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