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Congratulations! You and your partner have summer internships at an engineering firm. As part of your internship, you get to assist in the planning of a brand new roller coaster. For this assignment, you and your partner tackle the math behind some simple curves in the coaster's track and create a new ride for theme park patrons.
Part A (Complete Collaboratively)
Read the following statements with your partner:
A third-degree polynomial has four intercepts.
A third-degree function can have as many as three zeros only.

Is there a way for both statements to be correct? Explain your answer. Regardless of whether you and your partner agree, you must each submit an answer to this question in your own words.
Part B (Complete Individually)
The first part of your roller coaster is a curved pattern that can be represented by a polynomial function.
You are given has a list of possible functions. You need to pick one of the g(x) functions below, show how to find the zeros, and then describe the other key features of g(x). Each partner must choose a different function, so be sure to discuss with your partner which one you plan to select.
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 that you selected in Question 2.

The second part of the new roller coaster is a parabola.
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 (Complete with Your Partner)
After sketching portions of the roller coaster (see below), you and your partner need to collaborate and choose the four functions you have sketched that can best be assembled to create the coaster. Choose at least one third-degree polynomial function and one parabola based on the work you and your partner completed for

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

Create an ad campaign to promote the new 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.