Project
DESIGNING YOUR ROLLER COASTER
PROJECT
Completing this project will allow to assess your understanding of polynomials.
Your goal will be to design a roller coaster. The Roller Coaster Engineering project
is outlined in detail in the following pages. Please complete the project in
sequential order. Be sure to follow the directions for each question and show all
of your work in order to receive full credit. Write your work in the space provided
and attach extra sheets with your labeled work if needed. This project is due on I
look forward to seeing your awesome and creative designs!
Project: Designing your Roller Coaster
Congratulations! You have just been hired by Six Flags to design the next roller coaster that will
be built at Great Adventure. Before you begin your roller coaster design, however, Six Flags wants
to evaluate your knowledge of polynomial equations to make sure you are fit to design a roller
coaster of your own.
Part I – Analyzing Already Existing Roller Coasters (22points)
1. A portion of the graph of a particular quartic (degree 4)
polynomial model the path of the roller coaster Mako. The
graph of the path of Mako is shown at the right. The x-axis
represents time (in minutes) from when the ride begins, and the
y-axis represents the height (in hundreds of feet) of the roller
coaster at that given time.
a. This graph only includes positive values of x. Why is the
domain restricted to 𝑥 ≥ 0? (Hint: What do the x values
represent in the context of the problem?) Explain in a
complete sentence. (3 points)
b. Between 1 and 3 minutes the roller coasters height is negative. What does this
mean in the context of the problem? Is it possible – why or why not? Explain in a
complete sentence. (3 points)
c. If the polynomial is quartic, how many total roots must it have? How do you
know? Explain in a complete sentence. (4 points)
d. From looking at the graph of Mako, what are its roots? (3 points)
e. Describe the behavior of the graph of Mako at each root. Does the graph cross
the x-axis, or does it touch it and change direction? What does this mean about
the multiplicity of each root? Explain in complete sentences. (3 points)
f. Considering your answers to part c, d, and e, what is the multiplicity of each root
of Mako? List each zero and its corresponding multiplicity. (3 points)
g. Using the roots and their corresponding multiplicities from part f, write the
polynomial equation in factored form that models the path of Mako. (3 points)
P(x) =
2. (18 points) Another roller coaster, Phoenix, is modeled by a portion of the polynomial
equation
𝑷(𝒙) = 𝒙𝟓 − 𝟏𝟏𝒙𝟒 + 𝟒𝟓𝒙𝟑 − 𝟖𝟒𝒙𝟐 + 𝟔𝟖𝒙 − 𝟏𝟔
Once again, x values represent the time (in minutes) from when the ride begins, and the
y values represent the height (in hundreds of feet) of the roller coaster at that given time.
a. List all possible rational roots of this polynomial. (5 points)
b. Determine the actual rational roots by graphing the polynomial on your calculator.
What are their multiplicities based on the behavior of the graph? (5 points)
c. The park owners want to place a camera along the path of Phoenix where it enters
the underground tunnel (a little after 2 minutes on the ride). They want to take
pictures of people as they are enjoying the ride so they can make a profit by selling
them. In order for the camera to flash at the correct moment, you need to
determine the exact time the roller coaster will enter the tunnel. At what time
should the camera be programmed to flash? Show all your work. Round your
answer to the nearest thousandth. (Hint: You will need to use synthetic division).
(8 points)
3. (20 points) Another roller coaster, Montu, is modeled by the polynomial equation
𝑷(𝒙) = (𝟏. 𝟔𝟕𝟗𝟏 ∙ 𝟏𝟎−𝟏𝟒𝒙𝟐 − 𝟔. 𝟓𝟕𝟔 ∙ 𝟏𝟎−𝟏𝟐𝒙 + 𝟔. 𝟑𝟒𝟎𝟓 ∙ 𝟏𝟎−𝟏𝟎)
(𝒙 − 𝟕𝟎. 𝟒𝟎𝟎𝟕)𝟐(𝒙 − 𝟐𝟗𝟎. 𝟖𝟖𝟐)𝟐(𝒙 + 𝟑. 𝟎𝟕𝟔𝟏𝟖)𝟐
a. What are the roots of this polynomial? (5 points)
b. What is the highest point of Montu? (5 points)
c. What is the degree of this polynomial function? (5 points)
d. What is the end-behavior of this polynomial? Imagine that you want to describe
the movement of this roller coaster during a long time. Does it make sense? (5
points)
Part II – Designing Your Own Roller Coaster (40 points)
Good work on Part I of this project – you have been hired to engineer a new roller coaster!
Follow the directions below to design your own coaster and analyze its properties.
1. The polynomial representing the path of your roller coaster must meet the following
specifications:
a. Have integer coefficients (no decimals or fractions)
b. Be of degree 3 (or higher)
c. Have at least one rational, real root
d. Go underground through a tunnel at least once
2. Use desmos.com and the slider features to play around with the graphs of polynomials
and choose a portion of the graph of a polynomial you like that fits the criteria above.
3. What is your polynomial? Write the equation below. (5 points)
P(x) =
4. Determine when your roller coaster enters the tunnel(s) and exits the tunnel(s). (5
points)
5. Use the graph of your roller coaster drawn above and Desmos to answer the following
questions.
a. What are the relative minimum(s) and maximum(s) of your coaster? (5 points)
Rel Min(s): Rel Max(s):
b. When is the polynomial that represents your roller coaster increasing? Write this
using inequalities or interval notation. (5 points)
c. When is the polynomial that represents your roller coaster decreasing? Write
this using inequalities or interval notation. (5 points)
6. At what time(s) is your roller coaster at ground level? Can you find all the exact values
by hand? To see, break down your polynomial as much as possible by hand to find the
exact values of the roots. Estimate any remaining roots using Desmos. (5 points)
7. Do all roots of the polynomial make sense in regard to this problem? Why or why not?
Explain in a complete sentence. (5 points).
8. Draw your roller coaster on a sheet or graph paper OR print out your graph from
Desmos. Think of a creative name for your roller coaster. Place a dot and label the
important points of your roller coaster. (5 points)