Answer
Vertex: $(2,3)$
Focus: $\left(2,4\right)$
Directrix: $y=2$
See graph
Work Step by Step
We are given the parabola:
$(x-2)^2=4(y-3)$
The standard equation is:
$(x-h)^2=4p(y-k)$
Determine $h,k,p$:
$h=2$
$k=3$
$4p=4\Rightarrow p=1$
Determine the vertex:
$(h,k)=(2,3)$
Determine the focus:
$(h,k+p)=\left(2,3+1\right)=\left(2,4\right)$
Determine the directrix:
$y=k-p$
$y=3-1$
$y=2$
Determine the two points defining the latus rectum:
$y=4$
$(x-2)^2=4(4-3)$
$(x-2)^2=4$
$x-2=\pm 2$
$x-2=-2\Rightarrow x_1=0$
$x-2=2\Rightarrow x_2=4$
$\Rightarrow (0,4),(4,4)$
Plot the points, draw the directrix, and graph the parabola:
