Question

How do you find the equation of a parabola with vertex at the origin and directrix x=-2?

Answer 1

`y^2=8x`

Explanation:

The standard form of the parabola is `y^2=4ax`, giving a parabola with its axis parallel to the `x`-axis, vertex at the origin, focus `(a,0)` and directrix `x=-a`. So in your case `a=2`, giving `y^2=4ax`.

Alternatively, you can work from a definition of a parabola, which is the set of all points `(x,y)` such that the distance from the point to the directrix `x=-2` is the same as the distance to the focus (2,0)#.
(The vertex is half-way between the focus and the directrix.)

`(x-(-2))^2=(x-a)^2+y^2`
`cancel(x^2)+4ax+cancel 4=cancel(x^2)-4ax+cancel 4+y^2`
`y^2=4ax+4ax=8ax`