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