Chapter 7 - Review Exercises - Page 878: 44 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. 