Answer
The vertex form of the function is $f(x)=(x-1)^{2}+(-8)$. The vertex is $(1,-8)$.
Work Step by Step
$ g(x)=x^{2}-2x-7\qquad$ ...write in form of $x^{2}+bx=c$ (add $7$ to each side).
$ g(x)+7=x^{2}-2x\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{-2}{2})^{2}=(-1)^{2}=1\qquad$ ...complete the square by adding $1$ to each side of the expression
$ g(x)+7+1=x^{2}-2x+1\qquad$ ... write $x^{2}-2x+1$ as a binomial squared.
$ g(x)+8=(x-1)^{2}\qquad$ ...add $-8$ to each side of the expression
$ g(x)=(x-1)^{2}-8\qquad$ ...write in vertex form $y=a(x-h)^{2}+k$.
$g(x)=(x-1)^{2}+(-8)$
The vertex form of a quadratic function is $y=a(x-h)^{2}+k$ where $(h,k)$ is the vertex of the function's graph.
Here, $h=1,\ k=-8$, so the vertex is $(1,-8)$
