Question

Ray and Kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. Ray says the third-degree polynomial has four intercepts. Kelsey argues the function can have as many as three zeros only. Is there a way for the both of them to be correct? Explain your answer.

Kelsey has a list of possible functions. Pick one of the g(x) functions below and then describe to Kelsey the key features of g(x), including the end behavior, y-intercept, and zeros.
g(x) = (x + 2)(x − 1)(x − 2)
g(x) = (x + 3)(x + 2)(x − 3)
g(x) = (x + 2)(x − 2)(x − 3)
g(x) = (x + 5)(x + 2)(x − 5)
g(x) = (x + 7)(x + 1)(x − 1)

Create a graph of the polynomial function you selected from Question 2.

Part B

The second part of the new coaster is a parabola.

Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = (x − a)(x − b). Describe the direction of the parabola and determine the y-intercept and zeros.

Create a graph of the polynomial function you created in Question 4.

Part C

Now that the curve pieces are determined, use those pieces as sections of a complete coaster. By hand or by using a drawing program, sketch a design of Ray and Kelsey's coaster that includes the shape of the g(x) and f(x) functions that you chose in the Parts A and B. You do not have to include the coordinate plane. You may arrange the functions in any order you choose, but label each section of the graph with the corresponding function for your instructor to view.

Part D

Create an ad campaign to promote Ray and Kelsey's roller coaster. It can be a 15-second advertisement for television or radio, an interview for a magazine or news report, or a song, poem, or slideshow presentation for a company. These are just examples; you are not limited to how you prepare your advertisement, so be creative. Make sure to include a script of what each of you will say if you are preparing an interview or a report. The purpose of this ad is to get everyone excited about the roller coaster.

Answer 1

Part A) an n-degree polynomial can have, at most, n zeros

So a third-degree polynomial can only have a max of 3 zeros (Kelsey is right)

However, Ray can also be right. He says a max of 4 intercepts. If we include one y-intercept, then yes, the function can have four intercepts.

Both can be right.

Answer 2

For the functions, just pick one and graph it. You can use graphing software like desmos to make the job easier.

End behavior - do the ends of the function go to positive or negative infinity?

Zeroes: when does the function equal 0? What are the x-values?

Y-intercept: what is the value of the function when x = 0?

Answer 3

For part B) you’re making a brand new parabola by picking some a and b values (just pick some number to be a and some other number to be b) and fill them into the equation. Now, graph your parabola again. Does it open upwards? (Shaped like a U) or downwards (shaped like an upside down U). Find the y-intercepts and zeros using the same method you used in part A.

Part C) take the two graphs and put them together somehow to make a roller coaster path. You can include sections besides the parabolas, like a flat area, or a slope straight up, or down, etc. experiment/play around with the design, have some fun with it. Make sure to graph the design by hand or with software like desmos.

Part D) is more of a creative exercise, have fun and talk about the features of the roller coaster like the parabolas, etc.