The given equation is of a parabola: x^2 = -4y.
This is a standard form of a parabolic equation with a vertical axis. In these type of parabolas, the value '4p' from the right-hand side of the equation represents 4 times the distance from the vertex to the focus.
You can equate -4p to -4 (from the equation x^2 = -4y) and solve for p. Doing this gives you p = 1. However, since the equation is x^2 = -4y and not x^2 = 4y, the p value is negative. So, p = -1.
Now, let's find out the coordinates of the focus of the parabola. The formula for the focus is (h, k+p), where (h, k) are the coordinates of the vertex of the parabola. In this given equation, the vertex is at the origin (0, 0), so you can substitute these values into the formula to get F = (0, -1).
Next, let's find the equation of the directrix. The formula for the directrix is y = k - p. Since in this equation, the vertex (h, k) is at the origin (0, 0), substituting these values into the formula gives us the equation of the directrix: y = 1.
So, the focus of the parabola is at (0, -1) and the equation of the directrix is y = 1.