Answer
The vertex form of the function is $y=(x-4)^{2}+3.$ The vertex is $(4,3)$.
Work Step by Step
$ y=x^{2}-8x+19\qquad$ ...prepare to complete the square.
$ y+?=x^{2}-8x+?+19\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{-8}{2})^{2}=(-4)^{2}=16\qquad$ ...complete the square by adding $16$ to each side of the expression
$ y+16=x^{2}-8x+16+19\qquad$ ... write $x^{2}-8x+16+19$ as a binomial squared.
$ y+16=(x-4)^{2}+19\qquad$ ...add $-16$ to each side of the expression
$ y+16-16=(x-4)^{2}+19-16\qquad$ ...simplify.
$y=(x-4)^{2}+3$
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, $h=4,\ k=3$, so the vertex is $(4,3)$
