M 093 and M 117A - Quadratic Functions – Study Guide
y  ax 2  bx  c
Question-types:
1. Does the parabola open up or down? How do you know?
2. Does the parabola have a maximum or a minimum? How do you know?
3. Use algebra to find the vertex and interpret in context
4. Find the maximum/minimum value of the function and the value of x where it happens
5. Use the quadratic formula to solve quadratic equations
6. Use quadratic regression to find the model equation for a data set
7. Find the y-intercept of the model and interpret in context with correct units.
8. Find the x-intercept of the model and interpret in context with correct units
9. For a given x, find the y value
10. For a given y, find the x-value
11. Evaluate F(#) and interpret in context
12. Find x when F(x) = #, and interpret in context
13. Interpret in context any point (a, b) on the graph of the function
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Math 093 and 117 A - Quadratic Functions – Applications
1) A batter hits a baseball into the air. The height h in feet of the baseball after t seconds is given by the
quadratic function: ℎ = −16𝑡 2 + 64𝑡 + 3.
a) Complete the table of values, plot the ordered pairs and sketch the graph.
t
h
0
1
2
3
4
5
b)
c)
d)
e)
f)
What is the initial height of the ball?
How high is the ball after 1.5 second?
What is the maximum height of the ball?
When did it reach the maximum height?
When is the ball at a height of 63 feet?
I.
Solve graphically
II.
Solve algebraically
g) When will the ball hit the ground?
h) When is the ball at a height of 80 feet?
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2) A manufacturer determines that the profit in dollars for manufacturing x units is
P( x)  2 x 2  60 x  400
a) Enter the function in the Y = of your calculator and explore the table.
b) Scroll down on the table until you see that the profit values are positive.
c) Show some of those ordered pairs on the table.
X
P(x)
d) Plot the points on the grid and sketch the
graph.
e) Label axes in context.
f) Show how you find the value of P when x =
12.
g) For what value of x is P the largest?
h) What is the largest value of P?
i) Interpret in context the maximum point of the
graph.
j) For what values of x is P = 0?
k) Interpret in context the x-intercepts of the
graph.
l) What is P(13)? Interpret in context.
m) What is the p-intercept of the model? Interpret
in context.
n) What is x when P = 42? Interpret in context.
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3) A manufacturer determines that the profit in dollars for manufacturing n units is
𝑃(𝑛) = −2𝑛2 + 60𝑛 − 40 . Solve P(n) = 400 and interpret in context.
4) The following table shows the median number of hours of leisure time per week for Americans in
various years.
Year
1973 1980 1987 1993 1997
Median Number of Leisure Hours per Week
26.2 19.2 16.6 18.8 19.5
Let t represent the year 1973.
a) Use the calculator to sketch a scatterdiagram for the data.
b) Find the best model equation that fits the data; round to three decimal places.
c) Use algebra to determine the year in which the median amount of leisure time was the lowest.
d) What was the lowest median number of leisure hours per week?
5) An arrow is fired upward into the air with an initial velocity of 80 feet per second. The height in feet of
the arrow t seconds after it was shot into the air is given by the function
h(t )  16t 2  80t  96
a) What is the h-intercept? Interpret in context with correct units.
b) Find h(3). Interpret in context with correct units.
c) Solve h(t) = 160. Interpret in context with correct units.
(1) Solve this using a graphical approach – show graph with labels and indicate the answer in the graph
(2) Solve algebraically
d) Find the t-intercepts. Interpret in context with correct units.
e) Use algebra to find the coordinates of the vertex. Interpret in context with correct units. .
f) Use algebra to determine when will the arrow reach a height of 200 ft?
g) The point of coordinates (1.5, 180) is on the graph of this function. Interpret the coordinates in context
with correct units.
h) Use your knowledge to graph. Label axes in context. Label important points in the graph.
6) The quadratic function
models the median, or average, age, y, at which U.S.
men were first married x years after 1900.
a) In which year was this average age at a minimum? (Round to the nearest year.)
b) What was the lowest average age at first marriage? (Round to the nearest tenth.)
c) Use your knowledge to sketch graph; label axes in context.
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7) Not all murder cases are solved. The percentages of murder cases solved in various years are listed in
the table below.
(Source: Bureau of Justice Statistics)
Let f(t) represent the percent of murder cases solved at t years since 1980.
a) Sketch a scatter-diagram. Which model fits the data best; linear, exponential of quadratic?
b) Find the model equation for the data; round to three decimal places.
c) Show a rough sketch of the function; label axes in context.
d) You should be familiar with the typical questions for this type of function. Make up at least four
different types of questions and answer them showing all algebraic work. Questions should be
phrased within context.
8) A company started business in 1995. Assume that a company has a profit per
year shown in the table below. (Negative profits are losses.) Let the model
𝑃(𝑡) = −0.19𝑡 2 + 2.56𝑡 − 2.38 represent the amount of profit in millions of
dollars, where t is the number of years since 1995.
a) Find P(8). Write the meaning of your result in a complete sentence using correct
units.. b) Find t when 𝑃(𝑡) = 4.23. Write the meaning of your result in a
complete sentence using correct units. c) Find 𝑡 when 𝑃(𝑡) = 0. Write the
meaning of your result in a complete sentence using correct units. d) In which year
will the company have a maximum profit? e) Use the regression feature on your
calculator to determine the quadratic model equation for this data and compare the
calculator model to the given model.
Year
Profit
millions of dollars
1995
-2.4
1996
0
1997
2.0
1998
3.6
1999
4.8
2000
5.7
2001
6.2
9) A person standing on top of a building throws a stone into the air so that it hits the ground at the base of
the building. The stone’s height in feet after t seconds is given by the equation: h  16t 2  30t  200 .
a) Find the t-intercepts and interpret in context.
b) Find the h-intercept and interpret in context.
c) Find the vertex and interpret in context.
d) Use your knowledge to graph; label axes in context and important points on the graph.
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10) The sales for a gaming console for various years are listed in the table below.
Let f(t) represent the sales (in billions of dollars) at t years since 1990. A reasonable model is
a)
b)
c)
d)
Use quadratic regression to check this model.
When will the sales be 1.65 billion dollars?
According to the model, when were sales at a minimum?
What were the minimum sales?
11) John owns a hotdog stand. He has found that his profit is represented by the equation
with P being the profit in dollars, and x the number of hotdogs sold.
a) How many hotdogs must he sell to earn the most profit?
b) What is the largest profit?
c) How many hotdogs must he sell to have a profit of 571 dollars?
12) A projectile is thrown upward so that its distance above the ground after t seconds is
where h is the height in feet.
a) When will it hit the ground?
b) After how many seconds does it reach its maximum height?
c) What is the maximum height?
13) Southern Granite and Marble sells granite and marble by the square yard. One of its granite patterns is
price sensitive. If the price is too low, customers perceive that it has less quality. If the price is too high,
customers perceive that it is overpriced. Let B(x) be the number of buyers at a price of x dollars per
square yard. The number of buyers can be modeled by the quadratic model
a) Find B(40) and interpret in context.
b) Use the model to estimate the price which yields the largest number of buyers.
c) What is the largest number of buyers?
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14) After the semester is over, you discover that the math department has changed textbooks (again) so the
bookstore won't buy back your nearly-new book. You and your friend Herman decide to get creative.
You go to the roof of a twelve-story building and look over the edge to the reflecting pool 160feet
below. You drop your book over the edge at the same instant that Hermanchucks his book straight down
at 48feet per second. By how many seconds does his book beat yours into the water?
Our initial launch heights will be the same: we're both launching from 160feet above ground. And the
gravity number, since we're working in feet, will be 16. My initial velocity is zero, since I just dropped
my book, but my buddy Herman's velocity is a negative 48, the negative coming from the fact that he
chucked his book down rather than up. So our "height" equations are:
yours: s(t) = —16t2 + 160
his: s(t) = —16t2 — 48t + 160
Now you can answer the question: By how many seconds does his book beat yours into the water?
15) The total lengths of fiber-optic cable that telecom companies have for various years are shown in the
table.
Total Length of Fiber Optic Cable that Telcom Companies Have
Year
Total length of cable (millions of miles)
1996
21
1997
24
1998
31
1999
42
2000
60
2001
72
Let d  Q (t ) be the total length of cable (in millions of miles) that the telecom companies have at t
years since 1996.
a. Use the graphing calculator to draw the scatter diagram. Show a rough sketch of the scatterdiagram in your paper and label axes as usual.
b. What model fits the data well; linear, exponential or quadratic? Name all that apply.
c. Use the regression feature of the calculator to produce the model equation for the data; round to
three decimal places. If more than one model applies, find all of them rounding constants to three
decimal places.
d. For each model, solve Q(t )  90 . Show algebraic work. What does it mean in this situation?
e. For each model, solve Q(t )  200 . Show graphical approach. What does it mean in this
situation? Expand the window values to go beyond the year 2003. Which model grows faster
after 2003?
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SOME ANSWERS
1-Answers: b) 3 feet, (c) 63 ft., (d) 67 ft., (e) 2 sec., (f) at 1.5 sec. and 2.5 sec. (g) 4.05 sec. (h) never – (nonreal solutions)
3-Answers: n = 12.8, 17.2. We need to sell 13 units or 17 units for the profit to reach about $400.
4- Answers: b) 𝑦 = 0.040𝑥 2 − 1.215𝑥 + 26.026; © 1988; (d) 16.8 hours per week
5-Answers: a) 96 ft., (b) 192 ft., (c) 1 and 4 sec. The arrow is at a height of 160 feet twice; one second after
fired (on its way up) and 4 seconds after fired, on its way down. (d) -1 and 6. The arrow hits the ground after 6
seconds. t = -1 does not make sense. (e) (2.5, 196) the arrow reaches its maximum height of 196 feet after 2.5
seconds. (f) Never – non-real solutions. (g) 1.5 second after the arrow was fired upwards the arrow is at a height
of 180 ft.
6-Answers: a) 1951, (b) 26.1 ys-old
7-Answers: a) quadratic; b) 𝑓(𝑡) = 0.201𝑡 2 − 5.309𝑡 + 99.721; c) (i) In what year was the percent of solved
murder cases the minimum? t = 13.2; year 1993;(ii) What was the lowest percentage of solved murder cases?
64.7%; (iii) What was the percent of solved murder cases in the year 2000? 73.9%; (iv) When will 100% of
murder cases be solved? in 2006 (model breakdown)
8-Answers: a) 𝑃(8) = 5.94; In 2003, the profit was $5.94 million. b) 𝑡 ≈ 3, 10; In 1998 and 2005, the profit is
$4.23 million. c) 𝑡 = 1, 12; In 1996 and 2007, the company had 0 profit. d) vertex: (6.74, 6.24); A maximum
profit of 6.24 million was in the year 2002.
9-Answers: a) 𝑡 ≈ −2.7, 4.6; The ball will hit the ground in 4.6 seconds. -2.7 has no meaning in this problem.
b) ℎ = 200; The ball is at 200 feet when it is thrown. c) 𝑡 = 0.9375 and ℎ(0.9375) = 214.0625, vertex:
(0.9375, 214.0625), The ball reaches a MAXIMUM height of 214.0625 feet after 0.9375 seconds.
10-Answer: b) In the year 2000; c) 1995; d) $172 million
11-Answer: a) 30 hotdogs, b) $971; c) 10 or 50 hotdogs
12-Answer: a) 17.5 seconds, b) 8.75 sec ; c) 1225 ft;
13-Answer: a) At $40 per square yard we expect 64 buyers; b) $64 per square yard; c) 78 buyers.
14-Answer: 1.16 seconds
15-Answers: c) quadratic: 𝑄(𝑡) = 1.589𝑡 2 + 2.739𝑡 + 20.25; exponential: 𝐸(𝑡) = 19.481(1.301𝑡 ; (d) Around
2002 with both models; (e) In 2005 with the exponential model; in 2006 with the quadratic model. The
exponential model grows faster.
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