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 2*a*.

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

kis even, there are 2kpetals on the rose.When

kis odd, there arekpetals 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

kpetals of lengthb-a.The larger set will have

kpetals of lengthb+a.When

kis even, the smaller petals lie between the larger petals.When

kis 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