Assignment #1 ~ Is it a Circle or a Square?

By Kevin Mylod

 In high school geometry or trigonometry classes we learned that the equation of a circle is:

Below is the graph of the circle.

What happens if we raised the x and y variables to the third power? Will the graph reveal a circle? Using the software Graphing Calculator 2.7, we can generate the graph of the equation,

Hmmm, not much of a circle! In fact, it looks as if this graph only generates 1/2 of a circle.........or does it? A second question to ponder would be: will we get what seems to be a 1/2 circle if we raise the x and y variables to an odd power? Using the same software, let's generate the graphs for the following equations:

In fact, as the odd exponent increases, the graph apears to be taking on the shape of not a half-circle but a half-square.

With the observation made concerning odd exponents, will our whole circle turn into a whole square if we investigate the equation x^n + y^n = 1 where n is an even integer? Below are the equations where n = 4, 6, and 8.

As expected, the circle seems to be transforming into a square as n becomes larger and larger. This begs the question, will our circle eventually become a square if we continue to increase the value of the exponent? What if we take our x and y variables to the 24th power? the 25th power? What do you think will happen then?

What if we took our variables to the 100th power? the 101st power? Do you think the graphs will look similar, different or exactly the same as the when they were raised to the 24th and 25th powers? If it looks exactly the same, why have our graphs reached a limit on its structure? To ponder these questions further, check out the graphs when n=100 and 101 in Graphing Calculator 3.1. To see an animated version of the graphs from n = 0 to n = 100, click here.

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