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.  Let’s 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.  Let’s 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