Start off by building a basic coaster! Build a polynomial that: has an - intercept at x = 500. rises to a maximum, falls as it crosses the x-axis, and then rises into x = 1000. y = ax (x - ) (x - 1000)

Answer: the missing factor is (x - 500) and the whole function is:

y = a x (x - 500) (x - 1000)

Explanation:

The question, if complete, is to find the missing factor.

There are 4 factors: a, x, (x - ?), and x - 1000

- a is a constant, which may take any value, unitl more information is given.

- x factor means that there is an x-intercept at x = 0

- x - 1000 factor means that there is an x - intercept at x = 100.

- The condition of an x-intercept at x = 500 means that x - 500 is a factor.

So, the factor missing is x - 500.

And you can write y as:

y = ax (x - 500) (x - 1000).

See, that for x = 500 that becomes y = a (00) (500 - 500) (500 - 1000).

⇒ y = a500 (0) (-500) = 0, which is what x-intercept means.