Answer
$${y^2} = 16x$$
Work Step by Step
$$\eqalign{
& {\text{The parabola is symmetric about the }}x - {\text{axis, then equation is of}} \cr
& {\text{the form }}{y^2} = 4px \cr
& {\text{The parabola passes through the point }}\left( {1, - 4} \right).{\text{ Then,}} \cr
& {y^2} = 4px\,\,\, \Rightarrow \,\,\,\,{\left( { - 4} \right)^2} = 4p\left( 1 \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,16 = 4p \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p = 4 \cr
& {\text{An equation of the parabola is }}{y^2} = 4px \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{y^2} = 4\left( 4 \right)x \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{y^2} = 16x \cr} $$
