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 ##(a0)## 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 ##(xy)## such that the distance from the point to the directrix ##x=-2## is the same as the distance to the focus (20).
(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##