Derive the equation of the parabola with a focus at (2,-1) and a directrix of y=-1/2

keeping in mind that the vertex is "p" distance from either the focus or directrix, that means is half-way between both, check the picture below.

from -1 to -1/2 the distance is just 1/2 and half of that is 1/4.

now, if we go down from the directrix which is at -1/2, by 1/4 of a unit, we'll end up at

the distance "p" is obviously 1/4, however, since the parabola is opening downwards, is negative, therefore -1/4.