Chapter 1.9, Problem 5E

To determine

To graph: The quadratic equation $y=x^4+2$ in a graphing window.

Explanation of Solution

Given information:

The quadratic equation $y=x^4+2$ .

Graph:

The graph of the quadratic equation $y=x^4+2$ can be sketched in the cartesian plane,

Consider the quadratic equation, $y=x^4+2$ .

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 $x$ .

Substitute the value of $x=1$ in equation $y=x^4+2$ :

  $\begin{array}{c}y=x^4+2\\ y={\left(1\right)}^4+2\\ y=1+2\\ y=3\end{array}$

Substitute the value of $x=2$ in equation $y=x^4+2$ :

  $\begin{array}{c}y=x^4+2\\ y=2^4+2\\ y=16+2\\ y=18\end{array}$

Substitute the value of $x=3$ in equation $y=x^4+2$ :

  $\begin{array}{c}y=x^4+2\\ y=3^4+2\\ y=81+2\\ y=83\end{array}$

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

Steps to plot the graph of the equation $y=x^4+2$ 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 $x^4+2$ .

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

  $\begin{array}{c}X\min =-10\\ X\max =10\\ Y\min =-20\\ Y\max =400\end{array}$

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 $y=x^4+2$ represents a parabola.

The parabola opens upward.

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

Recall that the graphical approach to solve the equation simultaneously is observe the point where the graph of equation $y=x^4+2$ intersects the $y$ -axis at the points $\left[0,2\right]$ .

Therefore, in the equation, $y=x^4+2$ there is no $x-$ intercepts.

Therefore, the equation $y=x^4+2$ is symmetric about the $y-$ axis Initially the graph of the function $y=x^4+2$ increases when increase the value of $x$ ,the graph is a parabola shaped graph.

When the $x$ -axis ranges from $\left[-2,2\right]$ and $y$ -axis ranges from $\left[-2,2\right]$

1. their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^4+2$ 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 $x^4+2$ .

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

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

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 $\left[0,4\right]$ and $y$ -axis ranges from $\left[0,4\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^4+2$ 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 $x^4+2$ .

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

  $\begin{array}{c}X\min =0\\ X\max =4\\ Y\min =0\\ Y\max =4\end{array}$

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 a perfect viewing window.

c.

When the $x$ -axis ranges from $\left[-8,8\right]$ and $y$ -axis ranges from $\left[-4,40\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^4+2$ 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 $x^4+2$ .

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

  $\begin{array}{c}X\min =-8\\ X\max =8\\ Y\min =-4\\ Y\max =40\end{array}$

For better view of graph.

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

The result obtained on the screen is provided below,

  

This is the perfect viewing window.

d.

When the $x$ -axis ranges from $\left[-40,40\right]$ and $y$ -axis ranges from $\left[-80,800\right]$ their graph obtained in the screen is provided below.

Steps to plot the graph of the equation $y=x^4+2$ 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 $x^4+2$ .

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

  $\begin{array}{c}X\min =-40\\ X\max =40\\ Y\min =-80\\ Y\max =800\end{array}$

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 $y=x^4+2$ is option ‘c’ $\left[-8,8\right]by\left[-4,40\right]$ .