Answer
A parabola that opens to the right:
with vertex: $\quad (-2,2)$
and focus: $\quad (0,2)$
Work Step by Step
We group like terms (containing y, as it is squared)
$(y^{2}-4y)-8x-12=0$
$(y^{2}-4y)=8x+12$
We complete the square
$(y^{2}-4y +4)=8x+12+4$
$(y-2)^{2}=8x+16$
$(y-2)^{2}=8(x+2)$
This has the form of an equation of a parabola,
$y^{2}=4px ,\qquad$ (opens right), $p=2$,
the focus is 2 units to the right of the vertex at the origin.
When $x$ is replaced with $(x+2)$, a shift to the left is made.
When $y$ is replaced with $(y-2)$, a shift upward is made.
$(x,y)\rightarrow(x-2,y+2)$
So the equation $(y-2)^{2}=8(x+2)$ represents a shift of the center to $(0-2,0+2)= (-2,2)$ .
The focus is 2 units to the right, $(0,2)$.
