Question

Find the standard form of the equation of the parabola with a focus at (0, -2) and a directrix at y = 2.

Answer 1

Check the picture below.

so the parabola looks more or less like so, bearing in mind that the directrix is a horizontal line, thusu the parabola is a vertical one, so the squared variable is the "x".

The vertex is always half-way between the focus point and the directrix, as you see there, and the distance from the vertex to the focus is "p" distance, since the parabola is opening downwards, "p" is negative, in this case -2.

$\bf \textit{parabola vertex form with focus point distance} \\\\ \begin{array}{llll} 4p(x- h)=(y- k)^2 \\\\ \stackrel{\textit{we'll use this one}}{4p(y- k)=(x- h)^2} \end{array} \qquad \begin{array}{llll} vertex\ ( h, k)\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} h=0\\ k=0\\ p=-2 \end{cases}\implies 4(-2)(y-0)=(x-0)^2\implies -8y=x^2\implies y=-\cfrac{1}{8}x^2$