Final answer:
The equation of the parabola with a focus at (0, 4) and a directrix of y = -4 is x² = 16y.
Step-by-step explanation:
To find the equation of a parabola with a focus at (0, 4) and a directrix of y = -4, we can use the definition of a parabola as the set of all points that are equidistant from the focus and the directrix. The vertex of the parabola will be midway between the focus and directrix, which in this case is at (0, 0).
The standard form of a vertical parabola is (x - h)² = 4p(y - k), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus or directrix. The distance p is 4, as the focus is 8 units away from the directrix and the vertex is in the middle. Thus, the equation becomes x² = 16(y - 0) or simply x² = 16y.