Question

Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8.

a. f(x) = -1/8 (x - 2)2 + 6
b. f(x) = 1/8 (x - 2)2 + 6
c. f(x) = -1/8 (x + 2)2 + 8
d. f(x) = 1/8 (x + 2)2 + 8

Answer 1

The focus and directrix are equidistant from the vertex of the parabola. If the directrix runs through 8 and the focus is at 4, then the vertex is halfway between them at y=6 with the same ordinate as the focus, 2.

Answer 2

You've seen several of these done by now, you should be able to do them yourself!

Answer 3

For a parabola with equation
\[y=a(x-h)^2+k\]
a can be found with
\[a=\frac{1}{4p}\]
where p is the distance from focus to vertex.

Answer 4

Sketching a quick graph also helps in determining the sign of a.

and so on.