Assignment 11: Polar Equations

Presented by: Amanda Oudi

Problem: Investigate:

Discussion:

Let’s use Graphing Calculator to investigate the graph of the equation.

Begin by exploring the case when a = b and k is an integer. This type of graph has a specific name, often referred to as the “n-leaf rose” in textbooks.

Example 1:

The graph produced looks like it has two "petals". Let’s keep a = b = 1 and focus on changing the k-value.

Example 2:

Okay, now we see that we have 3 "petals". So, I predict if I change the k-value, then I will have k number of "petals". We can explore that this seems to be the case using Graphing Calculator.

Below are snapshots of when k = 4, 5, 6, and 7:

Now, what would happen if the the a and b values were not equal? I predict that the graphs would be different in shape. Let’s explore with examples.

Example 3:

This graph has a somewhat similar shape to Example 1, but the graph does not pass through the origin. The "petals" are now wider than the ones in previous examples. I think it kind of has a peanut shape to it!

Example 4:

This graph has 4 "petals" because the k-value in this case = 4. So, we see that altering the a and b values does affect the shape of the "petals" and whether the graph passes thru the origin. Explore other examples using Graphing Calculator.

Now let's investigate this case:

Suppose b = 1. Let's see what happens for k = 2, 4, 6.

It seems that the number of "petals" are twice the k-value. For instance, for k = 2, there are 4 "petals". For k = 4, there are 8 "petals" and for k = 6, there are 12 "petals". Does the same happen for odd k-values?

Suppose b = 1. Let's see what happens for k = 3, 5,

Seems as if for odd k-values, the number of "petals" is the same as the k-value.

What happens if we replace cos() with sin()? Here is a snapshot of r = cos() and r = sin(). It seems that the replacement of cos() with sin() changes orientation.

Explore other examples relating to replacing cos() with sin() and make a few conjectures. Why does the orientation change? What do we know about trigonometry that would help us understand why the orientation of these curves is different?

Extensions: What happens if a and b are opposite signs? Can you identify the maximums and minimums in the polar curves explored in this investigation?