Assignment #1

# By Amber Krug

Problem:

After graphing:

x2 + y2 = 1

x3 + y3 = 1

x4 + y4 = 1

x5 + y5 = 1

What is expected for the graphs of:

x24 + y24 = 1

or

x25 + y25 = 1?

First, letŐs analyze the original graphs:

x3 + y3 = 1

x4 + y4 = 1

x5 + y5 = 1

The graphs begin to show that as the exponents of x and y increase the curves centered about the origin begin to develop into right angles.  The original circle begins to develop into a square as the exponents increase in even numbers.  See the graph below of increasing even exponents:

x2 + y2 = 1

x4 + y4 = 1

As the exponents increase in odd numbers, the original circular break in the line y = -x becomes more square.  See the graph below of increasing odd exponents:

x3 + y3 = 1

x5 + y5 = 1

x7 + y7 = 1

According to these observations, the graph of

x24 + y24 = 1

should resemble a square, and the graph of

x25 + y25 = 1

should have a break resembling half of a square in the line y = -x.  Below is our predicted graph:

x24+y24 = 1

x25+y25 = 1

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