Exploring Exponents

By

Princess Browne

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

x^{3
}+ y^{3} = 1

x^{4
}+ y^{4} = 1

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 will start by graphing the first four equations
above to find a pattern. From the given equations we will notice that the
equations are always equal to 1 and the exponents of x and y are equal in each
equation. These factors may be useful when we compare the different equations.

Graph 1

Purple

Graph 2

Purple

Red

Graph 3

Green

Graph 4

Green

Blue

Graph number 1 represent the equation x^{2 }+
y^{2} = 1. As we can see this graph forms a circle. Graph number 2 is
the graph of x^{2 }+ y^{2} = 1 and x^{4 }+ y^{4}
= 1. As we compare the two graphs we will see that the even number exponents
create a shape that starts of as a circle and as the exponents increases the
corners of the circle are extended out into a shape of a square. On the other
hand, graph number 3 represent the equation x^{3 }+ y^{3} = 1. This graph forms a line
with a curve in the middle. Graph number 4 is the graph of x^{3 }+ y^{3} = 1 and x^{5
}+ y^{5} = 1. When we compare the two graphs we will notice that
as the odd exponent increases the curve in the middle of the graph extends to
form a square like shape. From the following equations and graphs I will guess
that the equations with the even exponents starts of as a circle and as the
exponents increases the graph will end up looking like a square shape.

Next,
we will look at x^{n}+y^{n}=1. Let n equal all even numbers,
starting with 2 and increasing by 2.
We will graph n=2, n=4, n=6, and n=8. If we compare graph number 5 with
our previous graphs we will see that our hypothesis is true.

Graph 5

Purple

Red

Blue

Green

Graph
number 5 confirms our prediction that
when n is an even exponent the graph will form a circle and as n increases the corners of the circle
are extended to form a square shape. Next we will look at the equation when the
exponents are odd. If we look at x^{o}+y^{o}=1, where o stands
for all odd numbers starting with 3 and increasing by 2. We will graph the
functions of o=3, o=5, o=7, and o=9. Using the observation from our previous
graphs, we will hypothesize that the equations with odd exponents will start of
as a line with a curve or hump in the middle. As the exponent increases the curve would extend to from a
square shape.

Graph 6

Green

Blue

Purple

Red

Graph
number 6 shows that as the odd exponent increases the middle of the line
changes from a curve shape to a square shape. If we compare the odd exponents
graphs with the even exponents graphs we can conclude that as the exponents
increases both graphs ends up with square like corners.

Now
we will look at x^{24}+y^{24 }= 1 and x^{25}+y^{25 }=
1 to see if the following conclusions are true. Graph number 7 shows the graph
of x^{24}+y^{24 }= 1.

Graph 7

Purple

This
graph shows that our hypothesis is correct and that as the exponent increases
the corner of the graph extends to from the shape of square. Next, we will look
at the graph of x^{25}+y^{25}=1 to also prove our hypothesis
about the odd exponents.

Graph 8

Blue

Graph
number 8 also shows that our assumption is correct. As the odd exponent increases,
the portion of the graph that is located in quadrant I forms a square like
shape.

Lets
try x^{100} + y^{100 } = 1 and x^{101} + y^{101 } = 1

x^{100} + y^{100
} = 1

Graph looks like a
square!

x^{101} + y^{101
} = 1

Middle looks like a
square also!