Final answer:
The correct equation of the parabola with focus at (0,-1) and directrix y=1 is x² = -4y, which indicates a parabola opening downward.
Step-by-step explanation:
The question at hand involves determining the equation of a parabola given its focus and directrix. The parabola is a unique curve where each point on the parabola is equidistant from the focus and the directrix.
We are given the focus (0,-1) and the directrix y=1. A parabola that opens upward or downward has the general form x² = 4py, where p is the distance from the vertex to the focus (and also from the vertex to the directrix).
For this question, we know that the vertex is in the middle of the focus and directrix which means it lies at y=0 since the distance between the focus and directrix is 2 (from y=-1 to y=1), and the vertex is the midpoint. Therefore, the value of p is ± 1. The negative value of p is used for a parabola that opens downwards.
The correct equation of the parabola given the focus and directrix is therefore x² = -4y. The coefficient -4 comes from 4p where p = -1. This reflects a parabola opening downward since the focus is below the directrix.