Answer
$$\eqalign{
& {\text{focus: }}\left( {0, - \frac{1}{{16}}} \right) \cr
& {\text{directrix: }}y = \frac{1}{{16}} \cr
& {\text{axis of symmetry: }}x = 0 \cr} $$
Work Step by Step
$$\eqalign{
& y = - 4{x^2} \cr
& - \frac{1}{4}y = {x^2} \cr
& {x^2} = - \frac{1}{4}y \cr
& {\text{This equation is written in the form }}{x^2} = 4py.{\text{ }} \cr
& {\text{represents a parabola with Vertical Axis }} \cr
& {\text{of Symmetry and Vertex }}\left( {0,{\text{ }}0} \right). \cr
& \underbrace {{x^2} = - \frac{1}{4}y}_{{x^2} = 4py} \cr
& - \frac{1}{4}y = 4py \cr
& p = - \frac{1}{{16}} \cr
& {\text{With focus}}\left( {0,p} \right){\text{ and directrix }}y = - p \cr
& {\text{focus: }}\left( {0, - \frac{1}{{16}}} \right) \cr
& {\text{directrix: }}y = \frac{1}{{16}} \cr
& {\text{axis of symmetry: }}x = 0 \cr} $$
