Answer
$${y^2} = - 4x$$
Work Step by Step
$$\eqalign{
& {\text{The parabola opens left, then the equation is of the form}} \cr
& {y^2} = 4px,\,\,\,p < 0 \cr
& {\text{The point }}\left( { - 2, - 2\sqrt 2 } \right){\text{ is on the graph, it must satisfy the equation}}{\text{.}} \cr
& {y^2} = 4px \cr
& {\left( { - 2\sqrt 2 } \right)^2} = 4p\left( { - 2} \right) \cr
& 8 = 4p\left( { - 2} \right) \cr
& p = - 1 \cr
& {\text{Then, the equation is:}} \cr
& {y^2} = 4\left( { - 1} \right)x \cr
& {y^2} = - 4x \cr} $$
