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 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.

they’re both correct! a zero is oftentimes used interchangeably to refer to “intercept”, but in fact it means x-intercept specifically. a third degree polynomial can have three zeroes (three degrees = three zeroes), but you must remember that it can also have a y-intercept - so four intercepts in total. in this way, they can both be correct.

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
create a graph of the polynomial function you selected from question 2.

i pick g(x) = x3 - x2 - 4x + 4.

to describe it:
when x is a large, positive number, g(x) as a function is also large and positive.

the same goes for the opposite, in that the function g(x) will be small and negative when x is also small and negative.

the zeroes of the function are -2, 2, and 1.

the y-intercept of the function is 4.

part b
the second part of the new coaster is a parabola.
ray needs 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 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.

create a graph of the polynomial function you created in question 4.



i've already done part a, as you can see, but i've included it because it'll be useful in part b.

! in advance!