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##