Answer
Vertex: $(-1,2)$
Focus: $\left(1,2\right)$
Directrix: $x=-3$
See graph
Work Step by Step
We are given the parabola:
$(y-2)^2=8(x+1)$
The standard equation is:
$(y-k)^2=4p(x-h)$
Determine $h,k,p$:
$h=-1$
$k=2$
$4p=8\Rightarrow p=2$
Determine the vertex:
$(h,k)=(-1,2)$
Determine the focus:
$(h+p,k)=\left(-1+2,2\right)=\left(1,2\right)$
Determine the directrix:
$x=h-p$
$x=-1-2$
$x=-3$
Determine the two points defining the latus rectum:
$x=1$
$(y-2)^2=4(2)(1+1)$
$(y-2)^2=16$
$y-2=\pm 4$
$y-2=-4\Rightarrow y=-2$
$y-2=4\Rightarrow y=6$
$\Rightarrow (1,-2),(1,6)$
Plot the points, draw the directrix, and graph the parabola:
