This write-up will involve forms of the equation:
where n is a natural number greater than 1.
The graphs of this equation form two distinct patterns. The first pattern is for when n is even and the second pattern when n is odd.
The first case to look at is when n=2. By substitution, that gives the equation which is the equation of a circle with a radius of 1. The graph of this equation is below.
As n increases, the graph starts to change shape. Instead of a circle, it starts to look like a square with rounded corners. This can be seen in the graphs of the following equations: (red),(purple),(blue).
As can be seeen by the graphs, as n increases the graphs look more and more like a square with sides of 1 unit. Now, what would the picture look like when n=24? Will it look like a square or will the corners still appear to be rounded? If they still appear to be rounded, when will the graph look like a square or will the rounded courners ever go away? Let's predict that the graph will still have rounded corners. Now, let's look at the graph of .
The graph shows the prediction to be correct that the corners are still rounded. Now, use graphing calculator to make the graphs of . This allows the program to run make the graph of any value of n. Doing this allows us to see that when n = 90 the graph does appear to be a square.
When n=odd integer greater than 1, the pattern is completely different. From the graphs above, one would think that these graphs would also be either circular or polygonal. This is not the case as can be seen when by the graph of .
This graph does not come back to itself as the graphs of the first case did. This graph has tails, curvature between x=0 and x=1, and it appears that there is an asymptote. Now, the question becomes what happens to this graph as n increases. In order to get some idea, we will look at the following cases in order to make a prediction: (purple),red,(blue).
From the above graphs, it appears that as n increases that the curvature of the graph between x=0 and x=1 decreases. That would lead to the prediction that as n increases the curve would go more and more towards a corner instead of a curve. So now, let's see find the graph of the equation when n=25.
As predicted, the curvature has again decreased. Now, one wonders if the corner appears to form a right angle when n is around 90 as it does in the even case. Which it appears to do when looking at the graph of which is below.
Click here for a movie of .