Answer
The graph of $f\left( x \right)={{\left( x-2 \right)}^{2}}+1$ opens upward. True.
Work Step by Step
For the standard equation of the parabola $f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$.
If $a>0$ , then the parabola opens upward and the function f has a minimum value that occurs at $x=-\frac{b}{2a}$.
Point $x=-\frac{b}{2a}$ is the lowest point of the parabolic function after that function’s value increases -- that is, the graph of the parabola increases.
If $a<0$ , then the parabola opens downward and the function f has a maximum value that occurs at $x=-\frac{b}{2a}$.
Point $x=-\frac{b}{2a}$ is the highest point before that function’s value decreases -- that is, the graph of the parabola increases in the negative side of the graph.
So, for the given function $f\left( x \right)={{\left( x-2 \right)}^{2}}+1$ , $a>0$ and the graph opens upward. It matches the statement given.
