Question

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)
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

Answer 1

Ray and Kelsey can both be correct about the number of intercepts, as a third-degree polynomial can have up to three zeros and one y-intercept. The function g(x) = (x + 2)(x - 1)(x - 2) has an end behavior extending outward and upward, a y-intercept at (0, 4), and zeros at x = -2, x = 1, and x = 2.

Step-by-step explanation:

In addressing whether Ray and Kelsey can both be correct in the assertion about the number of intercepts for a third-degree polynomial, it is important to clarify that a third-degree polynomial can indeed have up to three zeros. However, the term 'intercepts' can refer to both x-intercepts (zeros) and y-intercepts. A third-degree polynomial can have only one y-intercept and up to three x-intercepts, making a total of four possible intercepts on the graph if we consider both types of intercepts. Hence, Ray referring to four intercepts and Kelsey arguing for three zeros are both correct.

Now, let's discuss the key features of the function g(x) = (x + 2)(x - 1)(x - 2). First, end behavior of this cubic function is such that as x approaches negative infinity, g(x) approaches negative infinity; and as x approaches positive infinity, g(x) approaches positive infinity.

This is because the leading coefficient is positive, which suggests that both tails of the curve extend outward and upward. The y-intercept of the function is the value of g(x) when x is zero, which is obtained by substituting x with 0 in the function: g(0) = (0 + 2)(0 - 1)(0 - 2), resulting in a y-intercept at (0, 4). Lastly, the zeros (or x-intercepts) of the function are the solutions to the equation g(x) = 0, which are x = -2, x = 1, and x = 2.