**Assignment 1**

for

The following equations, x

^{2}+ y^{2}= 1 and x^{3}+ y^{3}= 1, produce the below graph:The equations with even numbered exponents, x

^{2}+ y^{2}= 1 and x^{4}+ y^{4}= 1, produce a square like figure with rounded edges, as seen below:The first equation, x

^{2}+ y^{2}= 1, is a circle. As the exponents increase to the next even integer (x^{4}+ y^{4}= 1), the circle appears to be taking on the resemblance of a square.

The equations odd numbered exponents, x

^{3}+ y^{3}= 1 and x^{5}+ y^{5}= 1, produce the following graph:The odd numbered exponent equations create a line containing a bend at around the point (1,1).

As the exponents of both the even and odd equations increase, their respective graphs are taking on sharper features ("corners"). The even equation also appears to fit better into the bend of the odd function as their exponents increase.

The following equations x

^{24}+ y^{24}= 1 and x^{25}+ y^{25}= 1 produce the below graph:The larger the exponents of the equations, the curves in the graph of each equation have much sharper angles that are beginning to look more and more like right angles. This exploration helps us discover the difference between even and odd equations and what we can expect them to look like as the exponents of the respective equations are increased.