Describing Transformations of Quadratic Functions

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. $$\begin{array}{c}y=(x+1{)}^2-1\\ \Updownarrow \\ y=(x-(\text{-}1){)}^2+(\text{-}1)\end{array}$$ In our case, we have $h=\text{-}1$ and $k=\text{-}1.$ 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.

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