Answer
$$\eqalign{
& {\text{focus: }}\left( {0,\frac{1}{{32}}} \right) \cr
& {\text{directrix: }}y = - \frac{1}{{32}} \cr
& {\text{axis of symmetry: }}x = 0 \cr} $$
Work Step by Step
$$\eqalign{
& {x^2} = \frac{1}{8}y \cr
& {\text{This equation is written in the form }}{x^2} = 4py.{\text{ }} \cr
& {\text{represents a parabola with Vertical Axis }} \cr
& {\text{of Symmetry and Vertex }}\left( {0,0} \right). \cr
& \underbrace {{x^2} = \frac{1}{8}y}_{{x^2} = 4py} \cr
& \frac{1}{8}y = 4py \cr
& p = \frac{1}{{32}} \cr
& {\text{With focus}}\left( {0,p} \right){\text{ and directrix }}y = - p \cr
& {\text{focus: }}\left( {0,\frac{1}{{32}}} \right) \cr
& {\text{directrix: }}y = - \frac{1}{{32}} \cr
& {\text{axis of symmetry: }}x = 0 \cr} $$
