Answer
$y = \frac{1}{8}~x^2$
Work Step by Step
We can write an expression for the distance from a point $(x,y)$ to the focus $(0,2)$:
$d_1 = \sqrt{(x-0)^2+(y-2)^2}$
$d_1 = \sqrt{x^2+(y-2)^2}$
We can write an expression for the distance from a point $(x,y)$ to the directrix $y = -2$:
$d_2 = \sqrt{(0)^2+[y-(-2)]^2}$
$d_2 = \sqrt{(y+2)^2}$
$d_2 = y+2$
We can equate these two distances to find the equation of the parabola:
$d_2 = d_1$
$y+2 = \sqrt{x^2+(y-2)^2}$
$(y+2)^2 = x^2+(y-2)^2$
$y^2+4y+4 = x^2+y^2-4y+4$
$8y = x^2$
$y = \frac{1}{8}~x^2$
