Chapter 7.5, Problem 30E

To determine

To write: an inequality for the shaded region shown in the figure.

Answer to Problem 30E

  $y\ge x^2-4$ .

Explanation of Solution

Given information:

Given graph

Calculation:

Consider the following figure,

According to the graph, a quadratic equation will fit the graph. Assume the equation to be

  $y=a{x}^2+bx+c$

From the graph, it is clear that $\left(-2,0\right),\left(2,0\right)$ and $\left(0,-4\right)$ satisfy the quadratic equation.

Hence, put $\left(-2,0\right),\left(2,0\right)$ and $\left(0,-4\right)$ in the equation,

  $\begin{array}{c}4a-2b+c=0\\ 4a+2b+c=0\\ c=-4\end{array}$

Solve the equations and obtain the values of the unknowns as done below,

  $\begin{array}{l}a=1\\ b=0\\ c=-4\end{array}$

Substitute the above values in the general quadratic equation and find th e equation as done below,

  $\begin{array}{c}y=a{x}^2+bx+c\\ ={\left(1\right)}^{2}x+\left(0\right)x+\left(-4\right)\\ =x^2-4\end{array}$

Hence, the equation of the graph is $y=x^2-4$ and the shaded region is towards the origin from the graph therefore the symbol will be more then equal to,

  $y\ge x^2-4$

Thus, the inequality shown in the above graph is $y\ge x^2-4$ .