Answer
$x^2= -8y$
Work Step by Step
We are told that the vertex is at the origin.
$4^2=(4)(p)(-2)$
So, $p=\dfrac{16}{-8}=-2$
Since the vertex and focus have the same y-coordinate, the parabola must have a horizontal axis.
Now, we will write the equation for a parabola that has a vertical axis using the vertex and $p$.
$x^2=4py \implies x^2=(4)(-2) \times y $
$\implies x^2= -8y$
