For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the coaster’s track.
Part A
The first part of Ray and Kelsey’s roller coaster is a curved pattern that can be represented by a polynomial function.
- 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.
(Hint: Are there only x-axis intercepts? How many times can a function cross the y-axis?)
- Kelsey has a list of possible functions. Pickone 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
| Show how to find the zeros:
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Describe to Kelsey the other key features of g(x). |
| Roots:
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| y:Intercept:
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| Does the function open upwards or downwards?
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- Create a graph, either by hand or using the using software of your choice, of the polynomial function you selected from Question 2. Copy and paste your graph below:
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.
| Create a unique parabola in the pattern f(x) = ax2 + bx + c. Pick integers other than one or zero for a, b and c.
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| Describe the direction of the parabola. (Hint: How can you tell if a parabola will open upwards or downwards)
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| Determine the y-intercept. (Hint: Using your equation, set the x’s equal to zero and solve for y)
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| Determine the zeros. (Hint: Using your equation, set y = zero, factor, and solve for x. |
- 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
| How do you find the x value of the vertex?
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| How do you find the y value of the vertex?
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| What is the vertex of you parabola? (Hint: You just found the x and y coordinates) |
| What is the axis of symmetry of your parabola? (Hint: You just found the vertex) |
- Create a graph, either by hand or using the using software of your choice, of the polynomial function you created in Question 4. Copy and paste you graph below:
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. (Note: Once you combine your sections, it should look like a roller coaster)
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.
