Graph the following: x^{2} + y^{2
}= 1

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

Now examine the graph: x^{3} + y^{3}
= 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
x^{4} + y^{4} =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 x^{5} + y^{5 }= 1.

The graph now looks very similar to x^{3}
+ y^{3} = 1. It is a line with an interruption around
(-1, 1) and (1, -1).

Now lets examine all the graphs together.

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

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

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

x^{5} + y^{5 }=
1

Notice the points at x = 1, -1 and y = 1, -1.
What can you assume from the graph of x^{24} + y^{24} =1?
Or x^{25} + y^{25}
=1?

Lets try even higher exponential values.

x^{100} + y^{100} =1

x^{101} + y^{101}
=1

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

Return to MBauers 6680 Home Page