Transformations of Quadratic Functions

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. $$\begin{array}{c}y=(x-1{)}^2+1\end{array}$$ 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.

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