Answer
Vertex: $(0,1)$
Focus: $\left(2,1\right)$
Directrix: $x=-2$
See graph
Work Step by Step
We are given the parabola:
$y^2-2y=8x-1$
Put the equation in standard form:
$y^2-2y+1=8x-1+1$
$(y-1)^2=8x$
The standard equation is:
$(y-k)^2=4p(x-h)$
Determine $h,k,p$:
$h=0$
$k=1$
$4p=8\Rightarrow p=2$
Determine the vertex:
$(h,k)=(0,1)$
Determine the focus:
$(h+p,k)=\left(0+2,1\right)=\left(2,1\right)$
Determine the directrix:
$x=h-p$
$x=0-2$
$x=-2$
Determine the two points defining the latus rectum:
$x=2$
$(y-1)^2=4(2)(2)$
$(y-1)^2=16$
$y-1=\pm 4$
$y-1=-4\Rightarrow y_1=-3$
$y-1=4\Rightarrow y_2=5$
$\Rightarrow (2,-3),(2,5)$
Plot the points, draw the directrix, and graph the parabola:
