Assignment 11:

Polar Equations

**In this assignment
we will be investigating the polar equation:**

**
r = a + b cos (k **
**q****)**

We can begin our investigation by looking at the case when a = b and k is an integer.

This is called the "n-leaf rose".

**
In the above case, a=b=2 and k=3.**

What do you think will happen if a and b are increased?

Click here to see if you are right!

Now Let's Look at what happens when a=b, but we change the value for k

**
In the above case, a=b=2 and k=6**

Are you starting to see a pattern for the value of k?

Click here to see if you are right!

What happens when a=b, but k is NOT an integer?

**
In the above case, a=b=2 and k=10.5**

Do you see how k being a non-integer changes the graph?

Click here to see if you are right!

What happens when a and b are not equal?

**
In the above case, a=7, b=3, and k=7**

**
Do you think you see what
happens when a and b are not equal?**

**
Click
here to see if you are right!**

**
OK ... Now Let's Look at the
Equation**

**
r = b cos (k **
**q****)**

**
In the above case, b=25 and k=10**

**
Do you see how this equation
differs from the first equation?**

**
Click
here
to see if we have noticed
the same thing!**

**
Do you think that if k a
non-integer the resulting graph will have half a loop? Click
here
to see if you are right!**

**
What Do You Think Would
Happen if We Replace cos with sin?**

**
In the above case, b=12 and k=4.5**

**
**NOTICE: When we used cos in the formula when b=12 and k=4.5, the
resulting graph had two half loops. Now when we use sin, there are nine
full loops (the two half loops have been put together).**

What do you think happens when k is an integer?

Click here to check your answer.

** **