For this assignment we will investigate the polar equation:

where we will vary a, b and k.

Lets look at a four different cases.

Please note that when you are looking at these graph pay attention to the scale.

1. When a and b are equal, and k is an integer, this is a textbook version of the "n-leaf rose"

Let a = 1, b = 1 and k = 2

Let a = 1 , b=1 and k = 3

Let a =1, b = 1 and k = 4

Notice that for k=2 there are two leaves, for k=3 there are 3 leaves and for k=4 there are four leaves. Lets see what would happen if we let k = 24.

Sure enough we got 24 leaves :)

 

2. If a < b and k is an integer.

Let a = 1, b = 2 and k = 3

The purple rose is the first case we looked at the the red rose is the current case. When we changed b from 1 to 2 we now have an inter "n-leaf rose"

Lets look at what happens as b increases

As b increases the inter rose is getting larger by 1. If b = 1 then the inter rose was at 0 on the x-axis, if b=2 then the inter rose was at 1, if b= 3 then the inter rose is at 2 and so on.

 

3. If a > b and k = integer.

Let a = 2, b=1 and k=3

Now that a is larger than b we still have three leaves; however, it does not meet at the origin anymore. Lets look at if a gets larger and b stays the same.

Let a =3, b = 1 and k = 3

Now that a is a little larger than b it looks as if the rose is getting further away from the origin. Lets make a even larger. Isn't this exciting:)

Let a = 10, b = 2 and k = 3

Let a = 20, b = 2 and k = 3

Let a = 50, b = 2 and k = 3

The further away a and b are from each other the further away the rose is from the origin.

 

4. Lets now replace cos with sin, lets take a look at an example from each of the previous cases. I have included the cos graph with the sin graph to make the comparsion easier.

Let a = 1, b = 1 and k =3

Green:

Purple:

 

Let a = 2, b = 1 and k = 3

Red:

Purple:

 

Let a = 2, b = 2 and k = 3

Red:

Blue:

 

For each of the previous cases the sin graph has just rotated from the cos graph.

 

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