Answer
The vertex for the parabola is $\left( 4,3 \right)$.
Work Step by Step
Consider the formula:
$y={{\left( x-4 \right)}^{2}}+3$
Now, simplify the above formula:
$\begin{align}
& y={{\left( x-4 \right)}^{2}}+3 \\
& ={{x}^{2}}-8x+{{4}^{2}}+3 \\
& ={{x}^{2}}-8x+19
\end{align}$
Now, consider the x-coordinate of the vertex for the quadratic formula:
$x=\frac{-b}{2a}$
Substitute the values in the above equation:
$\begin{align}
& x=\frac{-\left( -8 \right)}{2\left( 1 \right)} \\
& =4
\end{align}$
This shows that the vertex is located when x is equal to 4.
Substitute x as 4 in the given equation:
$\begin{align}
& y={{\left( 4 \right)}^{2}}-8\left( 4 \right)+19 \\
& =16-32+19 \\
& =3
\end{align}$
This shows that the vertex occurs at the coordinate $\left( 4,3 \right)$.
Thus, the vertex of the equation is $\left( 4,3 \right)$.
