I need help with this question. I cannot find any equations that work....

  see the attachments for the graph(s)

Step-by-step explanation:

1) The point at (3, 6) is on the vertical line that is halfway between the zeros at x=-3 and x=9, so it represents the vertex of the function. That knowledge, with any of the other points, lets you write the vertex form of the equation.

  y = a(x -3)^2 +6

Using the point (0, 4.5), we can find the value of 'a':

  4.5 = a(0 -3)^2 +6

  -1.5 = 9a

  -1.5/9 = a = -1/6

So, the vertex form of the equation is ...

  y = -1/6(x -3)^2 +6

A graph of this is shown in the attachment.

__

2) Now that we know the leading coefficient is -1/6, we can write the equation in "intercept form" (factored form) as ...

  y = -1/6(x +3)(x -9)

In this form, each zero (p) gives rise to a factor (x-p).

The second attachment shows the graph of this.

__

3) We can also write the equation in standard form, by expanding the one in (2) above:

  y = -1/6(x^2 -6x -27)

  y = -1/6x^2 +x +9/2

The third attachment shows the graph of this.