Answer
$${x^2} = - \frac{2}{3}y$$
Work Step by Step
$$\eqalign{
& {\text{The parabola is symmetric about the }}y - {\text{axis, then equation is of}} \cr
& {\text{the form }}{x^2} = 4py. \cr
& {\text{The parabola passes through the point }}\left( {2, - 6} \right).{\text{ Then,}} \cr
& {x^2} = 4py\,\,\, \Rightarrow \,\,\,\,{\left( 2 \right)^2} = 4p\left( { - 6} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4 = 4p\left( { - 6} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p = - \frac{1}{6} \cr
& {\text{An equation of the parabola is}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} = 4\left( { - \frac{1}{6}} \right)y. \cr
& \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} = - \frac{2}{3}y \cr} $$
