Answer
Parabola.
Vertex: $ \quad (2,-2)$
Focus: $ \displaystyle \quad (2,-\frac{3}{2})$
Directrix:$ \displaystyle \quad y=-\frac{5}{2}$
Work Step by Step
One variable is squared, the other is not: Parabola.
Complete the square on the LHS
$x^{2}-4x+4=2y+4$
$(x-2)^{2}=2(y+2)$
$(x-2)^{2}=4(\displaystyle \frac{1}{2})(y+2)$
$a=\displaystyle \frac{1}{2}$
Table 2:
$\begin{array}{lll}
{\text{Focus}}&{\text{Directrix}}&{\text{Equation}}\\\hline
{(h,k+a)}&{y=k-a}&{(x-h)^{2}=4a(y-k)}\\
\end{array}$
Vertex: $ \quad (h,k)=(2,-2)$
Focus: $ \displaystyle \quad (h,k+a)=(2,-\frac{3}{2})$
Directrix: $ \displaystyle \quad y=k-a \quad \Rightarrow\quad y=-\frac{5}{2}$
