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We want to match the given function with its graph. We 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⇕y=(x−(-1))_{2}+(-1) $
In our case, we have $h=-1$ and $k=-1.$ This means that the graph of $y=f(x+1)−1$ is obtained by translating the graph of $y=f(x)$ to the left $1$ unit and down $1$ unit.

$Parent Function$

$Horizontal Translation$

$Vertical Translation$

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