Derive the equation of the parabola with a focus at (2, –1) and a directrix of y = – .5

Check the picture below...the parabola, with that focus point and that directrix.. .looks more or less like so

now, notice that, the directrix is above the focus, meaning the parabola is vertical and it opens downwards

the distance "p" is the same distance from the vertex to the directrix, since the vertex is half-way between those two fellows

now the  focus point is at 2,-1 and the directrix up above at +0.5, from -1 to 0.5 over the y-axis, there are 1.5 units, the vertex is half-way through, so 1.5/2 or 0.75

so the vertex is from -1 up 0.75 units, or from 0.5 down 0.75 units, that places it at -0.25, the axis of symmetry is x = 2, so, the vertex is at 2, -0.25

because the parabola is opening downwards, the value for "p" is negative, so, we know the distance "p" is 0.75, but because is an opening downwards parabola, then  p = -0.75

now, let's plug all those guys in, let's use for -0.25 then -1/4, and for -0.75 the -3/4, which is just the fraction version

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