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

Keeping in mind that, there's a "p" distance from the vertex to the focus, and the same distance from the vertex to the directrix.  Therefore then, the vertex is half-way between those two fellows.

now, we know the focus point is at 2, -1, and the directrix is below it at y = -½, therefore, is a vertical parabola, opening upwards.

check the picture below.

from y = -1 to y = -½, there's only ½ units, and the vertex is right in the middle, and half of ½ is ¼, then the y-coordinate for the vertex must be at -1 - ¼, or -5/4 then.

since the parabola is opening upwards, the "p" unit is positive, ¼.

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