Chapter 1.9, Problem 7E

To determine

To graph: The quadratic equation $y=100-x^2$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=100-x^2$ .

Graph:

The graph of the quadratic equation $y=100-x^2$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=100-x^2$ .

Now put the values of $x$ in the equation to find the values of $y$ or to find the real roots of $y$ with the help of $x$ .

Solve the equation $y=100-x^2$ with the quadratic formula:

The quadratic formula is:

  $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Now solve the equation $y=100-x^2$ by quadratic formula:

Find the value of $a,b,c$ :

The value of $a,b,c$ are , $a$ is the coefficient of $x^2$ , $b$ is the coefficient of $x$ , $c$ is constant.

Thus, the value of:

  $\begin{array}{l}a=-1\\ b=0\\ c=100\end{array}$

Substitute the values of $a,b,c$ in quadratic formula:

The quadratic formula is:

  $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  $\begin{array}{c}\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{-0\pm \sqrt{-4\times -1\times 100}}{2\times -1}\\ =\frac{\pm \sqrt{400}}{-2}\\ =\frac{\pm 20}{-2}\end{array}$

Either the value $\frac{\pm 20}{-2}$ is positive or negative:

When the value $\frac{\pm 20}{-2}$ is positive, then the real root is $-10$ ,

When the value $\frac{\pm 20}{-2}$ is negative, then the real root is $10$ ,

So $x=-10$ or $x=10$ ,

Here observed the equation of parabola, the coefficient of $x^2$ is negative, so the graph is opened downward.

The roots and the shape of parabola is cleared.

The graph of the equation $y=100-x^2$ .

Substitute $x=0$ , in the equation $y=100-x^2$ ,

  $\begin{array}{l}y=100-x^2\\ y=100-{\left(0\right)}^2\\ y=100\end{array}$

Substitute $x=1$ , in the equation $y=100-x^2$ ,

  $\begin{array}{l}y=100-x^2\\ y=100-{\left(1\right)}^2\\ y=100-1\\ y=99\end{array}$

Substitute $x=2$ , in the equation $y=100-x^2$ ,

  $\begin{array}{c}y=100-x^2\\ y=100-{\left(2\right)}^2\\ y=100-4\\ y=96\end{array}$

Observe that as the value of $x$ increases there is the decrement in value of $y$ .

Steps to plot the graph of the equation $y=100-x^2$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=100-x^2$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-100\\ X\max =100\\ Y\min =-100\\ Y\max =150\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

Interpretation:

The equation of the function $y=100-x^2$ represents a parabola.

The parabola opens downward.

The $x-$ intercepts are the points on $x-axis$ where the graph of the equation touches $x-axis$ .

Therefore, in the equation, $y=100-x^2$ $x-$ intercepts are $\left[-10,10\right]$ .

Therefore, the equation $y=100-x^2$ is symmetric about the $y-$ axis Initially the graph of the equation $y=100-x^2$ decreases when increase the value of $x$ ,the graph is a parabola shaped graph.

a.

When the $x$ -axis ranges from $\left[-4,4\right]$ and $y$ -axis ranges from $\left[-4,4\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=100-x^2$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=100-x^2$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-4\\ X\max =4\\ Y\min =-4\\ Y\max =4\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

This window is not perfect viewing window.

b.

When the $x$ -axis ranges from $\left[-10,10\right]$ and $y$ -axis ranges from $\left[-10,10\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=100-x^2$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=100-x^2$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-10\\ X\max =10\\ Y\min =-10\\ Y\max =10\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

This window is not a perfect viewing window.

c.

When the $x$ -axis ranges from $\left[-15,15\right]$ and $y$ -axis ranges from $\left[-30,110\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=100-x^2$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=100-x^2$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-15\\ X\max =15\\ Y\min =-30\\ Y\max =110\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

This is the perfect viewing window.

d.

When the $x$ -axis ranges from $\left[-4,4\right]$ and $y$ -axis ranges from $\left[-30,110\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=100-x^2$ with the help of graphing utility are as follows:

Step 1: Press $\boxed{\text{MODE}}$ key.

Step 2: Use the down arrow key to reach $\boxed{\text{FUNC}}$ option.

Step 3: Press $\boxed{\text{ENTER}}$ key.

Step 4: Press $\boxed{\text{Y=}}$ key.

Step 5: Enter the function $y=100-x^2$ .

Step 6: Press $\boxed{\text{WINDOW}}$ key. Change the settings to

  $\begin{array}{c}X\min =-4\\ X\max =4\\ Y\min =-30\\ Y\max =110\end{array}$

For better view of graph.

Step 8: Press $\boxed{\text{GRAPH}}$ key.

The result obtained on the screen is provided below,

  

This window has large view, in this the graph points are not clear.

So perfect viewing window of the graph $y=100-x^2$ is option ‘c’ $\left[-15,15\right]by\left[-30,110\right]$ .

Thus, option ‘c’ $\left[-15,15\right]by\left[-30,110\right]$ has a perfect viewing window of the equation $y=100-x^2$ .