Sinje J. Butler

Write-up # 1

The following is the graph of the function

n .

Below is a graph of

n

n

n

and

n .

There appears to be several things occurring as the exponents of

are raised to higher powers.

1.      The graphs of the equations raised to an odd power

n

and

n

are no longer circles.  They open up and stretch out.  This occurs in an equation that is in the format of

for .

The reason is that for any real number value of x, when x is raised to an odd power, there exists a real number y such that when it is raised to that same power there sum is equal to one.  So it can be predicted that

n

will look similar to the graphs of the previous odd powered equations.

This is not the case for the equation

because x^2 and y^2 will always be positive numbers, therefore, x and y are limited to being values from –1 to 1 where they cannot both be equal to one at the same time, both be equal to negative one at the same time,  or equal to one and  negative one at the same time.

2.      When

n

is changed to

n

the curve begins to straighten out in order to take on more of a square shape.  Recall,

the concept of distance on the coordinate plane is based on the Pythagorean Theorem.  The theorem states

where C is the distance from the origin.  We can write

as

where 1 is the distance from the origin.  We can easily conclude that for any values of x and y from –1 to1, where x and y are both not equal to 1 , both not equal to –1, or not equal to –1 and 1 at the same time, the distance from the origin is one.  Thus making the graph a circle.

This is not the case for

where

> 1.

In these equations, the various (x,y) coordinates of a given graph will be different distances form the origin.  The distances from the origin will increase and decrease through various intervals. This gives the graph a more square shape.