**Assignment
#1**

**Graphing**

By

**Michelle
Nichols**

Look
at the following graphs of the form: x^{n} + y^{n} = 1

**x +
y = 1**

**x ^{2}
+ y^{2} = 1**

**x ^{3}
+ y^{3} = 1**

**x ^{4}
+ y^{4} = 1**

**x ^{5}
+ y^{5} = 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 x^{24}
+ y^{24} = 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…

**x ^{24}
+ y^{24} = 1**

The assumption was correct!

Now how about x^{25} + y^{25} = 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…

**x ^{25}
+ y^{25} = 1**

Correct again!

So given the previous graphs, the equation x^{n} + y^{n}
= 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