Chapter C, Problem 1E

To determine

The most appropriate graph of the equation $y=x^4+2$ in between the given viewing rectangle $\left(\text{a}\right)-\left(\text{d}\right)$.

Answer to Problem 1E

Solution:

The rectangular view $\left[-8,8\right]$ by $\left[-4,40\right]$ gives the appropriate view of the equation $y=x^4+2$ as shown in figure $\left(6\right)$.

Explanation of Solution

Given:

The given equation is,

$y=x^4+2$

The rectangular view given are,

(a) $\left[-2,2\right]$ by $\left[-2,2\right]$.

(b) $\left[0,4\right]$ by $\left[0,4\right]$.

(c) $\left[-8,8\right]$ by $\left[-4,40\right]$.

(d) $\left[-40,40\right]$ by $\left[-80,800\right]$.

Approach:

The graph of the equation using graphing calculator in the given rectangular view is constructed on the basis of the following steps:

Step 1: Press the key $\boxed{\text{Y}=}$ on the graphing calculator and write the given equation, $\boxed{\text{Y}=}$ screen in the graphing calculator is as shown below,

Figure $\left(1\right)$

Step 2: Press the key $\boxed{\text{WINDOW}}$ on the graphing calculator and write the given rectangular view, $\boxed{\text{WINDOW}}$ screen in the graphing calculator is as shown below,

Figure $\left(2\right)$

Step 3: Press the key $\boxed{\text{GRAPH}}$ on the graphing calculator and view the graph obtained, $\boxed{\text{GRAPH}}$ screen in the graphing calculator is as shown below,

Figure $\left(3\right)$

Calculation:

Case (a): $\left[-2,2\right]$ by $\left[-2,2\right]$.

Write the given equation in the graphing calculator,

$y=x^4+2$

Start with the viewing rectangle $\left[-2,2\right]$ by $\left[-2,2\right]$.

Set,

$\begin{array}{l}\text{Xmin}=-2\\ \text{Xmax}=2\\ \text{Ymin}=-2\\ \text{Ymax}=2\end{array}$

The result graph is as shown below,

Figure $\left(4\right)$

The resulting graph is blank. The graph lies entirely outside the view of window. So this rectangular view is not appropriate.

Case (b): $\left[0,4\right]$ by $\left[0,4\right]$.

Write the given equation in the graphing calculator,

$y=x^4+2$

Start with the viewing rectangle $\left[0,4\right]$ by $\left[0,4\right]$.

Set,

$\begin{array}{l}\text{Xmin}=0\\ \text{Xmax}=4\\ \text{Ymin}=0\\ \text{Ymax}=4\end{array}$

The result graph is as shown below,

Figure $\left(5\right)$

The resulting graph only shows some portion of the graph. Some turning point in the graph is not visible in the graph. So this rectangular view is also not appropriate.

Case (c): $\left[-8,8\right]$ by $\left[-4,40\right]$.

Write the given equation in the graphing calculator,

$y=x^4+2$

Start with the viewing rectangle $\left[-8,8\right]$ by $\left[-4,40\right]$.

Set,

$\begin{array}{l}\text{Xmin}=-8\\ \text{Xmax}=8\\ \text{Ymin}=-4\\ \text{Ymax}=40\end{array}$

The result graph is as shown below,

Figure $\left(6\right)$

The resulting graph gives the complete view of the graph. Thus, this rectangular view is appropriate.

Case (d): $\left[-40,40\right]$ by $\left[-80,800\right]$.

Write the given equation in the graphing calculator,

$y=x^4+2$

Start with the viewing rectangle $\left[-40,40\right]$ by $\left[-80,800\right]$.

Set,

$\begin{array}{l}\text{Xmin}=-40\\ \text{Xmax}=40\\ \text{Ymin}=-80\\ \text{Ymax}=800\end{array}$

The result graph is as shown below,

Figure $\left(7\right)$

The resulting graph does not show the $\text{x}$-intercept of the graph as it is visible in the previous graph. So this rectangular view is also not appropriate.

Therefore, the rectangular view $\left[-8,8\right]$ by $\left[-4,40\right]$ gives the appropriate view of the equation $y=x^4+2$ as shown in figure $\left(6\right)$.

Conclusion:

Hence, the rectangular view $\left[-8,8\right]$ by $\left[-4,40\right]$ gives the appropriate view of the equation $y=x^4+2$ as shown in figure $\left(6\right)$.