Answer
The graph is shown below:
Work Step by Step
Let us consider the inequality $ y\le {{x}^{2}}-1$; substitute the equals symbol in place of the inequality and rewrite the equation as given below:
$ y={{x}^{2}}-1$
To find the value of the x-intercept, substitute y = 0 as given below:
$\begin{align}
& y={{x}^{2}}-1 \\
& 0={{x}^{2}}-1 \\
& {{x}^{2}}=1 \\
& x=\pm 1
\end{align}$
To find the value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& y={{x}^{2}}-1 \\
& y={{0}^{2}}-1 \\
& y=-1
\end{align}$
And the intercepts show that it is a parabola with origin $\left( 0,-1 \right)$.
Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
$\begin{align}
& y\le {{x}^{2}}-1 \\
& 0\le {{0}^{2}}-1 \\
& 0\le -1
\end{align}$
So, the above expression is incorrect, which means the shaded region will not contain the test point; that is, the region away from the origin will be shaded. And the parabola passes through three points $\left( -1,0 \right),\left( 0,-1 \right)$ and $\left( 1,0 \right)$.
Plot the graph using the intercepts as given below:
Thus, the graph of the provided inequality is plotted and the parabola passes through points $\left( -1,0 \right),\left( 0,-1 \right)$ and $\left( 1,0 \right)$. Also, we see that the shaded region does not contain the origin.
