Answer
$${x^2} = - y$$
Work Step by Step
$$\eqalign{
& {\text{The parabola is symmetric with respect to the }}x{\text{ - axis, then }} \cr
& {\text{the equation is of the form }}{x^2} = 4py \cr
& {\text{The point }}\left( {2, - 4} \right){\text{ is on the graph, it must satisfy the equation}}{\text{.}} \cr
& {x^2} = 4py \cr
& {\left( 2 \right)^2} = 4p\left( { - 4} \right) \cr
& p = - \frac{1}{4} \cr
& {\text{Then, the equation is:}} \cr
& {x^2} = 4\left( { - \frac{1}{4}} \right)y \cr
& {x^2} = - y \cr} $$
