Answer
Vertex: $(2,-8)$
Focus: $\left(2,-7.75\right)$
Directrix: $y=-8.25$
See graph
Work Step by Step
We are given the parabola:
$x^2-4x=y+4$
Put the equation in standard form:
$x^2-4x+4=y+4+4$
$(x-2)^2=y+8$
The standard equation is:
$(x-h)^2=4p(y-k)$
Determine $h,k,p$:
$h=2$
$k=-8$
$4p=1\Rightarrow p=0.25$
Determine the vertex:
$(h,k)=(2,-8)$
Determine the focus:
$(h,k+p)=\left(2,-8+0.25\right)=\left(2,-7.75\right)$
Determine the directrix:
$y=k-p$
$y=-8-0.25$
$y=-8.25$
Determine the two points defining the latus rectum:
$y=-7.75$
$(x-2)^2=-7.75+8$
$(x-2)^2=0.25$
$x-2=\pm 0.5$
$x-2=-0.5\Rightarrow x_1=1.5$
$x-2=0.5\Rightarrow x_2=2.5$
$\Rightarrow (1.5,-7.75),(2.5,-7.75)$
Plot the points, draw the directrix, and graph the parabola:
