8! in two or more complete sentences, explain how you would find the equation of a parabola, given the coordinate of the focus and the equation of the directrix.

2 types of parabolas

left right or up down opening ones
left right ones are in form (y-k)²=4p(x-h)
up down ones are (x-h)²=4p(y-k)
in all of them, the vertex is (h,k)
p is the distance from the vertex to the focus, also the shortest distance from the vertex to the directix making p half of the distance of the shortest path from focus to directix

if p is positive, then the parabola opens up or right
if p is negative then the parabola opens down or left

if the directix is y=something, then it is a up down parabola
if directix is x=something, then it is a left right parabola
directix is outside the parabola, kind of at the back

so lets say we had
focus=(2,3) and directix is x=-4
dirextix is x =-4 so left right
from x=-4 to x=2 (focus), that is distance of 6
6/2=3
p=3
wait, ok, so directix is on oposite side of opening
hmm, -4 is to left of the 2 so
the direxix is at back so the parabola opens to the right
p=3, positive 3

then we have the vertex is halfway between those
so 3 back from focus is from (2,3) to (-1,3)
so vertex is (-1,3)

equation is
(y-3)²=4(3)(x-(-1))
(y-3)²=4(3)(x+1)