QCC MA-119 Final Review Questions.
HSI QCC MA-119
Spring 2021 Final Review Questions
Page 1 of 12
Graphs and Functions
1. Use the graph of the function, ? = ?(?), to answer the question.
a) Estimate ?(1)
Answer: ?(1) ≈ 5 (find a point on the graph that has ?-coordinate = 1).
) Estimate the solution(s) to ?(?) = 9
Answer: ? ≈ 1.75 (find all points on the graph that have ?-coordinate=9; in this case,
there's just one such point).
c) Estimate the coordinates of all intercepts.
Answer: ?-intercept is approximately(−8.5,0). (All ?-intercepts have ?-coordinate=
0.) There’s only one ?-intercept in this graph; ?-intercept is approximately(0,3). (All ?-
intercepts have ?-coordinate= 0. There’s only one y-intercept in this graph.)
d) Estimate the domain and range of the function
Answer: Domain: (−∞, ∞). (The a
ows indicate that the graph extends, without
eaks, indefinitely to the left and indefinitely to the right.) Range: (−∞, ∞). (The
a
ows indicate the graph extends, without
eaks, indefinitely up and indefinitely
down.)
e) Estimate the minimum or maximum ?-value
Answer: no minimum ?-value and no maximum ?-value. (The a
ows indicate that the
graph extends indefinitely up and indefinitely down.)
f) Estimate the ?-value at which the minimum or maximum is achieved?
Answer: no such ?-values because of the answer to part e).
g) Estimate the values of ? for which ? < 0
Answer: (−∞, 8.5) For these ?-values the graph is below the ?-axis so for these ?-
values, ?<0. See how this answer relates to part c)
h) When 1 < ? < 2, is the function increasing or decreasing?
Answer: Increasing. Move to the right within the stated interval of ?-values. The graph
is going upwards for those ?-values.
2. Use the graph of the function, ? = ?(?), to answer the question.
a) Estimate ?(−11)
Answer: ? = ?(−11) ≈ 7
) Estimate the solution(s) to ?(?) = −4
Answer: ? ≈ −13 and ? ≈ −6
c) Estimate the coordinates of all intercepts.
HSI QCC MA-119
Spring 2021 Final Review Questions
Page 2 of 12
Answer: ?-intercepts: (−14,0), (0,0), (9,0); ?-intercept: (0,0)
d) Estimate the domain and range of the function
Answer: Domain (−∞, ∞) Range [−7.75, ∞)
e) Estimate the minimum or maximum ?-value
Answer: minimum ?-value ≈ −7.75; no maximum ?-value
f) Estimate the x-value at which the minimum or maximum is achieved?
Answer: For minimum: ? ≈ −10 (The ?-coordinate of the point with the minimum ?-
value); For maximum: no such ?-value
g) Estimate the values of ? for which ? < 0
Answer: (−14,0) ∪ (0,9) (the symbol ∪ stands for union or “together with”. The
endpoints are not included because there we have ? = 0 not ? < 0
h) When 4 < ? < 5, is the function increasing or decreasing?
Answer: decreasing
3. Compute
?(4)−?(1)
4−1
for the given function.
a) ?(?) = ?2 − 3? + 1
Answer: 2
) ?(?) = √?
Answer:
1
3
Linear Functions
4. Suppose that a manufacturer wishes to use a linear function to model how the demand for its
product affects the product’s price. The unit price function has the form ?(?) = ?? + ? where
? is the number of units of the product in demand. Suppose that when 2400 units are
demanded then the unit price is $250 and when 3000 units are demanded then the unit price
is $300.
Number of units ? Price per Unit ?
XXXXXXXXXX
XXXXXXXXXX
a) Determine the equation of the function ?(?) = ?? + ?
b) According to this model, what is the unit price ?(?) if the demand is ? = 3600?
HSI QCC MA-119
Spring 2021 Final Review Questions
Page 3 of 12
Answer: a) ?(?) =
1
12
? + 50 b) $350
5. A linear function ? passes through (−1,5) and (2, −4). Find the function value ?(−2).
Answer: ?(−2) = 8
6. Markos cu
ently has 200 songs in his music collection. Every month, he adds 15 new songs.
Write a formula for the number of songs, ?, in his collection as a function of time, ?, the
number of months. How many songs will he own in a year?
Answer: ?(?) = 15? + 200; 380 songs
7. A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009
the population had grown to 8,100. Assume this trend continues.
a) Write a linear equation to model the population growth.
Answer: ?(?) = 380? + 6200, where ? is the number of years since 2004.
) Predict the population in 2013.
Answer: 9,620
c) Identify the year in which the population will reach 15,000.
Answer: Sometime around the year 2027
Quadratic Equations (Factoring and Quadratic Formula)
8. Factor completely. 40?4 + 22?3 − 6?2
Answer: 2?2(4? + 3)(5? − 1)
9. Solve the quadratic equations
a) 4?2 + ? = 3
Answer: ? = −1 or ? = −
3
4
) (? + 1)(? − 3) = 2
Answer: ? = 1 ± √6
c) ?2 + 7? + 12 = 0
Answer: ? = −3 or ? = −4
d) 2?2 + 1 = ?
Answer: ? =
1±?√?
4
e) 2(? XXXXXXXXXX = 0
Answer: ? = −3 ± 2?
f)
1
3
?2 + ? = −
1
2
Answer: ? =
−3±√3
2
g) 4?2 − ? − 3 = 0
Answer: ? = 1 or ? = −
3
4
h) (? −
1
3
)
2
−
4
9
= 0
Answer: ? = 1 or ? = −
1
3
HSI QCC MA-119
Spring 2021 Final Review Questions
Page 4 of 12
Quadratic Functions
10. Graph the following quadratic functions.
a) ?(?) = −?2 − 6? − 5
) ℎ(?) = −(? − 3)(? + 2)
c) ?(?) = 2?2 − 3? − 2
d) ?(?) = −2?2 − 4? + 5
e) ?(?) = (2? − 1)(? + 3)
f) ?(?) = 2?2 + 8
Before graphing the parabolas, state which open upwards and which open downwards.
For each parabola include
a) The coordinates of all intercepts
) The coordinates of the vertex
c) The equation and graph of the axis of symmetry
d) The domain and range in interval notation
e) The coordinates of an additional point on the graph
f) The maximum or minimum range value
g) The domain value at which the max or min is reached
h) ?-interval(s) at which the function is negative
i) ?-interval(s) at which the function is positive
Application of Quadratic Equations/Functions
11. The length of a rectangle exceeds twice its width by 3 inches. If the area is 10 square inches,
find the rectangle's dimensions. Round to the nearest tenth of an inch.
Answer: Approximately 1.6 in by 6.2 in.
HSI QCC MA-119
Spring 2021 Final Review Questions
Page 5 of 12
12. The hypotenuse of a right triangle is 6 feet long. One leg is 2 feet shorter than the other. Find
the lengths of the legs. Round to the nearest tenth of a foot.
Answer: Approximately 5.1 ft and 3.1 ft.
13. A retailer estimates that, by charging x dollars each for a particular phone case, she can sell
40 − ? units each week. The function ?(?) = ?(40 − ?), models the weekly revenue, ?(?),
eceived when the selling price is ?.
a) Interpret (0, ?(0)) and (1, ?(1)) in context.
Answer: When the charge is $0, the revenue is $0; When the charge is $1, the revenue
is $39
) Find the selling price that will give the maximum revenue, and then find the amount of
the maximum revenue.
Answer: The maximum revenue is $400 when the charge is $20
c) For which selling price(s) will the weekly revenue equal $300?
Answer: $10 and $30
14. A ball is thrown vertically upward, at the rate of 120 ft/sec, from a rooftop with a height of 50
feet. The quadratic function ℎ(?) = −16?2 + 120? + 50 models the height of the ball, from the
ground, at time ?.
a) Interpret (0, ℎ(0)) and (1, ℎ(2)) in context.
Answer: 0 seconds after the ball is thrown the ball is 50 feet from the ground; 1
second after the ball is thrown the ball is 154 feet from the ground.
) Find how long it will take the ball to reach its maximum height, and then find the
maximum height.
Answer: The maximum height of 275 feet is reached in 3.75 seconds
c) At which time(s) does the ball reach 200 feet above ground?
Answer: 1.58 and 5.92 seconds
d) When does the ball hit the ground?
Answer: After 7.896 seconds
Rational and Radical Functions
15. Find the domain of the following functions.
a) ?(?) =
?
?−3
− 2
Answer: (−∞, 3) ∪ (3, ∞)
HSI QCC MA-119
Spring 2021 Final Review Questions
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) ?(?) = 1 − √3? − 6
Answer: [2, ∞)
16. Given functions ?(?) = √? − 3, find:
a) ?(7) − 2
Answer: 0
) the domain of the function ? which is define as ?(?) = ?(?) −