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We want to match the given function with its graph. We will begin by noticing that the given function is a transformation of the form $y=a(x−h)_{2}+k,$ where $h$ represents a horizontal translation and $k$ represents a vertical translation.
$y=(x−1)_{2}+1 $
In our case, we have $h=1$ and $k=1.$ This means that the graph of $y=(x−1)_{2}+1$ is obtained by translating the graph of $y=y_{2}$ to the right $1$ unit and up $1$ unit.

$Parent Function$

$Horizontal Translation$

$Vertical Translation$

Thus, we conclude that the correct answer is option D.