Math 2 Honors
Lesson 2-2: Rollercoaster Quadratics
Name______________________________
Date _____________________________
Learning Goals:
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I can predict whether a quadratic will have a minimum or a maximum based on the value of a.
I can use the problem situation to explain where and why the quadratic is increasing or decreasing.
I can define “increasing interval” as a set of function inputs for which the output increases as the input increases and
“decreasing interval” as a set of function inputs for which the output decreases as the input increases.
I can define the maximum as a place where the quadratic transitions from increasing to decreasing and the minimum as
a place where the quadratic transitions from decreasing to increasing.
I can identify a parabola’s reflective symmetry in a table or graph (extend to a problem situation).
I can identify a quadratic expression, ax2+bx+c.
I can identify the increasing interval and the decreasing intervals of a table or graph.
If you love rollercoasters, thank parabolas! Parabolic hills provide great airtime. Floating over
maximums and dropping down into the minimums are what coaster enthusiasts crave! Below is a
parabolic hill from the Raging Bull coaster located at Six Flags, Illinois.
The following equation represents the hill:
Left support
column
y  .014 x 2  2.8x  160
Right support
column
1. a. The support column furthest to the left is ________ feet tall. The ordered pair for this point is
____________and its mathematical name is ________________________. Describe where this is
located in the equation of the parabola:
b. The shortest distance from the track to the ground is ________ feet and it occurs __________ feet
from the base of each support column. The ordered pair for this point is ______________ and the
mathematical name is _____________________.
c. As the distance from the left support column increases, the height of the track
___________________ at a(n) ___________________ rate, until you reach the ________________
point of (_______,_______). Then the height of the track will begin to ___________________ at
a(n) ___________________ rate.
d. What x-value goes with the second occurrence of a height of 160 feet? How did you determine that
value?
2. Graph the function of the roller coaster on the calculator. y  .014 x  2.8x  160
a. Use the x-values to describe the intervals of the track that are at least 100 feet above the ground.
2
b. Use the x-values to describe the intervals of the track that is at most 50 feet above the ground.
c. Below is a table of values representing some ordered pairs that make up the parabola.
x (feet)
0
40
60
80
100
140
160
180
200
y (feet)
160
109.6
42.4
25.6
20
25.6
70.4
109.6
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Underline the input/output pair of numbers that represent the y-intercept of the function.
From the table, how do you know this is the y-intercept?
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Circle the input/output pair of numbers that represent the minimum of the function. From
the table, how do you know this is the minimum?
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There are some values in the table that have not been filled in. Use the symmetric nature of
the parabola to fill in each of those values.
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What is happening to the rates of change as the values approach the minimum? How do you
know from the table?
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What is happening to the rates of change after the minimum? How do you know from the
table?
***Verify your intervals from 2a & 2b by making a table in the calculator. Also, confirm the values you
filled in the table
To make a table: /  T  b  2Table  5 Edit Table Settings . . .
3. Recall that the height function for dropping pumpkins was h(t ) 144 16t 2 and for a basketball long
shot was h(t )  8  40.8t 16t 2 .
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Describe the similarities between the graphs of the rollercoaster hill and those of the
pumpkin drop and basketball functions.
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Describe the differences between the graphs of the rollercoaster hill and those of the
pumpkin drop and basketball functions.
***How is the rule for the rollercoaster hill function similar to and how is it different from the rules of
the pumpkin drop and basketball functions?
In 1996, the first Tibetan Freedom Concert, regarded by many as the single greatest cultural event in
modern rock history, took place in Golden Gate Park in San Francisco. This was the first in a series of
benefit concerts organized by the Milarepa Fund to raise awareness about nonviolence and the Tibetan
struggle for freedom, as well as to encourage youth activism. The primary goal for the Tibetan Freedom
Concerts was to raise awareness, not money. However, careful planning was needed to ensure that the event
would reach a large audience and that it would not lose money. The profit from any event will be the
difference between income and operating expenses.
4. As organizers planned for the event, they had many variables to consider:
a. What factors will affect the number of tickets sold for the event?
b. What kinds of expenses will reduce profit from tickets sales, and how will those expenses depend on
the number of people who buy tickets and attend?
5. Suppose that a group of students decided to organize a local concert to raise awareness and funds for the
Tibetan struggle, and that planning for the concert led to this information:
 The relationship between number of tickets sold s and ticket price x in dollars can be
approximated by the linear function s( x )  4000  250 x.
 Expenses for promoting and operation of the concert will include $1,000 for advertising,
$3,000 for pavilion rental, $1,500 for security, and $2,000 for catering and event T-shirts
for volunteer staff and band members.
a. Make a table and graph of s ( x ).
x
0
8
12
14
16
s(x)
b. If the event organizers give away free tickets for the concert, how many tickets can be distributed?
______________
c. Where is this value located on the graph?__________________________________________
d. Where is this value located in the table?___________________________________________
e. Where is this value located in the function rule?_____________________________________
f. Describe the graph for the ticket sales: As the price of the tickets increases the number of tickets
sold ______________ at a(n) ________________ rate.
g. Find a function that can be used to predict income I for any ticket price x.
Income  (ticket price)  (# of tickets sold ) . Substitute expressions into this equation.
h. Find a function that can be used to predict profit P for any ticket price x.
Profit = Income – Expenses. Substitute expressions into this equation.
To answer parts (i) and (j), make both a graph and a table for the functions in parts (g) and (h)
i. How do predicted income and profit change as the concert organizers consider ticket prices ranging
from $1 to $20? How are those patterns of change shown in graphs of the income and profit
functions?
As the ticket prices range from $1 to $20, both of the graphs are _________________ at a(n)
________________ rate, until the ____________ is reached, then both graphs are _______________
at a(n) _________________ rate.
Examine the y values in the income and profit tables and describe any patterns that may exist.
j. What ticket price(s) seem likely to give maximum income and maximum profit for the concert?
What are those maximum income and profit values? How many tickets will be sold at the price(s)
that maximize income and profit?
The price of _______ gives both the maximum profit and maximum income. The maximum
income would be _______________, while the maximum profit would be _______________.
This profit and income would result from ______________ tickets being sold.
6. The break-even point is the ticket price for which the event’s income will equal expenses. Another
way to think of the break-even point is the ticket price when profit is $0.
a. Use the equation from (5h) and your calculator to write an equation that can be used to find the
break-even ticket price(s) for this particular planned concert.
b. Use the equation from (5h) and your calculator to find an inequality that represents the following
situations:
 Write an inequality that can be used to find ticket prices for which the planned concert will make
a positive profit.
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Write an inequality for the time period when profit is increasing.
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Write an inequality for the time period when profit is decreasing.
c. Use the equation from (5h) and your calculator to write an inequality that can be used to find ticket
prices for which the planned concert will lose money?
7. The rule for ticket sales is _________________________
The rule for income is ____________________________
The rule for profit is _____________________________
a. The function rule for ticket sales is a _______________________ function, while the function rule
for income is a _______________________ function, and the function rule for profit is
a___________________ function.
b. What similarities and differences do you see in the rules of all three functions?
c. What similarities and differences do you see in the graphs of all three functions?
d. What similarities and differences do you see in the tables of all three functions?
PRACTICE
1. Below is a picture of the third hill (taller one) of the Intimidator rollercoaster at Carowinds amusement
park, located in North Carolina.
Suppose the following equation represents the graph of
the hill, where x represents the horizontal distance in
feet and y represents the vertical distance in feet.
y   .00832 x 2  2.08 x  20
a. Complete the table of values and then graph the function.
x (feet)
0
25
50
75
100
125
150
175
200
225
250
y (feet)
b. What are the coordinates of the y-intercept? _________ How do you know from the table? From the
equation?
c. What are the coordinates of the point where the tallest support column meets the track? __________
What is the mathematical name of this point? ____________________________
d. Use a compound inequality to describe the x-values where the track is at least 140 feet above the
ground.
e. Use compound inequalities to describe the x-values where the track is at most 85 feet above the
ground.
2. Answer the following questions about the equation:
y   2 x 2  5x  20
***NO CALCULATOR***
a. Is this function linear or non-linear? How do you know?
b. Does this parabola have a maximum or a minimum (which one)? How do you know?
c. What is the y-intecept of this function? How do you know?
d. Complete the table of values.
x
y
0
.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-12
-6.5
1.5
5.5
6
4
1.5
-2
e. Underline the input/output pair of numbers that represent the y-intercept of the function.
f. Circle the input/output pair of numbers that represent the maximum of the function.
g. Describe what is happening to the rates of change as the values approach the maximum.
h. Describe what is happening to the rates of change of the values after maximum.