Question

Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. 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.

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.

2. 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

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

Answer 1

Let's talk about the first step...what can you do for it?

Answer 2

sorry for the late reply

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.

hint - "intercepts" can include both x-intercepts AND y-intercepts. so is there a way for a function to have 3 zeros (x-intercepts) and 4 total intercepts?

Answer 3

for 2.

2. 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

these are all degree three (the highest exponent is 3) and the leading coefficients are positive (all of the x^3 terms are positive). so taking a look at this chart, an odd degree + positive leading coefficient, what sort of end behavior do we get?



it also asks for y-intercept, so you just need to take your chosen function, plug in x = 0 to get the y-intercepts. to get the zeros, you will have to set the whole function equal to 0, factor, and solve for x

Answer 4

3. is the easiest part, you already have the intercepts + end behavior from part 2
for a third degree polynomial there should be 2 "turning points"

Answer 5

once you have your graph you should double check w/ graphing software to make sure you're on the right track