Chapter 1.9, Problem 19E

Solution

To describe: the error in the given graph of the quadratic inequality and correct it.

The only error is the shaded region will be inside of the parabola not the outside of the parabola.

Given information: 

The given quadratic inequality is $y\ge x^2+2$_.

The given graph of the quadratic inequality is

  

Formula used:

A quadratic function of the form $y=a{x}^2+bx+c$ has vertex at $\left(-\frac{b}{2a},y\left(\frac{b}{2a}\right)\right)$ .

If a quadratic inequality is less than or greater than type then the outline of the parabola is a dashed line.

If a quadratic inequality is less than equal to or greater than equal to type then the outline of the parabola is a solid line.

If $a>0$ then parabola is open upwards

If $a<0$ then parabola is open downwards.

Calculation:

First, $y\ge x^2+2$ is an inequality with greater than equal to sign.

So, outline of the parabola is a solid line.

Now, check for $x$ -coordinate of the vertex.

  $\begin{array}{c}-\frac{b}{2a}=-\frac{0}{2\left(1\right)}\\ =0\end{array}$

Now, check for $y$ -coordinate of the vertex.

  $\begin{array}{c}y=0+2\\ =2\end{array}$

So, the vertex is at $\left(0,2\right)$ .

Here, $a=1>0$

The parabola is open upwards

Now, take $\left(0,0\right)$ as check point.

  $\begin{array}{l}0\ge 0+2\\ 0\ge 2\end{array}$

Which is false.

So, the parabola is not satisfied by $\left(0,0\right)$ .

The shaded region will be inside and the outline of the parabola not the outside of the parabola.

Graph the correct function as shown.

  

The only error is the shaded region will be inside of the parabola not the outside of the parabola..