Chapter 7.4, Problem 12E

Solution

To Graph: The given inequality by using the hand sketch and state the boundary region.

  $y\ge x^2-3$

The boundary $y=x^2-3$ is included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $y\ge x^2-3$

Explanation:

Consider the given inequality,

  $y\ge x^2-3$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $y=x^2-3$ , which is a parabola and opening to upward.

Now, find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}0=x^2-3\\ x^2=3\\ x=\pm \sqrt{3}\end{array}$

And,

  $\begin{array}{c}y={\left(0\right)}^2-3\\ y=0-3\\ y=-3\end{array}$

Therefore, the intercept are $\left(\sqrt{3},0\right)\left(-\sqrt{3},0\right)\text{ and }\left(0,-3\right)$ .

The boundary point is included if the inequality containing $\le \text{ or }\ge$ symbols and the boundary point is excluded if the symbols $<\text{ or }>$  is used.

As the greater then equal to symbol was used, then the boundary is included in the graph and drawing it by using solid line. To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}\left(0\right)\ge {\left(0\right)}^2-3\\ 0\ge -3\end{array}$

The statement is true so shade the half-plane where  $\left(0,0\right)$ is located. The graph will be:

  

Hence, the required graph is shown above and the boundary $y=x^2-3$ is included in the graph.