ASSIGNMENT ONE
Problem #6
 

This assignment explores the graphs of X^a + Y^a = 1 and X^b + Y^b = 1 where a is an even number greater than or equal to two and b is an odd number greater than or equal to three.

The graph of X^2 + Y^2 = 1 is a circle with radius one as shown.

The graph of X^4 + Y^4 = 1 shows a rounded square.

Note that the distances from the origin to the x and y axis are one.



Continuing with the even exponents, we graph X^6 + Y^6 = 1.

We now observe the graph of X^3 + Y^3 = 1 below:

Note the semi-circle of radius one with open ends connected at the 'ends' to the identity equation.

Graphing the next odd exponent, we see the graph of X^5 + Y^5 = 1.

We now have a rounded semi-square with open ends connected to the identity equation with the distance from the origin to the x and y axis also as one.


If we continue with the odd exponents, the graph of X^7 + Y^7 = 1 is as follows:

The angles of the semi-square are less rounded this time, but the distance from the origin to the angles is one as in the previous graph. This figure is also connected at the open ends to the identity equation.



Now suppose a is 24 . The previous illustrations with even exponents suggest that this graph will be a square with the distance from the origin to the angles of the square equal to one.

Suppose b is 25. The previous illustrations with odd exponents suggest that this graph will be a semi-square with the distance from the origin to the angles of the semi-square equal to one, and that the open ends will be connected to the identity equation.

To test our predictions, we will graph both equations => X^24 + Y^24 = 1 and
X^25 + Y^25 = 1.

What is difficult to see in this particular version of the graphs is that the red square is partially hidden under the green semi-square. Knowing that, we can see that our predictions are verified.


To drive home this lesson to my students, I would have them graph several samples of their choosing for both even and odd exponents, first separately, then simultaneously. Next, I would also ask them about a = 0 and b = 1. I would also have them predict negative even and odd exponents before graphing, and then ask them to verify their prediction. They would answer such questions as 'Was your prediction verified?', 'If not, what did happen?', "Was there another pattern of prediction?' and so forth.