You have been tasked with designing a roller coaster. The amusement park you are designing for has given you the following criteria:

The ride must last 4 minutes.

The ride starts 250 feet in the air.

The ride will dive below the ground into a tunnel at least once.

To begin, draw a rough sketch of a roller coaster that matches the above criteria. Regard your design as a polynomial.

Now, the amusement park has, for some reason, decided that the zeroes of your polynomial must include at most one real root of multiplicity two and at least one real root of multiplicity one.

Create a new polynomial, (), that satisfies all of the original criteria and has the necessary zeroes.

Write () in factored form

Write () in standard form

Describe the end behavior of () and provide a real-world explanation.

How many turning points does () have? How is this justified mathematically?

What part of () guarantees criteria (i) is satisfied?

What part of () guarantees criteria (ii) is satisfied?

What part of () guarantees criteria (iiii) is satisfied?

What is the real-world explanation for the real root of multiplicity two?

What is the real-world explanation for the real root of multiplicity one?