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Subject
Mathematics
Date
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MTH 103A
Practice Exam 2 – Unit B
Version B
Name
:
Section
:
Instructor
:
This is a practice exam. It may not be representative of the actual exam as
there are many different ways to test each objective.
How to get the most out of the practice exam experience
:
•
Give yourself a time limit. The actual exam is 2 hours so we recommend you try to do this in 2 hours
or less.
•
Don’t use notes / friends. Try to take it under actual exam conditions to get an accurate gauge of
your current knowledge level.
•
Grade yourself and be a bit mean, especially with explanations
–
If you are saying “
yeah that’s what I meant to say
” this could be a bad sign. The graders don’t
know what you meant to say, only what you said. This could be an indicator that you need to
work on your explanations a bit.
–
Once you grade yourself use these grades for each objective to help guide your studying. If you
mark yourself a 2 or below for some objective this may be a sign that you should go back and do
more WebWork problems / practice problems / review the lesson content or explorations / etc.
to help solidify your knowledge.
•
Use your instructor’s office hours or the MLC for tougher questions you still need help with. Remember
we are here to help!
MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B1: Given any of the four representations of a function, student can determine
if it is linear and, if so, student can find the slope and intercepts.
1. For parts (a)-(d), determine if the function is linear. Explain your reasoning.
If it is linear, identify the slope,
y
-intercept, and
x
-intercept.
(a)
-
5
-
4
-
3
-
2
-
1
1
2
3
4
5
-
2
2
4
6
8
10
x
f
(
x
)
(b)
x
g
(
x
)
1
9
3
15
5
21
7
27
9
33
11
39
13
45
(c)
h
(
x
) = 2
x
(d) The forester left his truck at 9:00 a.m. and
walked due north at a pace of 3 miles per
hour.
Let
d
(
t
) represent the distance from
his truck after
t
hours.
Page 2
MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B2: Given one representation of a linear function, student can create or identify
the other three representations.
2. Under a proposed graduated income tax system, single taxpayers would owe $1500 plus 20% of the
amount of their income over $13,000. (For example, if your income is $18,000, you would pay $1500
plus 20% of $5000.)
(a) Create a table to represent the amount of income tax owed by single people making more than
$13,000. Use income values between $15,000 and $60,000.
(b) Create a graph to represent this context. Be sure to use an appropriate scale and label the axes.
(c) Write the function in symbolic form.
(d) Identify the domain and range of the function in the context of this situation.
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MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B3: Given data or a verbal description, student can find a linear model and
use it to answer questions about a situation.
3. A mountain climber feels that the air temperature decreases as his elevation increases. When his eleva-
tion is 2000 feet above sea level, the temperature is 60
o
F. The temperature decreases by 3
o
F for every
1000 feet the climber ascends.
(a) Find a linear model
f
(
x
) that can be used to determine the air temperature at an elevation of
x
feet above sea level.
(b) Identify the slope and vertical intercept of this function and explain their meanings in the context
of this situation.
(c) Find
f
(10000) and explain its meaning in the context of this situation.
(d) For which value of
x
is
f
(
x
) = 32? What does this mean in the context of this situation?
Page 4
MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B4: Given a function in any of the four representations, student can evaluate
the average rate of change (AROC) between two points.
4. Jodi and Jamal compete in a bicycle race.
The graphs below show the distance, in miles, each has
traveled as a function of time which is measured in hours.
1
2
3
4
5
10
20
30
40
50
60
hours
distance
Jamal’s bicycle race
1
2
3
4
5
10
20
30
40
50
60
hours
distance
Jodi’s bicycle race
(a) Use the graph to estimate Jamal’s average speed for the entire race and his average speed between
hours 2 and 4.
(b) Use the graph to estimate Jodi’s average speed for the entire race and her average speed between
hours 2 and 4.
(c) What do your answers to parts (a) and (b) tell you about the race?
(d) The equation that can be used to model Jodi’s disance after
t
hours is
s
(
t
) = 2
.
4
t
2
.
Use this
function to find Jodi’s average speed during the last hour of the race.
Page 5
MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B5: Student can find a solution, if one exists, to a system of linear equations
and use it to solve application problems.
5. Two containers are placed at opposite ends of a balance scale. Container A weighs 210 kg but is leaking
at a rate of 8 kg per minute. Container B weighs 93 kg but liquid is being added at a rate of 27 kg per
minute.
(a) Write a function
A
(
t
) that can be used to determine the weight of container A, at
t
minutes after
it is placed on the scale.
(b) Write a function
B
(
t
) that can be used to determine the weight of container B, at
t
minutes after
it is placed on the scale.
(c) Use your equations to determine after how many minutes the scale will reach equilibrium. (When
the objects have the same weight.) What will each container weigh at this time?
(d) Graph
A
(
t
) and
B
(
t
) on the same coordinate plane and explain how you could use the graph to
verify your answer to part (c).
Page 6
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MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B6: Given a graphical or symbolic (including piecewise) representation of an
absolute value function, student can find the other representations.
6. Given the absolute value function
f
(
x
) =
1
3
|
x
-
2
| -
6
(a) Sketch the graph of
f
(
x
). Be sure to use an appropriate scale and label the axes.
(b) Write
f
(
x
) as a piecewise function.
(c) Find the domain and range of
f
(
x
).
Page 7
MTH 103A
Practice Exam 2 – Unit B
Version B
Objective B7:
Student can use equation or graph to find solutions to absolute value
equations and inequalities.
7. Solve the following:
i)
-|
x
+ 1
|
+ 4 = 2
ii)
-|
x
+ 1
|
+ 4
≤
0
(a) Using a graph.
(b) Algebraically using the piecewise form of
f
(
x
).
(c) Algebraically using the absolute value form of
f
(
x
).
Page 8
MTH 103A
Practice Exam 2 – Unit B
Version B
Scoring Rubric:
Points
Description
4
I own this! I can explain how to do this and why it works.
3
I know how to do this and can do the problems independently, but I am not quite
sure why it works.
2
I’m starting to get it, but I made a couple of mistakes.
1
I have to look at examples to finish this problem.
0
I have no idea what you are talking about!
Page 9
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MTH 103A
Practice Exam 2 – Unit B
Version B
1. For parts (a)-(d), determine if the function is linear. Explain your reasoning.
(a)
-
5
-
4
-
3
-
2
-
1
1
2
3
4
5
-
2
2
4
6
8
10
x
f
(
x
)
Solution:
The function
f
(
x
) is linear be-
cause the graph is a straight line.
(b)
x
g
(
x
)
1
9
3
15
5
21
7
27
9
33
11
39
13
45
Solution:
This function is linear
because
Δ
g
(
x
)
Δ
x
is
al-
ways
6
2
or 3.
(c)
h
(
x
) = 2
x
Solution:
The function
h
(
x
) is NOT linear because
the variable is an exponent so it can’t be
written in the form
y
=
mx
+
b
.
(d) The forester left his truck at 9:00 a.m. and
walked due north at a pace of 3 miles per
hour.
Let
d
(
t
) represent the distance from
his truck after
t
hours.
Solution:
The function
d
(
t
) is linear because the dis-
tance is changing at a constant rate of 3
miles per hour.
Now For each of the functions in question 1 that is linear, identify the slope, vertical intercept, and
horizontal intercept.
Solution:
(a):
Slope is
rise
run
=
0
1
= 0.
The vertical intercept is where the line intersects the vertical axis, or (0,4).
The horizontal intercept is where the line intersects the horizontal axis. It does not exist.
(b):
Slope is
Δ
g
(
x
)
Δ
x
=
6
2
= 3.
The vertical intercept is the point where
x
= 0 so, using the slope to continue the pattern until
x
= 0, we get the point (0,6).
The horizontal intercept is the point where
g
(
x
) = 0 so, using the slope to continue the pattern
until
g
(
x
) = 0, we get the point (-2,0).
(c):
Not linear.
(d):
Slope is rate of change. Because the distance is increasing at a constant rate of 3 miles each hour,
the slope is 3.
Page 10
MTH 103A
Practice Exam 2 – Unit B
Version B
The vertical intercept is the initial amount. When
t
= 0 or the forester starts walking, his dis-
tance from the truck is also 0 so both the vertical and horizontal intercepts are (0,0).
2. Under a proposed graduated income tax system, single taxpayers would owe $1500 plus 20% of the
amount of their income over $13,000. (For example, if your income is $18,000, you would pay $1500
plus 20% of $5000.)
(a) Create a table to represent the amount of income tax owed by single people making more than
$13,000. Use income values between $15,000 and $60,000.
Solution:
x
,Income
15000
20000
25000
30000
35000
40000
45000
50000
55000
60000
f
(
x
),Tax
1900
2900
3900
4900
5900
6900
7900
8900
9900
10900
(b) Create a graph to represent this context. Be sure to use an appropriate scale and label the axes.
Solution:
10
20
30
40
50
60
2
4
6
8
10
12
x
, (income
x
1000)
f
(
x
), (tax
x
1000)
(c) Write the function in symbolic form.
Solution:
f
(
x
) = 1500 + 0
.
2(
x
-
13000) OR
f
(
x
) = 0
.
2
x
-
1100
Page 11
MTH 103A
Practice Exam 2 – Unit B
Version B
(d) Identify the domain and range of the function in the context of this situation.
Solution:
The domain of this function is theoretically [13000,
∞
). The rule applies to singles making over
$13000, it does not give an upper limit on the income.
The range of this function is theoretically [1500,
∞
). This is the range of taxes for incomes included
in the domain.
3. A mountain climber feels that the air temperature decreases as his elevation increases. When his eleva-
tion is 2000 feet above sea level, the temperature is 60
o
F. The temperature decreases by 3
o
F for every
1000 feet the climber ascends.
(a) Find a linear model
f
(
x
) that can be used to determine the air temperature at an elevation of
x
feet above sea level.
Solution:
Because the temperature decreases 3
o
F every 1000 feet of elevation, the slope of this function is
-
3
1000
or -0.003.
Using this along with the point (2000,60) that is given, we can use point-slope form to find the
function:
y
-
60 =
-
0
.
003(
x
-
2000)
y
-
60 =
-
0
.
003
x
+ 6
y
= 0
.
003
x
+ 66
So
f
(
x
) =
-
0
.
003
x
+ 66
(b) Identify the slope and vertical intercept of this function and explain their meanings in the context
of this situation.
Solution:
Slope = -0.003. This means that the temperature decreases 0.003
o
F for each 1 foot gain in eleva-
tion. We could also say that the temperature decreases 3
o
F each 1000 ft. gain in elevation.
Vertical intercept = 66. This means that the temperature at sea level is 66
o
F.
(c) Find
f
(10000) and explain its meaning in the context of this situation.
Solution:
f
(10000) =
-
0
.
003(10000) + 66
=
-
30 + 66
= 36
This means that the temperature at an elevation of 10,000 feet is 36
o
F.
Page 12
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MTH 103A
Practice Exam 2 – Unit B
Version B
(d) For which value of
x
is
f
(
x
) = 32? What does this mean in the context of this situation?
Solution:
32 =
-
0
.
003
x
+ 66
-
34 =
-
0
.
003
x
11333
.
33 =
x
The temperature will drop to freezing at 11,333.33 feet of elevation.
4. Jodi and Jamal compete in a bicycle race.
The graphs below show the distance, in miles, each has
traveled as a function of time which is measured in hours.
1
2
3
4
5
10
20
30
40
50
60
hours
distance
Jamal’s bicycle race
1
2
3
4
5
10
20
30
40
50
60
hours
distance
Jodi’s bicycle race
(a) Use the graph to estimate Jamal’s average speed for the entire race and his average speed between
hours 2 and 4.
Solution:
Entire race:
AROC
=
60
-
0
5
-
0
=
60
5
= 12
Average speed = 12 mph
Between hours 2 and 4:
AROC
=
48
-
24
4
-
2
=
24
2
= 12
Average speed = 12 mph
(b) Use the graph to estimate Jodi’s average speed for the entire race and her average speed between
hours 2 and 4.
Page 13
MTH 103A
Practice Exam 2 – Unit B
Version B
Solution:
Entire race:
AROC
=
60
-
0
5
-
0
=
60
5
= 12
Average speed = 12 mph
Between hours 2 and 4:
AROC
=
38
-
10
4
-
2
=
28
2
= 14
Average speed = 14 mph
(c) What do your answers to parts (a) and (b) tell you about the race?
Solution:
Jamal’s speed remained constant throughout the entire race while Jodi started out slower but
her speed gradually increased as the race went on. In the end, it was a tie because their overall
average speeds were the same.
(d) The equation that can be used to model Jodi’s disance after
t
hours is
s
(
t
) = 2
.
4
t
2
.
Use this
function to find Jodi’s average speed during the last hour of the race.
Solution:
AROC
=
s
(5)
-
s
(4)
5
-
4
=
2
.
4(5)
2
-
2
.
4(4)
2
5
-
4
=
60
-
38
.
4
5
-
4
=
21
.
6
1
= 21
.
6
Jodi’s average speed during the last hour of the race was 21.6 mph.
5. Two containers are placed at opposite ends of a balance scale. Container A weighs 210 kg but is leaking
at a rate of 8 kg per minute. Container B weighs 93 kg but liquid is being added at a rate of 27 kg per
minute.
(a) Write a function
A
(
t
) that can be used to determine the weight of container A, at
t
minutes after
it is placed on the scale.
Page 14
MTH 103A
Practice Exam 2 – Unit B
Version B
Solution:
A
(
t
) = 210
-
8
t
(b) Write a function
B
(
t
) that can be used to determine the weight of container B, at
t
minutes after
it is placed on the scale.
Solution:
B
(
t
) = 93 + 27
t
(c) Use your equations to determine after how many minutes the scale will reach equilibrium. (When
the objects have the same weight.) What will each contain weigh at this time?
Solution:
We want to know when
A
(
t
) =
B
(
t
):
210
-
8
t
= 93 + 27
t
210 = 93 + 35
t
117 = 35
t
117
35
=
t
They will reach equilibrium after approximately 3.34 minutes.
A
(
t
) = 210
-
8(
117
35
)
= 210
-
936
35
=
6414
35
After approximately 3.34 minutes, each container will weigh about 183.26 kg.
(d) Graph
A
(
t
)
and
B
(
t
)
on the same coordinate plane and explain how you could use the graph to
verify your answer to part (c).
Solution:
Page 15
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MTH 103A
Practice Exam 2 – Unit B
Version B
0
.
5
1
1
.
5
2
2
.
5
3
3
.
5
4
4
.
5
5
20
40
60
80
100
120
140
160
180
200
220
Time (minutes)
Weight (kg)
We can see that the
x
coordinate of the point of intersection is between 3 and 3.5 and the
y
coordinate of the point of intersection is just above 180. This aligns with our answer to part (c).
6. Given the absolute value function
f
(
x
) =
1
3
|
x
-
2
| -
6
(a) Sketch the graph of
f
(
x
). Be sure to use an appropriate scale and label the axes.
Solution:
-
10
-
8
-
6
-
4
-
2
2
4
6
8
10
-
10
-
8
-
6
-
4
-
2
2
4
6
8
10
x
f
(
x
)
(b) Write
f
(
x
) as a piecewise function.
Solution:
We can find the slopes of the two lines from the graphs or the ”
a
” value in the absolute value
equation, however the
y
-intercept is not an integer value so we have to use point slope form to
find the two linear equations. To find the left line, we use a slope of
-
1
3
and the point (2,-6) as
Page 16
MTH 103A
Practice Exam 2 – Unit B
Version B
follows:
y
-
(
-
6) =
-
1
3
(
x
-
2)
y
+ 6 =
-
1
3
x
+
2
3
y
=
-
1
3
x
-
16
3
To find the right line, we use a slope of
1
3
and the point (2,-6) as follows:
y
-
(
-
6) =
1
3
(
x
-
2)
y
+ 6 =
1
3
x
-
2
3
y
=
1
3
x
-
20
3
So the piecewise function is:
f
(
x
) =
-
1
3
x
-
16
3
:
x
≤
2
1
3
x
-
20
3
:
x >
2
OR
f
(
x
) =
-
1
3
x
-
16
3
:
x <
2
1
3
x
-
20
3
:
x
≥
2
(c) Find the domain and range of
f
(
x
).
Solution:
From the graph we can see that the domain contains all
x
values or
D
= (
-∞
,
∞
).
We can also see that the lowest y-value is -6 so
R
= [
-
6
,
∞
)
7. Solve the following:
i)
-|
x
+ 1
|
+ 4 = 2
ii)
-|
x
+ 1
|
+ 4
≤
0
(a) Using a graph.
Solution:
Page 17
MTH 103A
Practice Exam 2 – Unit B
Version B
-
10
-
8
-
6
-
4
-
2
2
4
6
8
10
-
10
-
8
-
6
-
4
-
2
2
4
6
8
10
x
f
(
x
)
-|
x
+ 1
|
+ 4 = 2:
As shown by the red points on the graph,
-|
x
+ 1
|
+ 4 = 2 when
x
=
-
3 and
x
= 1.
-|
x
+ 1
|
+ 4
≤
0:
As shown by the yellow highlight, the interval over which
-|
x
+ 1
|
+ 4
≤
0 is (
-∞
,
-
5]
∪
[3
,
∞
) .
(b) Algebraically using the piecewise form of
f
(
x
).
Solution:
The piecewise form of
f
(
x
) is
f
(
x
) =
x
+ 5
:
x
≤ -
1
-
x
+ 3
:
x >
-
1
-|
x
+ 1
|
+ 4 = 2:
To find all solutions, we have to find the
x
values for which each piece is equal to 2:
x
+ 5 = 2
x
=
-
3
-
x
+ 3 = 2
-
x
=
-
1
x
= 1
-|
x
+ 1
|
+ 4
≤
0:
To find all solutions, we have to find the
x
values for which each piece is less than or equal to 0:
x
+ 5
≤
0
x
≤ -
5
-
x
+ 3
≤
0
-
x
≤ -
3
x
≥
3
Combining these results we get the solution (
-∞
,
-
5]
∪
[3
,
∞
)
Page 18
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MTH 103A
Practice Exam 2 – Unit B
Version B
(c) Algebraically using the absolute value form of
f
(
x
).
Solution:
-|
x
+ 1
|
+ 4 = 2: We start by isolating the absolute value:
-|
x
+ 1
|
+ 4 = 2
-|
x
+ 1
|
=
-
2
|
x
+ 1
|
= 2
In order for this to be true, the value inside the absolute value must be either 2 or -2 so:
x
+ 1 = 2
x
= 1
x
+ 1 =
-
2
x
=
-
3
-|
x
+ 1
|
+ 4
≤
0: Again we start by isolating the absolute value:
-|
x
+ 1
|
+ 4
≤
0
-|
x
+ 1
| ≤ -
4
|
x
+ 1
| ≥
4
This means that the value of
x
+ 1 must be more than 4 spaces from 0.
-
6
-
5
-
4
-
3
-
2
-
1
0
1
2
3
4
5
6
So we set up the following inequalities:
x
+ 1
≤ -
4
x
≤ -
5
x
+ 1
≥
4
x
≥
3
So the solution is (
-∞
,
-
5]
∪
[3
,
∞
)
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