**Roxanne
Kerry**

I chose to do my first writeup on the following:

After
graphing and analyzing the first equations, we can see that if we
graph the even exponents, the graphs are transforming from the unit
circle to a square of sides 1, a “unit square”, it seems. By
graphing x^{24} + y^{24 }= 1 alongside the first two
graphs with even exponents, my hypothesis seemed to be confirmed (see
below figure). The equation with squared exponents is the unit
circle, represented by the blue graph, and the equation with the 4^{th}
power exponents is represented by the brown graph, and the equation
with the 24^{th} power exponents is represented by the green
graph.

I
would think that the equations with odd exponents would be somewhat
of an inverse or complement to the graphs with even exponents, and in
a way, they are, but not exactly as I had expected. It makes sense
that the graphs with odd exponents would have the ends of the graphs
approaching different directions, since any odd polynomial graphs do.
These graphs with odd exponents look sort of like a linear function
with slope of one going through the origin (or x=y) except for the
area around the origin, where it “bubbles up” as if it were
trying to wrap around the the unit circle. Just as the equations
with even exponents expanded from the unit circle towards a unit
square as the powers increased, with these equations with odd
exponents consist of a graph resembling x=y with a middle “bump”
around the origin wrapping around the unit circle approaching the
unit square as the powers increase here as well (see below). The
equation with exponents to the 3^{rd} power is represented by
the black graph, the equation with exponents to the 5^{th}
power is represented by the orange graph, and moving up to the
equation with exponents to the 25^{th} power the graph seems
to be getting closer to wrapping around the perimeter of the unit
square, and is represented by the graph in light blue.

I
think this would be a good activity for high school students to
explore exponents in a different way than polynomial or exponential
equations. I would consider also taking this exploration one step
further and see what happens when the equation is equal to a value
other than one. I would expect for a similar event to happen as did
with the equations equal to one, but for x^{n}+y^{n }=
t, I would expect the graphs with even exponents to approach a square
centered at the origin with side lengths of t as the values of n got
larger (expanding from a circle centered at the origin with a radius
of √t ) and I would expect the graphs with odd exponents to
approach a graph like x=y with a “bubble” approaching a square
going around the limit of the square formed by the even exponents,
similar to the original problem. The graph of x^{2}+y^{2
}= t did give a graph of a circle with radius of √t, as
expected, however the limit of these even exponent graphs still
approached the unit square, as demonstrated below. The odd exponent
graphs had a similar trend, approaching a graph resembling the line
x=y approaching a “unit square bubble” around the origin, rather
than a “t square bubble” as I predicted (also shown below).

The graph of functions with even exponents

The graph of functions with odd exponents

So why does this happen with these graphs? I can't say for sure, although the conclusion we can draw from this is that the limit of any x