In this essay, we will make use of Graphing Calculator 3.1 to examine a particular equation. We will set the parameters and check what happens to the graphs as we change those parameters. The equation to be looked at is :
Hopefully, given this equation, we will be able to see what happens to the graph as we change the value of n, ranging from negative numbers to positive numbers. Other things we could look at might be to change the numerator of each fraction, or possibly change the value of what the equation is set equal to. These are ideas we might look at later, but, most likely, their effect would be minimal.
Before we begin looking at individual n values, let's look at a movie where n ranges from -50 to 50. this will give us an idea of what we are looking for when we begin discussing the individual n values. See movie.
Let's begin by looking at the most basic equation, letting n=1 :
Looking at the graph, we get a clear picture to see that when n=1 in our equation we have a basic rational function, with one vertical asymptote and one horizontal asymptote. We should have expected this graph to be the result.
I guess the next step would be to increase n , letting n=-1:
So we see, that when n=-1 we get a line having slope m=-1 and y-intercept b=1. Again, this is something that any algebra student should be able to figure out for himself by simply solving for y and manipulating the equation. Let's move on.
Now, let's look at the case for n=-2. However, before we look at the graph, let's discuss what we think it might look like. We know by exponential properties that terms having negative exponents in the denominator can be flipped to the numerator with positve exponents. So, if we let n=-2, we could rewrite our equation as follows:
This is the equation of a circle, with radius 1 and center at the origin. Let's look at the graph given n=-2 and see what we expect:
Now, let's examine what happens if we let n=0 . Why don't we work this equation through algebraically:
We can see that when we let n=0, we get the invalid equation 2 = 1. In this case, we get no graph, because it is invalid at every point. In fact, this is the only value of n in which we would get no graph.
Now let's look at the cases where n=-100 and n=100 . These are cases which most of us would not be familiar with, so let's begin by looking at their graphs together on the same coordinate plane:
Looking at the graph above, we can see the case for n=-100 (in blue) and the case for n=100 (in green). This is a very interesting result to think on. When we let n=-100 it appears that we have created a square having sides of length 2, with vertices at (1,1), (-1,1), (-1,-1), and (1,-1).This would be curious by itself, however, when we put it together with the case of n=100 we can clearly see that the 4 branches of the second graph all appear to be right angles with their vertices on the vertices of the square.
This leads to a very interesting insight for the cases where n= an even number. Let's begin by discussing the negative evens first.
For the case of n= negative evens, we see that from n=-2 (the circle) to n=-100 we get closer and closer to a square. Let's look at a few of these on the same xy-plane (see graph):
We can see from the graph, that given n=-2 (green), n=-10 (purple), n=-30 (red), and n=-100 (blue), we are in fact getting closer and closer to a perfect square.
Now, let's exmaine the cases of n=positive evens. We have seen that when n=100 we get the four branches of a rational function, with the evrtices appearing to be at right angles. Well, recall when we let n=negative even numbers, we had a circle getting closer and closer to a square as n got smaller. I expect to see something similar for n=positive even numbers. However, in this case when n is small I expect to see curved branches that get more and more angular as n gets large. Let's look at the graphs of n=2, n=4, n=10, and n=100 on the same axes(see graph):
So, we can clearly see that what we speculated to be true, was so. In fact, we can now make a genaralization about the graph of this equation as the absolute value of n gets large. We can say that as the absolute value of n gets large we progress from curved graphs to angular graphs. This would be true regardless of whether or not we were letting n be odd or even. Just to show that this would be true, let's quickly take a look at the case for n= negative odd numbers and n= positive odd numbers. Let's first look at the negatives as n gets large (see graph):
Now let's look at the positives as n gets large (see graph):
So, we can see that for the cases where n is odd we have the same pattern of going from curved lines to angular lines. In both of the last 2 figures, the black graphs represent the largest n value.
For this investigation there are innumerable things to look at. Perhaps you could begin where I have left off to see further examinations of this equation.