Chapter 1.9, Problem 6E

To determine

To graph: The quadratic equation $y=x^2+7x+6$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=x^2+7x+6$ .

Graph:

The graph of the quadratic equation $y=x^2+7x+6$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=x^2+7x+6$ .

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 $x$ .

The method of middle term factorisation is used in the equation $y=x^2+7x+6$ :

  $\begin{array}{c}y=x^2+7x+6\\ y=x^2+6x+x+6\\ y=x\left(x+6\right)+1\left(x+6\right)\\ y=\left(x+1\right)\left(x+6\right)\end{array}$

Now either $x+1=0$ or $x+6=0$ .

So $x=-1$ or $x=-6$ ,

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

The roots and the shape of parabola is cleared.

The graph of the function $y=x^2+7x+6$ .

Substitute $x=0$ , in the equation $y=x^2+7x+6$ ,

  $\begin{array}{l}y=x^2+7x+6\\ y={\left(0\right)}^2+7\left(0\right)+6\\ y=6\end{array}$

Substitute $x=1$ , in the equation $y=x^2+7x+6$ ,

  $\begin{array}{l}y=x^2+7x+6\\ y={\left(1\right)}^2+7\left(1\right)\\ y=1+7\\ y=8\end{array}$

Substitute $x=2$ , in the equation $y=x^2+7x+6$ ,

  $\begin{array}{c}y=x^2+7x+6\\ y={\left(2\right)}^2+7\left(2\right)+6\\ y=4+14+6\\ y=24\end{array}$

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

Steps to plot the graph of the equation $y=x^2+7x+6$ 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 $y=x^2+7x+6$ .

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 $y=x^2+7x+6$ represents a parabola.

The parabola opens upward.

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

Therefore, in the equation, $y=x^2+7x+6$ there is no $x-$ intercepts.

Therefore, the equation $y=x^2+7x+6$ is symmetric about the $y-$ axis Initially the graph of the function $y=x^2+7x+6$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.

a.

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

Steps to plot the graph of the equation $y=x^2+7x+6$ 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 $y=x^2+7x+6$ .

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

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

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 $\left[0,10\right]$ and $y$ -axis ranges from $\left[-20,100\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^2+7x+6$ 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 $y=x^2+7x+6$ .

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

  $\begin{array}{c}X\min =0\\ X\max =10\\ Y\min =-20\\ Y\max =100\end{array}$

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 $\left[-15,8\right]$ and $y$ -axis ranges from $\left[-20,100\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation 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 $y=x^2+7x+6$ .

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

  $\begin{array}{c}X\min =-15\\ X\max =8\\ Y\min =-20\\ Y\max =100\end{array}$

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 $\left[-10,3\right]$ and $y$ -axis ranges from $\left[-100,20\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^2+7x+6$ 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 $y=x^2+7x+6$ .

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

  $\begin{array}{c}X\min =-10\\ X\max =3\\ Y\min =-100\\ Y\max =20\end{array}$

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 $y=x^2+7x+6$ is option ‘c’ $\left[-15,8\right]by\left[-20,100\right]$ .

Thus, option ‘c’ $\left[-15,8\right]by\left[-20,100\right]$ has a perfect viewing window.