Assignment #1

 

Graphing

 

By

 

Michelle Nichols

 

 

Look at the following graphs of the form: xn + yn = 1

 

x + y = 1

 

x2 + y2 = 1

 

x3 + y3 = 1

 

x4 + y4 = 1

 

x5 + y5 = 1

 

 

Notice the following about each of the graphs:

       The exponent n determines whether our graph will be continuous or closed.

       n as an even number produces a closed graph.

       n as an odd number produces a continuous graph.

       As n increase, the form, whether closed or continuous, forms a square-like shape around the origin.

 

Look at the equations together on one graph.

 

 

 

 

What do you think the graph will look like for the equation x24 + y24 = 1?

Given the data collected from the previous graphs, one can assume the graph will be closed, and near the shape of a square. Lets see

 

 

x24 + y24 = 1

 

The assumption was correct!

 

Now how about x25 + y25 = 1? For this graph, if the path follows as before with odd n characters, it should be a continuous graph, curved much like a square around the origin. Lets see

 

 

x25 + y25 = 1

 

Correct again!

 

So given the previous graphs, the equation xn + yn = 1, follows the form of a line for odd n, with a square-like shape around the origin, and for even n, a closed square-like shape around the origin. To see this graph animated in Graphing Calculator, follow this link: animation

 

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