Chapter 1.9, Problem 18E

To determine

To graph: The quadratic equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ .

Graph:

The graph of the quadratic equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ .

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 $x$ .

Rewrite the equation as $y=0$ :

  $\begin{array}{l}x\left(\left(x+6\right)\left(x-9\right)\right)=y\\ x\left(\left(x+6\right)\left(x-9\right)\right)=0\end{array}$

Either $x=0$ or $x+6=0$ , $x-9=0$ .

Hence $x=0$ and $x=-6$ , $x=9$

Now substitute the value of $x=0$ in equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ .

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=0\left(\left(0+6\right)\left(0-9\right)\right)\\ y=0\end{array}$

Now put the value of $x=1$ in the equation, $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ .

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=1\left(\left(1+6\right)\left(1-9\right)\right)\\ y=1\left(\left(7\right)\left(-8\right)\right)\\ y=-56\end{array}$

Substitute the value $x=2$ in the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ :

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=2\left(\left(2+6\right)\left(2-9\right)\right)\\ y=2\left(\left(8\right)\left(-7\right)\right)\\ y=2\cdot -56\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=2\left(\left(8\right)\left(-7\right)\right)\\ y=2\cdot -56\\ y=-112\end{array}$

Substitute the value $x=3$ in the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ :

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=3\left(\left(3+6\right)\left(3-9\right)\right)\\ y=3\left(\left(9\right)\left(-6\right)\right)\\ y=3\cdot -54\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=3\left(\left(9\right)\left(-6\right)\right)\\ y=3\cdot -54\\ y=-162\end{array}$

Substitute the value $x=-1$ in the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ :

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=-1\left(\left(-1+6\right)\left(-1-9\right)\right)\\ y=-1\left(\left(5\right)\left(-10\right)\right)\\ y=50\end{array}$

Substitute the value $x=-2$ in the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ :

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=-2\left(\left(-2+6\right)\left(-2-9\right)\right)\\ y=-2\left(\left(4\right)\left(-11\right)\right)\\ y=-2\cdot -44\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=-2\left(\left(4\right)\left(-11\right)\right)\\ y=-2\cdot -44\\ y=88\end{array}$

Substitute the value $x=-3$ in the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ :

  $\begin{array}{l}y=x\left(\left(x+6\right)\left(x-9\right)\right)\\ y=-3\left(\left(-3+6\right)\left(-3-9\right)\right)\\ y=-3\left(\left(3\right)\left(-12\right)\right)\\ y=-3\cdot -36\end{array}$

Now the value of $y$ is:

  $\begin{array}{l}y=-3\left(\left(3\right)\left(-12\right)\right)\\ y=-3\cdot -36\\ y=108\end{array}$

Observed that when the value of $x$ increases then the value of $y$ decreases slightly.

When the value of $x$ decreases then the value of $y$ increases slightly.

Steps to plot the graph of the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ 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 $x\left(\left(x+6\right)\left(x-9\right)\right)$ .

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

  $\begin{array}{c}X\min =-50\\ X\max =50\\ Y\min =-250\\ Y\max =200\end{array}$

For better view of graph.

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

The result obtained on the screen is provided below,

  

Interpretation:

The equation of the function $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ represents a wave graph.

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

At $x$ -axis $y$ is always zero.

Recall that the graphical approach to solve the equation simultaneously.

Therefore, the equation $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ is symmetric about the $y-$ axis Initially the graph of the function $y=x\left(\left(x+6\right)\left(x-9\right)\right)$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.