Chapter 3.2, Problem 85E

To determine

To sketch: the graph of the function $y=x^2,y=x^3,y=x^4,andy=x^5$ on the same viewing window. The conjecture about the graph of $y=x^{100},andy=x^{101}$ on the same interval.

Explanation of Solution

Graph of the quadratic function $y=x^2$ , for $-1\le x\le 1$ is as follows:

  

Graph of the function, $y=x^3$ , for $-1\le x\le 1$ is as follows:

  

Graph of the function, $y=x^4$ , for $-1\le x\le 1$ is as follows:

  

Graph of the function, $y=x^5$ , for $-1\le x\le 1$ is as follows:

  

That means, the graph of $y=x^n$ is having a general shape, when $n$ is odd, and the graph of $y=x^n$ is also having a same general shape, when $n$ is even. However, as the degree of $n$ becomes larger, the graphs become flatter around the origin and steeper elsewhere.

Therefore the graph $y=x^{100}$ , on the same interval, is having the same shape as $y=x^n$ , when $n$ is even.

It is a flatter around the origin that the smaller degree.

The graph of $y=x^{100}$ is as follows:

  

The graph $y=x^{101}$ , on the same interval is having the same shape as $y=x^n$ , where $n$ is odd.

It is flatter around the origin that the smaller degree.

The graph of $y=x^{101}$ is as follows:

  

Some values which satisfies the equation is given below:

$x$	$y=x^{100}$
0	0
$\frac{1}{2}$	${\left(\frac{1}{2}\right)}^{100}$
$-\frac{1}{2}$	${\left(-\frac{1}{2}\right)}^{100}$

And

$x$	$y=x^{101}$
0	0
$\frac{1}{2}$	${\left(\frac{1}{2}\right)}^{101}$
$-\frac{1}{2}$	${\left(-\frac{1}{2}\right)}^{101}$