G(x) = x3 − x2 − 4x + 4



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  1. 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 4 intercepts. Kelsey argues the function can have as many as 3 zeros only. Is there a way for the both of them to be correct? Explain your answer.

Yes, they both can be correct. Ray is correct because you can have 4 intercepts. Only 3 can be zeros and 1 can be the Y-Intercept.


  1. 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) = x3 − x2 − 4x + 4

    • g(x) = x3 + 2x2 − 9x − 18

    • g(x) = x3 − 3x2 − 4x + 12

    • g(x) = x3 + 2x2 − 25x − 50

    • g(x) = 2x3 + 14x2 − 2x − 14



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


The zeros of function g(x)= x^3-x^2-4x+4 is -2,2, and 1. The other key features of polynomial functions include end behavior, Y- intercept and the axis of symmetry, and the vertex.

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) = ax2 + bx + c. Describe the direction of the parabola and determine the y-intercept and zeros.

The direction of the parabola would be both sides going up because they are the leading coefficient and the degree is both even and positive numbers. The zeros are 1.5 and -2. The Y-intercept is 6.

The safety inspector notes that Ray also needs to plan for a vertical ladder through the center of the coaster's parabolic shape for access to the coaster to perform safety repairs. Find the vertex and the equation for the axis of symmetry of the parabola, showing your work, so Ray can include it in his coaster plan. The axis of symmetry is -1.75 and the vertex is 1.75 and 12.37



X= B/2A

X=7/2(2)


X= 1.75

2x^2+7X+6

2(1.75)^2+7(1.75)+6

2(-3.06)-12.25+6

-6.12-12.25+6

=12.37
Create a graph of the polynomial function you created in Question 4.

Part C


  1. 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

  1. 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.

*Reporter: So I hear you guys have this amazing roller-coaster that will be ready soon? Tell us all about it.

*Kelsey: Yes, We are so excited! We really took our time to make sure we build the best roller coaster ever and to see that Ray made sure everything is ready it's so wonderful and I know everyone will love it.



*Ray: Yea Kelsey is right we certainly took our time with it, It's gonna be the most epic ride yet! so everyone tell your friends!

*Reporter: Well we all can't wait to see it and take a ride!

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