Clay Bennett

Assignment  1.6

In this assignment, I am required to graph a series of equations and make predictions as to what a later series of equations will look like when graphed.  The equations I am supposed to graph are as follows:

I will start by graphing and commenting on them individually.  The first graph,

, will obviously be a circle.

No real surprise there.

I am not sure what the rest of the graphs will look like.  I would not think that the equations would be circles.  Now I will graph

I did not expect this to be the out come.  I was expecting a closed figure with a shape other than a circle.

Now I will move on to graphing

I really did not have an idea of what to expect from this equation, but this graph does not really surprise me.

Next, I will graph

In this graph, it seems to show that as the odd exponents get larger; its asymptotes approach the -(x)=y line.  I will show more on this a little later.  Now I will graph the final equation,

This graph seems to show that the higher the even exponents get the more square-shaped the graph becomes.  Now I am going to graph all of the equations together so I can see more trends.  I am also going to graph the

-(x) = y line.

After graphing all of the equations on the same plane it is much easier to see trends.  As you can see the -(x) = y line is a line of symmetry for the equations with even exponents.  As I stated earlier, the equations with odd exponents has asymptotes that approach the -(x) = y line.  They seem to converge on it sooner as the odd exponents get larger.  The odd equations also seem to be getting closer to the to forming the top half the equations with even exponents.  The equations with the even exponents start with a circle and as the exponents get larger they seem to form a 1 x 1 square.  Each time the exponent increases, its corner gets closer to the point (1,1).  My prediction for the equations for when the exponents are 24 and 25 are that they even exponent will be almost a perfect square and the odd exponent will be very close to being the top half of the even exponent and will almost converge at the -(x) = y line at points (-1,1) and (1,-1).  Now lets graph

As you can see my predictions were correct.  When I began this investigation I did not expect this to be the outcome.  One other thing I stumbled across due to a typing error on my part was what happens when one exponent is odd and the other is even.  Here is a graph of it for no extra charge.

I changed my mind.  If you looked at those last two graphs then it will be \$5.00.  Cash only.  You are on the honor system.