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We want to match the given function with its graph. We will begin by noticing that the function is a transformation of the form $y=a(x-{\color{#0000FF}{h}})^2+{\color{#009600}{k}},$ where ${\color{#0000FF}{h}}$ represents a horizontal translation and ${\color{#009600}{k}}$ represents a vertical translation.
$\begin{gathered}
y=(x-{\color{#0000FF}{1}})^2
\end{gathered}$
In our case, we have ${\color{#0000FF}{h}}={\color{#0000FF}{1}}.$ This means that the graph of $y=(x-1)^2$ is obtained by translating the graph of $y=x^2$ by ${\color{#0000FF}{1}}$ unit to the right.

$\text{Parent Function}$

$\text{Translation}$

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