Question

What is the equation of the quadratic graph with a focus of (3,4) and a drirectrix of y =8

Answer 1

check the picture below.  So the parabola looks more or less like so.

bearing in mind that the distance "p" is the same from the vertex to the directrix or the focus point, so the vertex is therefore half-way between those two fellows, now, from 3,4 up to y = 8, there are 4 units, and the vertex is half-way, thus is at y = 6, and x = 3, (3, 6) as you see there in the picture.

the parabola is vertical, meaning the squared variable is the "x", and is opening downwards, meaning the "p" distance is negative, so since "p" is 2 units, then p = -2.

$\bf \textit{parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{array}{llll} vertex\ ( h, k)\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} h=3\\ k=6\\ p=-2 \end{cases}\implies 4(-2)(y-6)=(x-3)^2\implies -8(y-6)=(x-3)^2 \\\\\\ y-6=\cfrac{(x-3)^2}{-8}\implies \blacktriangleright y=-\cfrac{1}{8}(x-3)^2+6 \blacktriangleleft$