How do you find the vertex, focus and directrix of y - 2 = -1/8 (x+2)^2?

1 Answer
Jun 26, 2018

Vertex: (-2,2)
Focus: (-2,0)
Directrix: y=4

Explanation:

For future reference: Formula for a parabola facing down

y-k=-1/(4p)(x-h)^2
(This is the exact same as y=-1/(4p)(x-h)^2+k. k is just on the other side.)

The vertex can be found by looking at our equation. The vertex is (h,k). The vertex is (-2,2):

y-2=-1/8(x-(-2))^2
h=-2, k=2

To find our focus and directrix, we need to know p first. You may have noticed it in our formula. It is the distance from the vertex of a parabola to both its focus and its directrix. To find p, all you have to do is set our scale factor (-1/8) equal to -1/(4p):

-1/(4p)=-1/8
-4p=-8
p=2

Our focus will be 2 units down from the vertex. This is because the focus is always contained within the parabola and the parabola faces down. The focus is (-2,0).

The directrix, which is a line, will also be 2 units away, but in the opposite direction. (In our case, it's above.) The directrix is y=4.

Hope this helped!