how do i determine a equation for a porabola given the vertex and the y intercept

The general form is

y = a ( x -  h)^2 + k     where (h, k) is the vertex  and (x, y) is the y intercept

Since we know the y intercept is a point (0, b), we can plug this into the "formula" to find "a"

Example......suppose we have a parabola whose vertex lies at (2, -2) with a y intercept of (0, 0)

Then we have

0 = a (0 - 2)^2  - 2   add 2 to both sides

2 = a(-2)^2

2 = a(4)    divide both sides by 4

2/4 = 1/2 = a

So...our equation is  y = (1/2)(x - h)^2 + k  =  

y = (1/2)(x - 2)^2 - 2