Question

How is the graph of y = –x2 – 1 different from the graph of y = –x2?

Answer 1

Every transformation like

$f(x)\mapsto f(x)+k$

translated the graph vertically, k units up if k is positive, k units down if k is negative.

So, in your case, we have that $-x^2-1$ is the same graph as $-x^2$, except it's shifted 1 unit down.