Answer
$x^2=8y$
Latus rectum points: $(-4,2),(4,2)$
See graph
Work Step by Step
We are given the parabola:
Focus: $(0,2)$
Vertex: $(0,0)$
Because the vertex and the focus have the same $x$-coordinate, the parabola is vertical. Its standard equation is:
$(x-h)^2=4p(y-k)$
Use the coordinates of the vertex to determine $h,k$:
$(h,k)=(0,0)$
$h=0$
$k=0$
Determine $p$ using the coordinates of the focus:
$(h,k+p)=(0,2)$
$(0,0+p)=(0,2)$
$p=2$
Determine the parabola's equation:
$(x-0)^2=4(2)(y-0)$
$x^2=8y$
Determine the two points defining the latus rectum:
$y=2$
$x^2=8(2)$
$x^2=16$
$x=\pm 4$
$\Rightarrow (-4,2),(4,2)$
Plot the points and graph the parabola:
