Question

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.(Hint: Are there only x-axis intercepts? How many times can a function cross the y-axis?)

Answer 1

Let's start by answering the question posed on the Hint.

Any given function can only cross the y-axis one time. Otherwise it would go against the definition of a function, that is, every element of the image of a function is related to a single element from the domain.

If the "function" crossed the y-axis two or more times, that would mean that there exist two values, a and b such that

`f(0)=a`

and

`f(0)=b,`

which contradicts the definition of a function.

Now, a third degree polinomial gunction can have, at most, 3 zeroes, that is, we could have three values (at most), x₁, x₂ and x₃ such that

`f(x_1)=0,`
`f(x_2)=0`

and

`f(x_3)=0`

Now, Ray claims the third degree polinomial has 4 intercepts. This could be true if the graph they ploted crosses the y-axis, in which case it would have three x-intercepts and one y-intercept.

Kelsey's claim is indeed true, as that is a property of third degree polinomials.

In conclussion, a way for both of their claims to be true would be if the graph they plot crosses the x-axis three times and the y-axis one time.