Chapter 1.9, Problem 9E

To determine

To graph: The quadratic equation $y=10+25x-x^3$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=10+25x-x^3$ .

Graph:

The graph of the quadratic equation $y=10+25x-x^3$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=10+25x-x^3$ .

Rewrite the given equation:

  $\begin{array}{l}y=10+25x-x^3\\ y=-x^3+25x+10\\ y=x\left(25-x^2\right)+10\end{array}$ ,

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 $x$ .

Substitute the value of $x=-3$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=-3\left(25-{\left(-3\right)}^2\right)+10\\ y=-3\left(25-9\right)+10\\ y=-3\times 16+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=-3\times 16+10\\ y=-48+10\\ y=-38\end{array}$

Substitute the value of $x=-2$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=-2\left(25-{\left(-2\right)}^2\right)+10\\ y=-2\left(25-4\right)+10\\ y=-2\times 21+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=-2\times 21+10\\ y=-42+10\\ y=-32\end{array}$

Substitute the value of $x=-1$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=-1\left(25-{\left(-1\right)}^2\right)+10\\ y=-1\left(25-1\right)+10\\ y=-1\times 24+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=-1\times 24+10\\ y=-24+10\\ y=-14\end{array}$

Substitute the value of $x=0$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=0\left(25-{\left(0\right)}^2\right)+10\\ y=0\left(25\right)+10\\ y=10\end{array}$

Substitute the value of $x=1$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=1\left(25-{\left(1\right)}^2\right)+10\\ y=1\left(25-1\right)+10\\ y=1\times 24+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=24+10\\ y=34\end{array}$

Substitute the value of $x=2$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=2\left(25-{\left(2\right)}^2\right)+10\\ y=2\left(25-4\right)+10\\ y=2\times 21+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=2\times 21+10\\ y=42+10\\ y=52\end{array}$

Substitute the value of $x=3$ in equation $y=x\left(25-x^2\right)+10$ :

  $\begin{array}{c}y=x\left(25-x^2\right)+10\\ y=3\left(25-{\left(3\right)}^2\right)+10\\ y=3\left(25-3\right)+10\\ y=3\times 24+10\end{array}$

The value of $y$ is:

  $\begin{array}{l}y=3\times 24+10\\ y=72+10\\ y=82\end{array}$

When the value of $x$ is $-3$ then value of $y$ is $-38$ .

Same as the value of $x$ is $-2$ then value of $y$ is $-32$ .

The value of $x$ is $-1$ then value of $y$ is $-14$ .

The value of $x$ is $0$ then value of $y$ is $10$ .

The value of $x$ is $1$ then value of $y$ is $34$ .

The value of $x$ is $2$ then value of $y$ is $52$ .

The value of $x$ is $3$ then value of $y$ is $82$ .

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

Same as the value of $x$ decreases there is a decrement in the value of $y$ .

Steps to plot the graph of the equation $y=-x^3+25x+10$ 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 $y=-x^3+25x+10$ .

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

  $\begin{array}{c}X\min =-20\\ X\max =20\\ Y\min =-800\\ Y\max =1000\end{array}$

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 $y=-x^3+25x+10$ represents a parabola.

The equation has a wave formed graph.

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

Recall that the graphical approach to solve the equation simultaneously is observe the point where the graph of equation $y=-x^3+25x+10$ intersects the $y$ -axis.

Therefore, the equation $y=-x^3+25x+10$ is symmetric about the $y-$ axis Initially the graph of the equation $y=-x^3+25x+10$ increases when increase the value of $x$ .

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 $y=-x^3+25x+10$ 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 $y=-x^3+25x+10$ .

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 $y=-x^3+25x+10$ 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 $y=-x^3+25x+10$ .

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 $\left[-20,20\right]$ and $y$ -axis ranges from $\left[-100,100\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=-x^3+25x+10$ 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 $y=-x^3+25x+10$ .

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

  $\begin{array}{c}X\min =-20\\ X\max =20\\ Y\min =-100\\ Y\max =100\end{array}$

For better view of graph.

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

The result obtained on the screen is provided below,

  

This is the perfect viewing window.

d.

When the $x$ -axis ranges from $\left[-100,100\right]$ and $y$ -axis ranges from $\left[-200,200\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=-x^3+25x+10$ 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 $y=-x^3+25x+10$ .

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

  $\begin{array}{c}X\min =-100\\ X\max =100\\ Y\min =-200\\ Y\max =200\end{array}$

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 the perfect viewing window of the graph $y=-x^3+25x+10$ is option ‘ $c$ ’ $\left[-20,20\right]by\left[-100,100\right]$ .

The perfect viewing window is $\left[-20,20\right]by\left[-100,100\right]$ , option ‘ $c$ ’.