Chapter 1.9, Problem 10E

To determine

To graph: The quadratic equation $y=\sqrt{8x-x^2}$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=\sqrt{8x-x^2}$ .

Graph:

The graph of the quadratic equation $y=\sqrt{8x-x^2}$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=\sqrt{8x-x^2}$ .

Rewrite the given equation by square both sides:

  $\begin{array}{l}y=\sqrt{8x-x^2}\\ y^2=8x-x^2\end{array}$ ,

Add both side of the equation $4^2$ to make a square of two functions, ${\left(a+b\right)}^2$ .

  $\begin{array}{c}{y}^2=8x-x^2\\ y^2+x^2-8x=0\\ y^2+x^2+4^2-8x=4^2\\ y^2+{\left(x-4\right)}^2=4^2\end{array}$

As the observation says this equation is an semicircle equation.

  $y$ always positive, roots given always positive value.

Either $x-4=0$ or $y-0=0$ :

Hence $x=4$ and $y=0$ .

Steps to plot the graph of the equation $y=\sqrt{8x-x^2}$ 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 $y=\sqrt{8x-x^2}$ .

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,

  

Interpretation:

The equation of the function $y=\sqrt{8x-x^2}$ represents a semicircle.

The equation has a semicircle graph.

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

The equation of circle is ${\left(x-a\right)}^2+{\left(y-1\right)}^2=r^2$ .

The center of circle is either $x-a=0$ or $y-b=0$ .

Thus $x=a$ , $y=b$ .

Thus $y$ is always positive in case of circle then roots give always positive value.

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 $y=\sqrt{8x-x^2}$ 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 $y=\sqrt{8x-x^2}$ .

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 $\left[-5,5\right]$ and $y$ -axis ranges from $\left[0,100\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=\sqrt{8x-x^2}$ 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 $y=\sqrt{8x-x^2}$ .

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

  $\begin{array}{c}X\min =-5\\ X\max =5\\ Y\min =0\\ Y\max =100\end{array}$

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 $\left[-10,10\right]$ and $y$ -axis ranges from $\left[-10,40\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=\sqrt{8x-x^2}$ 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 $y=\sqrt{8x-x^2}$ .

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

  $\begin{array}{c}X\min =-10\\ X\max =10\\ Y\min =-10\\ Y\max =40\end{array}$

For better view of graph.

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

The result obtained on the screen is provided below,

  

This is not the perfect viewing window.

d.

When the $x$ -axis ranges from $\left[-2,10\right]$ and $y$ -axis ranges from $\left[-2,6\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=\sqrt{8x-x^2}$ 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 $y=\sqrt{8x-x^2}$ .

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

  $\begin{array}{c}X\min =-2\\ X\max =10\\ Y\min =-2\\ Y\max =6\end{array}$

For better view of graph.

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

The result obtained on the screen is provided below,

  

This window is the perfect viewing window.

So the perfect viewing window of the graph $y=\sqrt{8x-x^2}$ is option ‘ $d$ ’ $\left[-2,10\right]by\left[-2,6\right]$ .

The perfect viewing window is $\left[-2,10\right]by\left[-2,6\right]$ , option ‘ $d$ ’.