Chapter 10.9, Problem 13E

To determine

To graph: the given inequality.

Answer to Problem 13E

  

Explanation of Solution

Given:

  $y>x^2+1$

Calculation:

Consider the inequality

  $y>x^2+1$ .

Graph the inequality $y>x^2+1$ .

To graph the inequality $y>x^2+1$ , simply graph the boundary line $y=x^2+1$ , which is a parabola.

Clearly, the points on the line itself do not satisfy the inequality because, the inequality contains $>$ symbol, so, graph the boundary line with a dashed curve.

To determine whether the inside or outside region of the parabola satisfies the inequality, use the test points $\left(3,3\right)$ on the outside and $\left(-1,4\right)$ on the inside of the parabola.

Substitute the coordinates of each point in the inequality and check whether the result satisfies the inequality.

The computations are tabulated as shown below:

Test point	$y>x^2+1$	Conclusion
$\left(3,3\right)$	$\begin{array}{l}3>3^2+1\\ 3>9+1\\ 3\cancel{>}10\end{array}$	Not a part of the graph
$\left(-1,4\right)$	$\begin{array}{l}4>{\left(-1\right)}^2+1\\ 4>1+1\\ 4>2\end{array}$	Part of the graph

From the above table, observe that the points inside the parabola are satisfies the inequality.

So, shade the region inside the parabola on the graph.

The graph of the inequality $y>x^2+1$ is shown below:

  

Conclusion:

Therefore, the graph of inequality is drawn.