Brandie Thrasher

Functions

When we examine the graph of the function x2 + y2 = 1, we see that it is the equation for a circle.

Next, we examine the function x3 + y3 = 1, and we find that due to its exponential value, its graph looks like a linear equation with a slight bulge.

See graph

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.

See graph

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 x2 + y2 = 1), where the odd powered equations continue to converge towards the solution set.

See graph

In conclusion, when graphing the equation x24 + y24 = 1, with its even exponent, it should follow the pattern and to have a formation of a shape similar to a square.  The equation x25 + y25 = 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.

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