*Brandie Thrasher*

__Functions__

When
we examine the graph of the function x^{2 }+ y^{2 }= 1, we see
that it is the equation for a circle.

Next,
we examine the function x^{3} + y^{3} = 1, and we find that due
to its exponential value, its graph looks like a linear equation with a slight
bulge.

As we increase the numerical value of our exponent, we start to see a slight trend, as the degree increases (by an even number), the graph begins to expand into a shape resembling a square. When the degree is odd, we see that the shape slightly shifts downward and the bulge angling more into a square.

Next, I took a look at the parameters to see how the graphs
of the functions would change, and noticed that the graphs of the even
exponents have a solution of x = ±1,
y = 0; where the odd exponents have solutions of (1, 0) and (0, 1). The even
powered equations mold into a shape resembling a square (starting from a circle
at x^{2} + y^{2} = 1), where the odd powered equations continue
to converge towards the solution set.

In conclusion, when graphing the equation x^{24} + y^{24}
= 1, with its even exponent, it should follow the pattern and to have a
formation of a shape similar to a square.
The equation x^{25} + y^{25} = 1 should follow the pattern
of the odd exponents and should eventually overlap the even equations (due to
the large quantity of the exponent) especially at points closest to their
solutions.