Math 107 Final Exam Review
1. Find the x- and y-intercepts of 2x − 5y = 15 and use them to sketch a graph of the line.
Explain your reasoning.
2. Find the slope and y-intercept of 5x + 3y = 2 and sketch a graph of the line.
3. Find the midpoint of the segment with endpoints ( 2,1) and ( −3, 7 ) .
4. Find the distance between the two points ( 2,1) and ( −3, 7 ) .
5. Do the points ( −3,1) , ( 2, −1) , and ( 6, 9 ) form the vertices of a right triangle? Explain your
reasoning.
6. Find the equation of the circle that has its center at ( −1, 4 ) passing through the point ( 3, 7 ) .
7. The points
(
)
7, −4 and
(
)
2, 3 are the endpoints of the diameter of a circle. What is the
center of the circle?
8. Given the graph of the following function, f, determine the values f (1) , f ( 3) , and f ( −1) .
9. An un-stretched spring of length 7 cm can be used to measure the mass of objects. As
masses are added to the end of the spring, it stretches. The length of the spring when 50 g is
added to the end is 8.1 cm.
a) Find a linear function l ( x ) where l represents the length of the spring and m the mass of
the object.
b) How long will the spring be when 25 g is hanged from the spring?
c) When a certain mass is hanged on the end of the spring, it stretches to a length of 9.2 cm.
What is the mass of the object?
d) Explain the meaning of the slope from your function.
10. Consider the following data on the average monthly Social Security benefit for retired
workers. Here the years are given in years since 1970 and the benefit is given in US dollars.
Use linear regression to model the data. Use your model to predict the average monthly
benefit for retired workers in 1993, 2009, and 2011.
Years
Avg. Social Security Benefit ($)
0
10
20
30
33
34
35
36
37
124
321
551
845
922
930
955
1011
1044
11. A private airplane leaves Midway airport and flies due east at a speed of 180 km/h. Two
hours later, a jet leaves Midway and flies due east at a speed of 900 km/h. How far from the
Midway airport will the jet be when it overtakes the private plane?
12. The Two and a Half Men and a Truck moving company charges $100 plus $30 per hour to
move a household across town. Clumsy Movers does not have a fixed fee in their price
scheme, but rather just charges $55 per hour for the same job. For what lengths of time does
it cost less to hire Clumsy Movers?
13. Graph the function f ( x ) = −x 3 + 6x 2 − 9x − 4 and find the interval(s) on which it is
increasing or decreasing and find any relative maxima or minima.
14. A rectangular piece of cardboard is to be used to make a rectangular box with open top by
cutting equal size squares out of the corners and folding up the resulting “flaps”. If the
original size of the cardboard rectangle is 3 feet by 5 feet. What must be the dimensions of
the squares to be cut out, if you wish to maximize the volume of the resulting box?
15. If f ( x ) = x 2 and g ( x ) = 2x − 1 , what is f ( g ( x )) and what is its domain?
16. A stone is thrown into a pond creating a circular ripple that spreads over the pond in such a
way that the radius is increasing at the rate of 3 ft/sec.
a) Find a function r ( t ) for the radius in terms of the time, t.
b) Find a function A ( r ) for the area of the ripple in terms of the radius, r.
c) Find ( A  r ) ( t ) . Explain the meaning of this function.
17. Determine if the function f ( x ) = 3x 2 + x is even, odd, or neither.
18. Determine if the function f ( x ) = x 10 − x 2 is even, odd, or neither.
19. Consider the graph of the following function, f. Sketch each of the following:
g ( x ) = f ( x − 1) , g ( x ) = f ( x ) + 2 , g ( x ) = 2 f ( x + 1) − 3 .
20. Write an equation for a function that is the same shape as y = x 3 , but is upside down and is
shifted 3 units to the left.
21. y varies directly with
x . If y = 1.62 when x is 1.1, what is y when x is 2.3?
22. The time t required to empty a tank varies inversely as the rate r of pumping. If a pump can
empty a tank in 31 minutes at a rate of 400 kL/min., how long will it take to empty the same
tank at the rate of 1000 kL/min.?
23. Given that the following parabola has a vertex of (1, 3) and also passes through the point
( 2, 5 ) , find the algebraic expression for the parabola.
2
f ( x ) = a ( x − h ) + k ].
[Hint: Use the form
24. A potato is launched vertically upward from the edge of a 20-meter cliff using a butane
fueled potato cannon. The height of the potato as measured from the ground below the cliff
is given by h ( t ) = −4.9t 2 + 47t + 20 . As the potato falls back to earth, it just misses the edge
of the cliff and falls to the floor of the canyon below. How long was the potato in the air?
What was the maximum height of the potato? At what time did the potato pass by the edge
of the cliff as it fell to the canyon floor?
25. To find the depth of a well, Sarah drops a stone and listens for the sound of the splash. If the
distance traveled by a falling object as a function of time is d = 4.9t 2 and if sound travels at
a rate of 335 m/sec and Sarah hears the splash 2.1 seconds after she drops the stone, how
deep is the well? [Use the fact that the sound must travel a distance equal to the depth of the
well after the stone hits. Therefore the combined time for the stone to drop to the water and
then for the sound to travel back up the well must be 2.1 seconds.]
26. A farmer is enclosing a rectangular pen for her cattle using 240 feet of fence. She is going to
use the side of the barn as one side of the rectangular pen so that she doesn’t need to use as
much fence. If she uses the 240 feet of fencing material for the other three sides of the pen,
what should the dimensions of the pen be so that she maximizes the area of the pen?
27. Recall that a parabola with vertex ( h, k ) can be written in the form f ( x ) = a ( x − h ) + k . If
2
we assume a is positive ( a > 0 ), explain why ( h, k ) must be the vertex (i.e. the lowest point
on the graph).
28. Use a graph to find the real zeros for f ( x ) = x 6 − 10x 5 + 13x 3 − 4x 2 − 5 . Sketch the graph
and explain how you obtained the zeros from the graph.
29. Given the graph below, solve the equation x 3 − 2x 2 + x − 1 = 2x − 3 . Explain how you
obtained the solution(s) from the graph.
30. Give a polynomial of lowest degree with integer coefficients that has roots x =
1
, − 3, and 5 .
2
31. Give a polynomial of lowest degree with real coefficients that has roots x = 1, − 2, and 3 + i .
32. Give a polynomial of lowest degree with integer coefficients that has roots
x = −1, 1 + 3, and 2 .
33. You are visiting the Detroit Science Center and observe the large pendulum swinging near
the entrance. Using your watch you time the pendulum and it takes 5.5 seconds to swing
back and forth once (this is called the period). If the period, T, of a pendulum of length, l, is
l
given by T ( l ) = 2π
, how long was the pendulum at the Detroit Science Center?
9.8
34. If f ( x ) =
5
2x − 7
, find f −1 ( x ) .
3x + 4
35. Consider the graph of the following one-to-one function, f. On the same axis, sketch the
graph of f −1 .
36. Consider the fact that the amount of money in your bank account at time, t, in years is given
nt
r⎞
⎛
by A ( t ) = P ⎜ 1 + ⎟ where P is the amount of money invested initially, r is the interest rate
⎝
n⎠
as a decimal (e.g. 4% would be 0.04), and n is the number times the interest is compounded
each year. If you invest $2000 at 3% interest compounded monthly, how long would it take
to have $2400 in your account? Sketch a graph of the amount in your account as a function
of time.
37. Students in a College Algebra class took a final exam and then took equivalent forms of the
exam at monthly intervals later. The average score s ( t ) , as a percent after t months was
found to be modeled by the function, s ( t ) = 78 − 15 log10 ( t + 1) where t ≥ 0 .
a) What was the average score when the students initially took the exam?
b) What was the average score after 4 months?
c) How many months will it take for the average score to drop below 50%.