## Polar Equations

### Marianne Parsons

By using Graphing Calculator 3.2, we can explore different types of polar equations. A polar equation is the equation of a curve expressed in polar coordinates.

Let's investigate graphs of the following equation, for different values of a, b, and k. Please note that our value for theta must be between 0 and 2pi inorder for the graphs to be closed. See what happens to the graph of the equation, if theta is defined over different intervals.

### Values of a

To investigate different values of a, let b=1 and k=1. Let's see what happens to our graph when a changes.

As the value of a gets larger, our graph looks more and more like a circle.

View the animation as the value of a goes from 0 to 5.

### Values of b

To investigate different values of b, let a=1 and k=1. Let's see what happens to our graph when b changes.

It seems as though with increasing values of b, the loops on our graphs get larger. Notice when b is greater than or equal to 2, the graph actually has two loops. One on the inside, and one on the outside.

View the animation as the value of b goes from 0 to 10.

### Values of k

To investigate different values of k, let a=1 and b=1. Let's see what happens to our graph when k changes. It is easier to view the effect of a changing k value if these graphs are viewed individually.

It seems as though the value of k determines how many "pedals" are graphed. For example, the graph when k=4 above shows four pedals around the origin. These graphs represent what is known as the "n-leaf rose," where n is the value of k. These graphs are drawn when a=b (in this case, both equal 1), and k is an integer.

View the animation as the value of k goes from 0 to 25 and watch as more "pedals" are generated.

Watch what happens when the values of a, b, and k all change together from 0 to 10. See how the "n-leaf rose" is generated this way.

### The n-leaf rose when a=0

Let's compare our previous findings for k when a=0 and b=1. So, our equation now looks like:

Let's now explore this new equation for different values of k and compare the results to the graphs above.

Immediately, we notice the domain and range of our graphs have changed. The graphs generated by this new equation fall between -1 and 1 on the x-axis, and -1 and 1 on the y-axis.

Also, we see other variations between the number and shape of the pedals of this new graph. Here, k does not always represent the number of pedals we are going to have in our rose. When k is even, there are 2k pedals graphed. For example when k=4, 8 pedals are graphed as shown above. So n=2k for our "n-leaf rose."

When k is odd, however, it seems to follow the examples from our original equation. So when k is odd, k directly represents the number of pedals graphed. As shown for k=3, there are three pedals graphed. When k is odd, n=k for our "n-leaf rose." Does this always work for odd and even values of k? Let's look at more values of k to find out.

View the animation as the value of k goes from 0 to 25 for this new equation. Can you see how this animation is different from the one above?

Watch what happens when the values of b, and k all change together from 0 to 10 for our new equation. See how the "n-leaf rose" is generated this way. How is this different from the animation above when b, k, and a changed together?

### What if cos (k theta) was replaced by sin (k theta)?

Let's look at the graphs of this new equation as the values of a, b, and k change. Our equation is now:

How do you think our graphs would change over the same interval for theta? Click to see an investigation of our new equation.