Question

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

Answer 1

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!