Chapter 1.9, Problem 18E

Solution

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

The only error is the outline of the parabola is a solid line not a dashed line.

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.

Graph the correct function as shown.

  

The only error is the outline of the parabola is a solid line not a dashed line.