Chapter 4.1, Problem 47E

(a)

To determine

To sketch: a graph of the given function using graphing calculator and to identify the symmetry of the graph.

Answer to Problem 47E

The graph of $y=x,y=x^3,y=x^5$ is symmetric about origin and the graph of $y=x^2,y=x^4,y=x^6$ is symmetric about $y$ axis.

Explanation of Solution

Given information:

Given function

  $y=x^n$

When $n=1,2,3,4,5,\text{and}\text{6}$

The graph of a function is symmetric with respect to the $y$ axis if it 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:

Input the function $Y_1=X$ using keys X and press GRAPH key.

  

Input the function $Y_1=X^2$ using keys $X{x}^2$ and press GRAPH key.

  

Input the function $Y=X^3$ using keys $X\wedge 3$ and press GRAPH key.

  

Input the function $Y=X^4$ using keys $X\wedge 4$ and press GRAPH key.

  

Input the function $Y=X^5$ using keys $X\wedge 5$ and press GRAPH key.

  

Input the function $Y=X^6$ using keys $X\wedge 6$ and press GRAPH key.

  

From the above six graphs, it is very clear that graph of $y=x,y=x^3,y=x^5$ is symmetric about origin and the graph of $y=x^2,y=x^4,y=x^6$ is symmetric about $y$ axis.

(b)

To determine

To find: the symmetry the graphs of $y=x^{10}\text{and}y=x^{11}$ .

Answer to Problem 47E

The graph is symmetric with respect to origin.

Explanation of Solution

Given information:

Given function

  $y=x^n$

When $n=1,2,3,4,5,\text{and}\text{6}$

The graph of a function is symmetric with respect to the $y$ axis if it 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:

Because the exponent of $y=x^{10}$ is even, so graph is symmetric with respect to $y$ axis.

Because the exponent of $y=x^{11}$ is odd, so graph is symmetric with respect to origin.