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.

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

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) = (x + 2)(x − 1)(x − 2)

Answer 1

Both Ray and Kelsey can be correct depending on how they interpret the term "intercepts".

Let's analyze the polynomial function:

g(x)=(x+2)(x−1)(x−2)

Key features of g(x):

End behavior: As x approaches positive or negative infinity, the value of g(x) approaches positive infinity. This is because the leading term, x^3 , grows much faster than any other term in the polynomial as x increases or decreases without bound.

y-intercept: The y-intercept is the point where the graph of the function crosses the y-axis. This occurs when x=0. Substituting x=0 into the function, we get:

g(0)=(0+2)(0−1)(0−2)=−4

Therefore, the y-intercept is (0,-4).

Zeros: The zeros of the function are the x-values for which g(x)=0. To find the zeros, we can factor the function:

g(x)=(x+2)(x−1)(x−2)

Setting each factor equal to zero and solving for x, we find that the zeros are x=-2, x=1, and x=2.

Addressing Ray and Kelsey's statements:

Ray is correct in saying that the third-degree polynomial can have four intercepts.

This is because the number of intercepts of a polynomial is equal to its degree, which is 3 in this case.

However, Kelsey is also correct in saying that the function can have as many as three zeros only.

This is because the number of zeros of a polynomial is equal to the number of times it can be factored into linear factors.

In this case, the function can be factored into three linear factors, so it has three zeros.

Therefore, both Ray and Kelsey can be correct depending on how they interpret the term "intercepts".