Explorations With Some Exponential Functions

by Oktay Mercimek

I will investigate some exponential functions like


, where c is from 2 to 25.

When c is 2, It is well known that is a circle.


If c is 3, then function is  and its graph is



For , graph is


For , graph is

After these graphs, I have a feeling about a difference between odd c values and even c values.

I want to look at some even c values first.

Do you imagine what will happen when c is 1000?

It is not difficult to say it will seem like a square.

Let's look at some odd ones



I will add x=-y (yellow line) to show you some similarity.

little closer look to graph

When c is odd , x>0 and y>0, It tend to behave similar to ones that c is even.

For x<-1 ,y>1 and for x>1, y<-1 , It is very near to x=-y line.

After this investigations,We have learned that similar exponential functions don't have similar graphs and being odd or even is a key to understand the behavior of exponential function.