Leaves

by Zack Kroll


Investigate the function:

Note: Explore what happens to the graphs when a and b are not equal. When a and b are equal and k is an integer, this is a version of the "n-leaf rose."

When analyzing polar coordinates, a frequent approach is to produce fun and exciting images. These are some of the examples of graphs that can be created by exploring the function:

 

 
 

                            

 

                           

 

As stated, when a and b are equal and k is an integer the graph is what is described as an “n-leaf rose”. This means that the value of k is equal to the number of pedals on the graph.

                              

                                                                                   

 

The images above show the graphs of the function . As we stated before, when the parameter k is an integer the number of pedals in the graph is equal to the absolute value of k. This applies as long as a and b are equal. The images above represent one set of a certain number of pedals. The graph of is one set of 4 pedals and is 6 pedals.

 

If we maintain that a and b would happen if k were not an integer? Does the idea of the “n-leaf rose” still apply?

In fact, the graph of the function does not maintain the same qualities that it does when k is an integer. As we can see in following graphs:

                  

 

The graphs of the functions above appear to be much different than those where k is an integer. However, upon closer examination we determine that they are more similar than they originally appear. 4.5 is equivalent to 9/2. When we look more intently at our graph we realize that there are in fact 2 sets of 9 pedals. The same rule applies to the graph

.

                 

                                  

 

We can see a similar pattern in the graphs of the these two functions as well. The value 2.8 = 28/10 = 14/5. Therefore, there are 5 patterns of 14 pedals in the graph of the function .

Now what happens when a and b are not equal? How does that affect the graph of the function?

      

                                                           

 

When the values of a and b are not equal we create graphs such as the one above. The graph above shows two sets of three leaves, one smaller set inside of the larger one. There are multiple other variations of these types of graphs and a variety of ways that they can be explored.