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
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