Find the equation of the parabola with focus at the origin and with directrix equation y=-12.

The vertex lies halfway between the focus (0,0) and the point on the directrix given by (0, -12). So, using the mid-point formula, we have......

((0 + 0) / 2 , (0 + (-12)) / 2 ) =  ( 0, -6)

So, the equation of the parabola, where a = 1, and the vertex is (h, k)  = (0, -6) =

y = a(x - h)² + k   =   (x - 0)² +  (-6)  =  x² - 6

Here's a graph......https://www.desmos.com/calculator/ixkwz7bips