19.
(a)
(b)
(c)
20.
(a)
(b)
(c)
(d)
21.
(a)
(b)
(c)
(d)
2
Math 107 Final Examination
Spring, 2019
SHORT ANSWER with Work Shown. Record your answers and work.
Problem
Number
Solution
Answers:
(a)
(b)
Work/for part (a) and explanation for part (b):
22
Answers:
(a)
(b)
(c)
Work for part (a):
23
3
Math 107 Final Examination
Spring, 2019
Answer:
Work:
24
Answer:
Work:
25
Answers:
(a)
(b)
Work for part (a) and for part (b):
26
4
Math 107 Final Examination
Answer:
Work:
27
Answer:
Work:
28
Spring, 2019
5
Math 107 Final Examination
Answers:
(a)
(b)
Work for (b):
29
Answer:
Work:
30
Spring, 2019
6
College Algebra MATH 107
Spring, 2019, V4.9
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
1. ______
4
2
-4
-2
2
4
-2
-4
A.
B.
C.
D.
Domain [–2, 2];
Domain [–3, 1];
Domain [–2, 0];
Domain [–1, 1];
2. Solve:
A.
B.
C.
D.
21 − 4x = − x
Range [–3, 1]
Range [–2, 2]
Range [0, 2]
Range [0, 2]
2. ______
–7
–7, 3
7
No solution
Page 1 of 11
College Algebra MATH 107
3. Determine the interval(s) on which the function is decreasing.
A.
(– 5.3, 5.3)
B.
(– 3.5, 3.5)
C.
(– 2, 2)
D.
(–, –3.5)  (0, 3.5)
Spring, 2019, V4.9
3. ______
4. Determine whether the graph of y = 2 x − 1 is symmetric with respect to the origin,
the x-axis, or the y-axis.
4. ______
A. symmetric with respect to the origin only
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. not symmetric with respect to the y-axis, not symmetric with respect to the x-axis, and
not symmetric with respect to the origin
5. Solve, and express the answer in interval notation: | 5 – 6x |  13.
A.
B.
C.
D.
5. ______
[–4/3, 3]
(–, −4/3]  [3, )
(–, –4/3]
[3, )
Page 2 of 11
College Algebra MATH 107
Spring, 2019, V4.9
6. Which of the following represents the graph of 3x − 8y = 24 ?
A.
B.
C.
D.
6. ______
Page 3 of 11
College Algebra MATH 107
Spring, 2019, V4.9
7. Write a slope-intercept equation for a line parallel to the line x – 6y = 12 which passes through
the point (–18, 7).
7. ______
A.
y = − 6x −101
B.
y=
1
x + 10
6
C.
y=
1
x+7
6
D.
1
y = − x+4
6
8. Which of the following best describes the graph?
A.
B.
C.
D.
8. ______
It is not the graph of a function.
It is the graph of a parabolic relation.
It is the graph of a function and it is one-to-one.
It is the graph of a function but not one-to-one.
Page 4 of 11
College Algebra MATH 107
9. Express as an equivalent expression: 3 log (x + 2) + log 1 – log y
A.
log ( 3 x + 7 − y )
B.
 x3 + 8 
log 

 y 
C.
6 log( x) + 1
log y
D.
 ( x + 2 )3 
log 



y


10. Which of the functions corresponds to the graph?
Spring, 2019, V4.9
9. ______
10. ______
A. f ( x ) = e x + 1
B. f ( x ) = e− x + 2
C. f ( x ) = e x + 2
D. f ( x ) = −e x + 2
Page 5 of 11
College Algebra MATH 107
Spring, 2019, V4.9
11. Suppose that for a function f , the equation f (x) = 0 has exactly three real-number solutions.
Which of the following statements MUST be true?
A.
B.
C.
D.
f
f
f
f
11. ______
is an invertible function.
has exactly three x-intercepts.
has exactly three y-intercepts.
is a polynomial of degree 3.
12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No
formulas are given.) What is the relationship between g(x) and f (x)?
12. ______
-4
4
4
2
2
2
-2
4
-4
2
-2
-2
-2
-4
-4
y = f (x)
4
y = g(x)
A.
B.
C.
D.
g(x) = f (x + 1) – 2
g(x) = f (x + 2) + 2
g(x) = f (x – 2) – 1
g(x) = f (x + 2) – 1
Page 6 of 11
College Algebra MATH 107
Spring, 2019, V4.9
SHORT ANSWER:
13. Multiply and simplify: (9 − 5i)(7 −2i).
Write the answer in the form a + bi, where a and b are real numbers.
Answer: ________
14. Solve, and write the answer in interval notation:
x+4
0.
x −1
Answer: ________
15. A can of soda at 79 F. is placed in a refrigerator that maintains a constant temperature of 36
F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by
T(t) = 36 + 43 e – 0.058 t
Find the temperature of the soda 10 minutes after it is placed in the refrigerator. (Round to the
nearest tenth of a degree.)
Answer: ________
1
16. Find the value of the logarithm: log 4   .
 16 
Answer: ________
17. Solve: 73 x− 4 = 49 .
Answer: ________
18. Suppose $7,200 is invested in an account at an annual interest rate of 4.3% compounded
continuously. How long (to the nearest tenth of a year) will it take the investment to double in
size?
Answer: ________
19. Let f (x) = x2 − 10x + 18.
(a) Find the vertex.
Answer: ________
(b) State the range of the function.
Answer: ________
(c) On what interval is the function decreasing?
Answer: ________
Page 7 of 11
College Algebra MATH 107
Spring, 2019, V4.9
20. Consider the polynomial P(x), shown in both standard form and factored form.
1
3
3
13
1
P( x) = x 4 − x3 + x 2 + x − 6 = ( x + 2)( x − 1)( x − 3)( x − 4)
4
2
4
2
4
(a) Which sketch illustrates the end behavior of the polynomial function?
A.
B.
vvvv
C.
vvvv
D.
vvvv
vvvv
Answer: ________
(b) State the y-intercept.
Answer: ________________
(c) State the zeros of the function.
Answer: ________________
Answer: ________
(d) State which graph below is the graph of P(x).
GRAPH A
GRAPH B
GRAPH C
GRAPH D
Page 8 of 11
College Algebra MATH 107
21. Let f ( x) =
Spring, 2019, V4.9
3x 2 − 3
.
x2 − 4
(a) State the domain.
Answer: _________________
(b) State the horizontal asymptote.
Answer: _________________
(c) State the vertical asymptote(s).
Answer: _________________
(d) Which of the following represents the graph of f ( x) =
3x 2 − 3
?
x2 − 4
GRAPH A.
GRAPH B.
GRAPH C.
GRAPH D.
Answer: ______________
Page 9 of 11
College Algebra MATH 107
Spring, 2019, V4.9
SHORT ANSWER, with work required to be shown, as indicated.
22. Let f ( x) = 9 − x and g(x) = x − 1.
f
(a) Find  (−7) . Show work.
g
(b) Find the domain of the quotient function
f
. Explain.
g
23. Points (–7, 2) and (5, 6) are endpoints of the diameter of a circle.
(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.
Show work.
(b) What is the center point C of the circle?
(c) Given the point C you found in part (b), state the point symmetric to C about the x-axis.
24. Find the equation for a line which passes through the points (3, 4) and (5, –8). Write the
equation in slope-intercept form. Show work.
25. A salesperson earns a base salary of $1,340 per month and a commission of 5.9% on the
amount of sales made. If the salesperson has a paycheck of $3,965.50 for one month, what was
the amount of sales for the month? Show work.
26. Let f (x) = 4×2 – 8 and g(x) = x – 5.
(a) Find the composite function ( f  g )(x) and simplify. Show work.
(b) Find ( f o g ) (3) . Show work.
27. Find the exact solutions and simplify as much as possible: 2×2 = 4x + 7. Show work.
1
28. Given the function f ( x) = 6 − x , find a formula for the inverse function. Show work.
9
Page 10 of 11
College Algebra MATH 107
Spring, 2019, V4.9
29. The Travel Time bus company has determined that when x tourists are given a particular bus
tour, the profit P, in dollars, is given by
P(x) = –0.25 x2 + 27.50x – 240
(a) What is the company’s profit if 34 tourists are given the tour?
(b) How many tourists should be given the tour in order to maximize the company’s profit?
Show work.
30. Solve:
x −3
20
= 2
. Show work.
x+2 x −4
Page 11 of 11
College Algebra MATH 107
Spring, 2019, V4.9
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
1. ______
4
2
-4
-2
2
4
-2
-4
A.
B.
C.
D.
Domain [–2, 2];
Domain [–3, 1];
Domain [–2, 0];
Domain [–1, 1];
2. Solve:
A.
B.
C.
D.
21 − 4x = − x
Range [–3, 1]
Range [–2, 2]
Range [0, 2]
Range [0, 2]
2. ______
–7
–7, 3
7
No solution
Page 1 of 11
College Algebra MATH 107
3. Determine the interval(s) on which the function is decreasing.
A.
(– 5.3, 5.3)
B.
(– 3.5, 3.5)
C.
(– 2, 2)
D.
(–, –3.5)  (0, 3.5)
Spring, 2019, V4.9
3. ______
4. Determine whether the graph of y = 2 x − 1 is symmetric with respect to the origin,
the x-axis, or the y-axis.
4. ______
A. symmetric with respect to the origin only
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. not symmetric with respect to the y-axis, not symmetric with respect to the x-axis, and
not symmetric with respect to the origin
5. Solve, and express the answer in interval notation: | 5 – 6x |  13.
A.
B.
C.
D.
5. ______
[–4/3, 3]
(–, −4/3]  [3, )
(–, –4/3]
[3, )
Page 2 of 11
College Algebra MATH 107
Spring, 2019, V4.9
6. Which of the following represents the graph of 3x − 8y = 24 ?
A.
B.
C.
D.
6. ______
Page 3 of 11
College Algebra MATH 107
Spring, 2019, V4.9
7. Write a slope-intercept equation for a line parallel to the line x – 6y = 12 which passes through
the point (–18, 7).
7. ______
A.
y = − 6x −101
B.
y=
1
x + 10
6
C.
y=
1
x+7
6
D.
1
y = − x+4
6
8. Which of the following best describes the graph?
A.
B.
C.
D.
8. ______
It is not the graph of a function.
It is the graph of a parabolic relation.
It is the graph of a function and it is one-to-one.
It is the graph of a function but not one-to-one.
Page 4 of 11
College Algebra MATH 107
9. Express as an equivalent expression: 3 log (x + 2) + log 1 – log y
A.
log ( 3 x + 7 − y )
B.
 x3 + 8 
log 

 y 
C.
6 log( x) + 1
log y
D.
 ( x + 2 )3 
log 



y


10. Which of the functions corresponds to the graph?
Spring, 2019, V4.9
9. ______
10. ______
A. f ( x ) = e x + 1
B. f ( x ) = e− x + 2
C. f ( x ) = e x + 2
D. f ( x ) = −e x + 2
Page 5 of 11
College Algebra MATH 107
Spring, 2019, V4.9
11. Suppose that for a function f , the equation f (x) = 0 has exactly three real-number solutions.
Which of the following statements MUST be true?
A.
B.
C.
D.
f
f
f
f
11. ______
is an invertible function.
has exactly three x-intercepts.
has exactly three y-intercepts.
is a polynomial of degree 3.
12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No
formulas are given.) What is the relationship between g(x) and f (x)?
12. ______
-4
4
4
2
2
2
-2
4
-4
2
-2
-2
-2
-4
-4
y = f (x)
4
y = g(x)
A.
B.
C.
D.
g(x) = f (x + 1) – 2
g(x) = f (x + 2) + 2
g(x) = f (x – 2) – 1
g(x) = f (x + 2) – 1
Page 6 of 11
College Algebra MATH 107
Spring, 2019, V4.9
SHORT ANSWER:
13. Multiply and simplify: (9 − 5i)(7 −2i).
Write the answer in the form a + bi, where a and b are real numbers.
Answer: ________
14. Solve, and write the answer in interval notation:
x+4
0.
x −1
Answer: ________
15. A can of soda at 79 F. is placed in a refrigerator that maintains a constant temperature of 36
F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by
T(t) = 36 + 43 e – 0.058 t
Find the temperature of the soda 10 minutes after it is placed in the refrigerator. (Round to the
nearest tenth of a degree.)
Answer: ________
1
16. Find the value of the logarithm: log 4   .
 16 
Answer: ________
17. Solve: 73 x− 4 = 49 .
Answer: ________
18. Suppose $7,200 is invested in an account at an annual interest rate of 4.3% compounded
continuously. How long (to the nearest tenth of a year) will it take the investment to double in
size?
Answer: ________
19. Let f (x) = x2 − 10x + 18.
(a) Find the vertex.
Answer: ________
(b) State the range of the function.
Answer: ________
(c) On what interval is the function decreasing?
Answer: ________
Page 7 of 11
College Algebra MATH 107
Spring, 2019, V4.9
20. Consider the polynomial P(x), shown in both standard form and factored form.
1
3
3
13
1
P( x) = x 4 − x3 + x 2 + x − 6 = ( x + 2)( x − 1)( x − 3)( x − 4)
4
2
4
2
4
(a) Which sketch illustrates the end behavior of the polynomial function?
A.
B.
vvvv
C.
vvvv
D.
vvvv
vvvv
Answer: ________
(b) State the y-intercept.
Answer: ________________
(c) State the zeros of the function.
Answer: ________________
Answer: ________
(d) State which graph below is the graph of P(x).
GRAPH A
GRAPH B
GRAPH C
GRAPH D
Page 8 of 11
College Algebra MATH 107
21. Let f ( x) =
Spring, 2019, V4.9
3x 2 − 3
.
x2 − 4
(a) State the domain.
Answer: _________________
(b) State the horizontal asymptote.
Answer: _________________
(c) State the vertical asymptote(s).
Answer: _________________
(d) Which of the following represents the graph of f ( x) =
3x 2 − 3
?
x2 − 4
GRAPH A.
GRAPH B.
GRAPH C.
GRAPH D.
Answer: ______________
Page 9 of 11
College Algebra MATH 107
Spring, 2019, V4.9
SHORT ANSWER, with work required to be shown, as indicated.
22. Let f ( x) = 9 − x and g(x) = x − 1.
f
(a) Find  (−7) . Show work.
g
(b) Find the domain of the quotient function
f
. Explain.
g
23. Points (–7, 2) and (5, 6) are endpoints of the diameter of a circle.
(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.
Show work.
(b) What is the center point C of the circle?
(c) Given the point C you found in part (b), state the point symmetric to C about the x-axis.
24. Find the equation for a line which passes through the points (3, 4) and (5, –8). Write the
equation in slope-intercept form. Show work.
25. A salesperson earns a base salary of $1,340 per month and a commission of 5.9% on the
amount of sales made. If the salesperson has a paycheck of $3,965.50 for one month, what was
the amount of sales for the month? Show work.
26. Let f (x) = 4×2 – 8 and g(x) = x – 5.
(a) Find the composite function ( f  g )(x) and simplify. Show work.
(b) Find ( f o g ) (3) . Show work.
27. Find the exact solutions and simplify as much as possible: 2×2 = 4x + 7. Show work.
1
28. Given the function f ( x) = 6 − x , find a formula for the inverse function. Show work.
9
Page 10 of 11
College Algebra MATH 107
Spring, 2019, V4.9
29. The Travel Time bus company has determined that when x tourists are given a particular bus
tour, the profit P, in dollars, is given by
P(x) = –0.25 x2 + 27.50x – 240
(a) What is the company’s profit if 34 tourists are given the tour?
(b) How many tourists should be given the tour in order to maximize the company’s profit?
Show work.
30. Solve:
x −3
20
= 2
. Show work.
x+2 x −4
Page 11 of 11