Answer
$(x-1)^{2}=8(y+7)$
New focus: $\quad (1, -5)$
New directrix: $\quad y=-9$
New vertex:$ \quad (1,-7)$
Work Step by Step
Shift right by $1 \Rightarrow$ in the equation, replace $x$ with $x-1$
Shift down by $7 \Rightarrow$ in the equation, replace $y$ with $y+7$
New equation:$\quad (x-1)^{2}=8(y+7)$
$x^{2}=8x\quad $is of the form
$x^{2}=4py $ ,$\qquad$ (opens up), $p=2$
focus: $\quad(0,p)$= $\quad(0,2)$
directrix: $ y=-p\Rightarrow \quad y=-2$
vertex:$\quad (0,0)$
The translations are such that $(x,y)\rightarrow(x',y')$, where
$\left\{\begin{array}{ll}
x'=x+1 & \text{... shift right}\\
y'=y-7 & \text{... shift down}
\end{array}\right.$
New focus: $\quad(0+1,2-7) = (1, -5)$
New directrix: $ y=-2-7\Rightarrow \quad y=-9$
New vertex:$\quad (0+1,0-7) = (1,-7)$
