Answer:
To find the equation of the parabola with the given focus and directrix, we can use the standard formula for a parabola with a vertical axis of symmetry:
\[ (x - h)^2 = 4p(y - k) \]
Where (h, k) is the vertex, and 'p' is the distance from the vertex to the focus (or from the vertex to the directrix).
Since the directrix is y = -1, and the focus is (0, -2), the vertex lies halfway between them, so the vertex is (0, -1). Also, since the parabola opens upward, p is positive.
So, using the vertex (0, -1) and the focus (0, -2), we find p = 1.
Plugging these values into the equation:
\[ (x - 0)^2 = 4(1)(y + 1) \]
\[ x^2 = 4(y + 1) \]
Thus, the equation of the parabola is \( x^2 = 4(y + 1) \).
