Question

How do you find the equation of a parabola when given Focus (-2, 6) Vertex (-2, 9)?

Answer 1

`y= -1/12(x+2)^2+9`

Explanation:

[There may be a more direct route to this solution, but the following uses the tools I remember.]

A parabola in vertex form has the general form:
`color(white)("XXXX")` `y = m(x-a)^2+b`
with a vertex at `(a, b)`

So a parabola with a vertex at `(-2,9)`
has `a= -2` and `b=9` in the general form

The focus of a parabola given the general vertex form (above) is at
`color(white)("XXXX")` `(a, b+1/(4m))`

With the given focus at `(-2, 6)`, this gives us
`color(white)("XXXX")` `b+1(/4m) = 6`
and since `b=9`
`color(white)("XXXX")` `9+1/(4m) = 6`

`color(white)("XXXX")` `1/(4m) = -3`

`color(white)("XXXX")` `-3m = 1/4`

`color(white)("XXXX")` `m = -1/12`

Substituting the derived values for `m, a,` and `b` gives
`color(white)("XXXX")` `y= -1/12(x+2)^2+9`