Assignment 11:Polar Equations

By Dorothy Evans

 

Letís investigate the polar equations below

 

Purple

Red

 

 

 

 

Here is the first case where a = 1, b = 1, and k = 1.As you can see itís ok, but letís see when we change k.

Here a = 1, b = 1, and k = 2.As you can see a flower like shape emerges.Notice the length of the red petals are 1 and the length of the purple are 2.Also notice the number of petals.At this point there are 4 red petals and 2 purple.Letís see what happens next.

 

Now a = 1, b = 1, and k = 3.Notice the shape now has 3 petals that are still 1 long in the red and 2 long in the purple.So what do you think k changes?Letís try another.

Here a = 1, b = 1, and k = 4.It would appear when k is even we get 2k petals in the red graph and k petals in the purple graph.Now letís see what happens as we change a.

 

In this case a = 1, b = 2, and k = 4.So, it would appear we now have 2k petals in both the red and purple, but in the purple graph the petals are different sizes.In the red graph the petals are b long and in the purple graph the petals are k-a and a in length.

In this case a = 1, b = 3, and k = 4.The flower got bigger.Notice the larger purple petals are 4.So maybe the length of the larger purple leaves is actually a+b instead of k-a.Also it appears the red are still b in length and the smaller purple petals are not a, but b-a in length.Letís try another and see if our equations hold true.

 

Now a = 1, b = 4, and k = 4.From our previous investigation we hypothesized:

# of petals: 2k

Size of red petals = b = 4true

Size of larger purple petals = a+b = 1+4 = 5 true

Size of smaller purple petals = b Ė a = 4-1 = 3 true

Looks like we got the pattern figured out.Now for those of you wondering how I verified the length of the smaller purple petals because none of them are on an axis.Itís simple, the good old Pythagorean Theorem.I took an estimate of my (x,y) coordinates and calculated the distance to the origin.Now in this case I used an approximation.How would you prove it?For that matter how would you prove any of the distances we hypothesized?

 

To explore these graphs further click here for the Graphing Calculator file.