EMAT 6680 Assignment 1

Problems and explorations with graphing functions and relations.

x2 + y2 = 1

Melissa Bauers Summer 2002

Graph the following: x2 + y2 = 1

Notice the circle has a radius of 1. It is a regular shape.

Now examine the graph: x3 + y3 = 1

The equation has now changed from a circle to a line with a rounded shape between the coordinates (-1,1) and (1,-1).

Let's increase the exponents again and graph x4 + y4 =1

The graph now exhibits the shape of a square with s=1.

How will an exponent of 5 changed the graph?

Let's examine it. Graph x5 + y5 = 1.

The graph now looks very similar to x3 + y3 = 1. It is a line with an interruption around (-1, 1) and (1, -1).

Now lets examine all the graphs together.

x2 + y2 = 1

x3 + y3 = 1

x4 + y4 =1

x5 + y5 = 1

Notice the points at x = 1, -1 and y = 1, -1. What can you assume from the graph of x24 + y24 =1? Or x25 + y25 =1?

Lets try even higher exponential values.

x100 + y100 =1

x101 + y101 =1

We can now justify an assumption that as the exponents get higher the image becomes sharper.