Question

How do I find vertex, focus and directrix for the following functions?

y^2−16y=12x−8^2

x^2+16x=4y−24

(x−2)^2=12(y−2)

Answer 1

You just need to remember the characteristics of parabolas.

y^2?16y=12x?8^2
y^2-16y+64 = 12x-64+64
(y-8)^2 = 12x

Now recall that for y^2 = 4px, we have
vertex = (0,0)
focus = (p,0)
directrix is x = -p

So, for your parabola, that means that it is shifted up 8, and p=3:

vertex = (0,8)
focus = (3,8)
directrix is x = -3

http://www.wolframalpha.com/input/?i=parabola+y%5E2%E2%88%9216y%3D12x%E2%88%928%5E2

Now massage the other equations till they fit the general formula for

(x-h)^2 = 4p(y-k)
or
(y-k)^2 = 4p(x-h)

and then apply the required shifts