Assignment 11:

Exploring

by

Margo Gonterman

Exploring Values of a, b, k

Case 1: Let a=b

Specifically, let a=2 and b=2.

Let's start by exploring integer values of k.

Note that when k=4 there are 4 "petals" on the flower. When k=2 there are 2 "petals".

In general, when k is an integer and a=b, the graph is called a "k-leaf rose"

What happens if k is not an integer?

The initial graph looks like this. Note that the values of theta are.

However, changing the values of theta to be yields the following graph.

When k is a fraction m/n, notice that the flower has m petals, but the values of theta must be changed toto view the entire graph.

Below is an animation as k varies from -10 to 10. Notice that as k increases, the number of petals increase.

Notice that when a=b=2 and k=6, the graph is a 6 leaf rose with petals that have length 4.

What a=b=4, the graph is a 6 leaf rose with petals that have length 8.

What a=b=0.5, the graph is a 6 leaf rose with petals that have length 1.

Generally, when a=b, the length of the petals of the rose is 2a.

Below is an animation of a 6 leaf rose as a and b vary from -5 to 5. Note that the petals vary in size from 0 to 10.

Case 2: Let a<b

What if a=0?

Notice that when a=0, the petals have length b.

Also, when k is even, there are 2k petals on the rose.

When k is odd, there are k petals on the rose.

Let's see if this holds when a≠0.

Notice that there are two sets of petals. There are 3 larger petals with length 5 and 3 smaller petals within the larger petals that have length 1.

Let's look at another example.

Here again, there are two sets of petals. There are 4 larger petals with length 5 and 4 smaller petals between the larger petals with length 3.

In general, when a<b, the flower will have two sets of petals.

The smaller set will have k petals of length b-a.

The larger set will have k petals of length b+a.

When k is even, the smaller petals lie between the larger petals.

When k is odd, the smaller petals line inside the larger petals.

Below is an animation of a flower with smaller petals of length 1 and larger petals of length 5 as the number of petals, k, varies.

Case 3: Let a>b

When a>b, the petals do not meet at the origin.

Notice the relationship between the two graphs above.

When a>b the largest part of the graph is still a+b and the smallest is a-b.

Below is an animation as k varies.

Below are a few more animations of the behavior of polar equations of the form

A Pretty Flower

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