Answer
Vertex: $(2,-2)$
Focus: $\left(2,-1.5\right)$
Directrix: $y=-2.5$
See graph
Work Step by Step
We are given the parabola:
$x^2-4x=2y$
Put the equation in standard form:
$x^2-4x+4=2y+4$
$(x-2)^2=2(y+2)$
The standard equation is:
$(x-h)^2=4p(y-k)$
Determine $h,k,p$:
$h=2$
$k=-2$
$4p=2\Rightarrow p=0.5$
Determine the vertex:
$(h,k)=(2,-2)$
Determine the focus:
$(h,k+p)=\left(2,-2+0.5\right)=\left(2,-1.5\right)$
Determine the directrix:
$y=k-p$
$y=-2-0.5$
$y=-2.5$
Determine the two points defining the latus rectum:
$y=-1.5$
$(x-2)^2=2(-1.5+2)$
$(x-2)^2=1$
$x-2=\pm 1$
$x-2=-1\Rightarrow x_1=1$
$x-2=1\Rightarrow x_2=3$
$\Rightarrow (1,-1.5),(3,-1.5)$
Plot the points, draw the directrix, and graph the parabola:
