Chapter 2.2, Problem 53E

Solution

$\left(a\right)$

To Graph: Use the graphing calculator to graph the function $y=x^n$ , when $n=1,2,3,4,5$ and $6$ . In each case identify the symmetry graph has.

Given:

The graph of a function is symmetric with respect to the y-axis if for each point $\left(a,b\right)$ on the graph, $\left(-a,b\right)$ is also a point on the graph. The graph of a function is symmetric with respect to the origin if for each point $\left(a,b\right)$ on the graph, $\left(-a,-b\right)$ is also a point on the graph.

Calculation:

The given function is $y=x^n$ , when $n=1,2,3,4,5$ and $6$ .

On substituting the values of $n$ in the given function, we get:

  $\begin{array}{l}y=x^1=x\\ y=x^2\\ y=x^3\\ y=x^4\\ y=x^5\\ y=x^6\end{array}$

Sketch the above graph using the graphing calculator and check for the symmetry as follows:

Graph:

  

  

Interpretation:

The graph of the function $y=x$ is symmetric about the origin as for point $\left(1,1\right)$ on the graph, $\left(-1,-1\right)$ is also a point on the graph.

The graph of the function $y=x^2$ is symmetric about the y-axis as for point $\left(1,1\right)$ on the graph, $\left(-1,1\right)$ is also a point on the graph.

The graph of the function $y=x^3$ is symmetric about the origin as for point $\left(1,1\right)$ on the graph, $\left(-1,-1\right)$ is also a point on the graph.

The graph of the function $y=x^4$ is symmetric about the y-axis as for point $\left(1,1\right)$ on the graph, $\left(-1,1\right)$ is also a point on the graph.

The graph of the function $y=x^5$ is symmetric about the origin as for point $\left(1,1\right)$ on the graph, $\left(-1,-1\right)$ is also a point on the graph.

The graph of the function $y=x^6$ is symmetric about the y-axis as for point $\left(1,1\right)$ on the graph, $\left(-1,1\right)$ is also a point on the graph.

$\left(b\right)$

Predict the symmetry of graphs of $y=x^{10}$ and $y=x^{11}$ . Explain the reasoning and then confirm the prediction by graphing.

Given:

The graph of a function is symmetric with respect to the y-axis if for each point $\left(a,b\right)$ on the graph, $\left(-a,b\right)$ is also a point on the graph. The graph of a function is symmetric with respect to the origin if for each point $\left(a,b\right)$ on the graph, $\left(-a,-b\right)$ is also a point on the graph.

Explanation:

The given functions are $y=x^{10}$ and $y=x^{11}$ :

To check the graph of the equation is symmetric about the y-axis, replace $x$ with $-x$ :

If $f\left(-x\right)=f\left(x\right)$ , the graph is symmetric about y-axis and if $f\left(-x\right)=-f\left(x\right)$ graph is symmetric about the origin.

For $y=x^{10}$ :

  $\begin{array}{l}f\left(x\right)=x^{10}\left[\text{Replace }x\text{ with }-x\right]\\ f\left(-x\right)={\left(-x\right)}^{10}=x^{10}\\ \therefore f\left(-x\right)=f\left(x\right)\end{array}$

Hence, the graph of $y=x^{10}$ will be symmetric about the y-axis.

For $y=x^{11}$ :

  $\begin{array}{l}f\left(x\right)=x^{11}\left[\text{Replace }x\text{ with }-x\right]\\ f\left(-x\right)={\left(-x\right)}^{11}=-x^{10}\\ \therefore f\left(-x\right)=-f\left(x\right)\end{array}$

Hence, the graph of $y=x^{11}$ will be symmetric about the origin.

Sketch the above graph using the graphing calculator and check for the symmetry:

Graph:

  

Interpretation:

The graph of the function $y=x^{11}$ is symmetric about the origin as for point $\left(1,1\right)$ on the graph, $\left(-1,-1\right)$ is also a point on the graph.

The graph of the function $y=x^{10}$ is symmetric about the y-axis as for point $\left(1,1\right)$ on the graph, $\left(-1,1\right)$ is also a point on the graph.

$\left(c\right)$

Determine the value of $n$ for which the graph of $y=x^{-n}$ is symmetric with respect to the y-axis as well as the values of $n$ for which the graph is symmetric with respect to origin.

For value of $n=3$ , the graph of $y=x^{-n}$ is symmetric about the origin.

For value of $n=2$ , the graph of $y=x^{-n}$ is symmetric about the y-axis.

Given:

If $n$ is a positive integer, determine the value of $n$ for which the graph of $y=x^{-n}$ is symmetric with respect to the y-axis as well as the values of $n$ for which the graph is symmetric with respect to origin.

Explanation:

The given function is $y=x^{-n}$ :

To check the graph of the equation is symmetric about the y-axis, replace $x$ with $-x$ :

If $f\left(-x\right)=f\left(x\right)$ , the graph is symmetric about y-axis and if $f\left(-x\right)=-f\left(x\right)$

graph is symmetric about the origin:

Since the graph of odd function is symmetric with respect to origin, take value of n as odd number:

Let assume $n=3$ :

Substitute the above value in the given function is $y=x^{-n}$ :

  $y=x^{-3}$

  $\begin{array}{l}f\left(x\right)=x^{-3}\\ f\left(-x\right)={\left(-x\right)}^{-3}\left[\text{Replace }x\text{with}-x\right]\\ f\left(-x\right)=-f{\left(-x\right)}^{-3}\end{array}$

Hence, for $n=3$ the graph of $y=x^{-n}$ is symmetric about the origin.

Since the graph of even function is symmetric with respect to y-axis, take value of n as even number:

Let assume $n=2$ :

Substitute the above value in the given function is $y=x^{-n}$ :

  $y=x^{-2}$

  $\begin{array}{l}f\left(x\right)=x^{-2}\\ f\left(-x\right)={\left(-x\right)}^{-2}\left[\text{Replace }x\text{with }-x\right]\\ f\left(-x\right)=f\left(x\right)\end{array}$

Hence for value of $n=2$ , the graph of $y=x^{-n}$ is symmetric about the y-axis.