Answer
Shift $4$ units to the right and $1$ unit upward.
Work Step by Step
The graph of
$$
h(x)=\sqrt{x-4}+1
$$ is similar to the graph of the parent function, $f(x)=\sqrt{x}$, which is a curve that starts at the $x$-axis and increases from left to right. The graph of $h(x)$, though, has all the points of the parent function shifted $4$ units to the right and $1$ unit upward.
