What is the equation of the parabola with a directrix at y = -4 and focus (0, 4)?

Step-by-step explanation:

We are given the directrix equation and the location of the focus as a means to determine what the vertex (h, k) of the parabola is.  Since the directrix is a horizontal line AND since the directrix is below the focus, AND since the parabola ALWAYS wraps itself around the focus, this is a positive parabola of the form

We also know that since the vertex is directly in between the focus and the directrix, it has coordinates of (0, 0).  So h = 0 and k = 0.

By definition, p is the distance between the vertex and the focus (and also the vertex and the directrix since they are both the same).  Therefore, p = 4.

Filling in our equation with those values gives us

and solving for y gives us