**Assignment 1: Exploring Graphs
with a degree of greater than or equal to 2
**

We
will be exploring many graphs of the following equation:

First
let’s graph this equation when n = 2.

When
we graph this equation with n = 2, , we noticed that it looks like a circle. This makes sense
because in mathematics, we have a unit circle that is a circle with a radius of
1. Thus since _{} for all x, and since the
reflection of any point on the unit circle about the x or y-axis is also on the
unit circle, the above equation holds for all points (x, y) on the unit circle
not just those in the first quadrant.

Now
let’s see what happens when n = 3? Will it have a similar shape as the graph of
when n = 2?

Wow, this graph is shaped completely different than when n =
2. This is becoming interesting. The graph maintains a similar shape around
0<x<1 but then curves out along the y = -x line. Before I visit this idea
further, let’s graph some more equations to see if they follow the trend of the
first two graphs.

Let’s see what the graph will look like when n = 4? Will it
look more like n = 2 or n = 3? But then I will become even more curious to see
what n = 5 looks like.

Let’s view these four equations together graphed on the same
Cartesian plane and see if we can notice a pattern and/or relationship.

What
patterns are visible from these graphs?

We
noticed that as n increases and is an even number, the graph looks like a
circle. It starts as a circle but continues to stretch out as n increases.
However, as n increases and is odd, the graph follows the trend of n = 3. As a
result we see extended curves that don’t close and then it wraps along line and
is asymptotic to the y = -x line. It’s so interesting to see that when n is odd
the graph almost becomes the y = -x line except near the region 0< x <1
it stretches and curves around that line.

We
are guessing that if we continue in this manner, for even powers of the
equation, we will get a shape between a circle and square and for the odd
powers we will get a curve passing through (0, 1) and (1, 0). Also each curve
is above the previous curve between x = 0 and x = 1, but below the curve for
other values of x.

So,
let’s explore further. What do we expect to see when n = 24 and n = 25?

So
as we observe these different graphs, we began to ask ourselves why do odd
graphs during this specific interval have an asymptote of y = -x? We noticed
that the graph will always be less than 1 but greater than 0 because you are always
subtracting the independent variable from 1; consequently the value of y
(dependent variable) will always be less than 1.

So
let’s test this assumption. Consider the point (3, -3). Let’s replace the x
value with 3 and the y with -3. We notice that it does not satisfy the equation
anywhere because the y will be less than 1.

Explore
the animation to the right to see if the pattern holds true for larger values
of n.

Notice
that if n is not an integer what happens to the graph.

In
conclusion, we can discover more intriguing results by changing the values of
n.

Last
but definitely not least, let’s look at what happens when we graph the
following equation:

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