Answer
$(x+3)^{2}=6(y+2)$
New focus: $\displaystyle \quad (-3,-\frac{1}{2})$
New directrix: $\displaystyle \quad y=-\frac{7}{2}$
New vertex:$ \quad (-3,-2)$
Work Step by Step
Shift left by $3 \Rightarrow$ in the equation, replace $x$ with $x+3$
Shift down by $2 \Rightarrow$ in the equation, replace $y$ with $y+2$
New equation:$\quad (x+3)^{2}=6(y+2)$
$x^{2}=6y\quad $is of the form
$x^{2}=4py $ ,$\qquad$ (opens up), $p=\displaystyle \frac{6}{4}=\frac{3}{2}$
focus: $\quad(0,p)$= $\displaystyle \quad(0,\frac{3}{2})$
directrix: $ y=-p\Rightarrow \displaystyle \quad y=-\frac{3}{2}$
vertex:$\quad (0,0)$
The translations are such that $(x,y)\rightarrow(x',y')$, where
$\left\{\begin{array}{ll}
x'=x-3 & \text{... shift left}\\
y'=y-2 & \text{... shift down}
\end{array}\right.$
New focus: $\displaystyle \quad(0-3,\frac{3}{2}-2)= (-3,-\frac{1}{2})$
New directrix: $ y=-\displaystyle \frac{3}{2}-2\Rightarrow \displaystyle \quad y=-\frac{7}{2}$
New vertex:$\quad (0-3,0-2) = (-3,-2)$
