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QUESTION

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

y^2=8x

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 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