Exercise 61 Page 287

Let's graph the two quadratic inequalities separately first.

Graphing $y\ge x^2-4$

The graph of $y\ge x^2-4$ is a region bounded by the graph of $y=x^2-4.$ Notice that this parabola is a vertical translation down $4$ units of the graph of its parent function $y=x^2.$

To determine the region to be shaded, we will test a point not on the graph. For simplicity, we will test $(0,0).$ If the substitution produces a true statement, we will shade the region that contains the point. If not, we will shade the opposite region. Since $0$ is greater than $\text{-}4,$ we shade the region containing $(0,0).$

Graphing $y\le \text{-}{x}^2+4$

Similarly, the graph of $y\le \text{-}{x}^2+4$ is a region bounded by the graph of $y=\text{-}{x}^2+4.$ Notice that this parabola is a vertical translation up $4$ units of the graph of $y=\text{-}{x}^2.$

To determine the region to be shaded, we test the point $(0,0)$ as we did for the previous inequality. In this case, again, this point produces a true statement and therefore we shade the region containing $(0,0).$

Finding the Intersection

Let's draw both inequalities on the same coordinate plane.

The solution is the overlapping region.