Answer
$(x-3)^2=-8y$
Latus rectum points: $(-1,-2),(7,-2)$
See graph
Work Step by Step
We are given the parabola:
Focus: $(3,-2)$
Vertex: $(3,0)$
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 $h,k$:
$(h,k)=(3,0)$
$h=3$
$k=0$
Determine $p$ using the coordinates of the focus:
$(h,k+p)=(3,-2)$
$(3,0+p)=(3,-2)$
$p=-2$
Determine the parabola's equation:
$(x-3)^2=4(-2)(y-0)$
$(x-3)^2=-8y$
Determine the two points defining the latus rectum:
$y=-2$
$(x-3)^2=-8(-2)$
$(x-3)^2=16$
$x-3=\pm 4$
$x-3=-4\Rightarrow x_1=-1$
$x-3=4\Rightarrow x_2=7$
$\Rightarrow (-1,-2),(7,-2)$
Plot the points and graph the parabola:
