EMAT 6680

Assignment 1

Stephanie K Lewis

6. Graph (a) x

^{2 }+ y^{2 }= 1

Graph (b) x^{3 }+ y^{3} = 1

Graph (c) x^{4} + y^{4 }= 1

Graph (d) x^{5 }+ y^{5 }= 1

What do you expect for the graph of x^{24} + y^{24}
= 1 or x^{25 }+ y^{25} = 1?

I believe that the graph of x^{24} + y^{24} = 1 will
have a similar pattern to that of figure c, while the graph of x^{25
}+ y^{25} = 1 will have a similar pattern to that of figures
d. Let’s see!!

The graph of x^{24} + y^{24} = 1 looks like the figure
below.

Using the laws of exponents, x^{24} + y^{24} = 1 can
be written as (x^{6})^{4} + (y^{6})^{4}=1.
Note that x^{4} + y^{4 }= 1 and (x^{6})^{4}
+ (y^{6})^{4}=1 resemble one another. Below, both equations
are graphed simultaneously.

While the graphs are different, they share a close resemblance to one
another and share common x- and y-intercepts. The graph of x^{4}
+ y^{4} = 1(in red) is rounder than x^{24} + y^{24}
= 1(in purple). I graphed several equations of the type x^{n}
+ y^{n} = 1 using even numbered
exponents. Each graph had likeness to a circle and/or quadrilateral .The
graph, at times, seemed to be a hybrid of the two. The higher the even
exponent, the more like a quadrilateral (square), the graph resembled.
The lower the even exponent, the more round the corners became. At x^{100
}+ y^{100} = 1, the graph was a square. As the even numbered
exponent increased or decreased respectively, differences were noted in
the sharp or rounded nature of the arcs or corners or each graph.

Below, the graph of x^{25 }+ y^{25} = 1 (in purple)
and x^{5 }+ y^{5 }= 1 (in red) have been drawn simultaneously.
Note the similarities.

The general pattern of each graph is the same. After graphing several
equations of the type x^{n}
+ y^{n} = 1, where the exponents
are the same and odd, the graph of the equation always had the same pattern.
The differences can be seen in the rounded or sharp nature of the concavity
of the graph.