Investigating equations in the form of


n=even positive integer

Consider the following equations of the form , where n is an even positive integer, and their graphs:

Notice when the indices changed from 2 to 4, the corners have become sharper and more squared off. If we continue this when n = 6, we notice the corners becoming even more square like and less rounded.

If this pattern continues, we would expect for the equation that the corners would be sharper and the graph would look similar to that of a square.

This leads to an interesting question: Do we conclude as n gets larger the corners will eventually square off and the graph of the equation will become a square? Consider the graph of .

The graph suggests that of a square. We can zoom in on one of the corners to uncover the true nature of the corners. Magnifying the corner allows one to verify that indeed the corners are still rounded and not square like as the graph had originally suggested.

We can continue this process for n being very large only to conclude the same results:

As n continues to grow as an even positive integer, the graph may suggest a square. Yet, further analysis verifies that the corners will always be rounded.

Think about this:

For this picture to represent a square, the corner would have to be represented by the ordered pair (1,1).

Consider the general equation : .

If we let n=any positive even integer and x = y =1, the equation does not hold true.

Therefore, (1,1) can never be a solution to the equation. Thus the graph will never represent that of a square.


n=ODD positive integer

Consider the following equations of the form , where n is an odd positive integer, and their graphs:

Similarly to Case 1, notice as n increases the graph is becoming sharper and square-like at the corners.

Focusing on the following graphs where n continues to increase, the same pattern will emerge as in Case 1; that is, the corners will become sharper and eventually emerge as 90 degree angle (or so it seems).

We can magnify the top right corner of the graph to actually see what is occuring and to disuade our original assumptions about the corners forming 90 degree angles.

As in Case 1, to obtain a half-square, the points (-1,1), (1,1), and (1,-1) would have to be solutions to the equation under investigation. Indeed, they are NOT solutions. Therefore, we can continue to increase n as large as possible, regardless of n being odd or even, and the graph may look as if it is transforming into squares or half-squares. In reality, however, when the graph is magnified, the corners are curved, not forming 90-degree angles.

This leads to the discussion that one must always consider the mathematics behind the technology being used to lead students to discover and generalize. If one had not magnified the corners, students would have easily assumed that the graph on the computer monitor says it is a square therefore it is a square. The use of technology in the mathematics classroom embarks on a new era of investigation and discovery. Yet, teachers and students must be responsible for further questioning, deliberation, and justification before making final conclusions.