Exploration in Polar Equations

Hello sports fans! And welcome to another episode of "Math stuff you never never knew anything about". Today we will be exploring the exciting and interesting world of polar equations. (Try to contain your excitement.)

But seriously, what we have here is another exploration where I try to make sense of a new type of equation. I'll be looking for the relationship between the given variables and the graph they produce. If you liked me in "Explorations with tangent circles", or "Parametric Equations and Linear Functions", you are sure to enjoy this as well.

 

Today we will be looking at equations in the form of This is known as a polar equation. Polar equations can be recognized as they express r as a function of theta Lets begin our exploration by choosing values of a, b and k, such that a and b are both equal to 3, and k is 4. The graph of this equation is:

As you can see, this is the traditional rose graph many of us saw in high school algebra II/ trig. In the case above, our rose has 4 leafs which reach a maximum of r = 6 . Our first glance at conjecture might be that k represents the number of leaf for given graph. Let's change the k in our equation to 5, and see if we get five leafs.

What do you know, it looks like our conjecture holds. Let us now look at varying a and b and see what they do to the graph.

Our next equation will beWe believe that based on our conjecture that the graph will have four leafs. Let's see...

Uh oh! We don't have four leaves, we actually have 8! The larger leaves have a maximum of r = 7, the shorter leaves have a maximum of what appears to be r = 3. We need to rethink our hypothesis. Why would we have more leaves when the values for a and b are not equal? It would seem that a and b have something to do with the leaves. Let's go back to our earlier equations and graphs.

 

When k was 4, and a and b were both 3, we know that we had four leaves that reached a maximum of r = 6. when k was five and a and b were both three, we had five leaves that had a maximum of r = 6. If we add the values for a and b we get six as well. Perhaps this is a relationship we can test out with our latest graph.

When k was 4, and a = 2 and b = 5, we got eight leaves, four of which had a maximum of r = 7. Seven is the sum of a and b, so our hypothesis holds. But it doesn't explain why did we also get four leaves whose maximum was 3. Aha! - the difference of b and a is three. Perhaps that is where the other leaves come from. Looking back it makes sense, that if a an b are the same, that their difference would be zero. That would explain why the other leaves were not there.

Okay, so our newest conjecture for the formis that a+b will represent the length of k leaves of the rose, and b - a will represent k leaves of the rose. Let's look at another equation and make some predictions.

For our next example, let's takeAccording to our conjecture, our graph should have six leaves of maximum length 5, and six leaves of maximum length 1. Survey says...

We are correct!

There is one thing that bothers me, what would happen if a was greater than b. This would mean that the difference would be negative. How could we have negative maximums? Let's look at what might happen when we graph

This is not good. It appears that we have four leaves with a maximum of r = 6, but these leaves are not closed like the other graphs. They do not go through the origin as the others do. Instead, they have a minimum value of r = 2. Two is the difference of a - b in our equation, so the minimum values makes logical sense. If we look at another equation, we should now be able to predict its shape completely.

How about the equation Our graph should have five leaves that have a maximum vale of r = 8, and should have minimum values of r = 2. Let's see what happens...

 

One again, we are correct!

In summary, we now can predict the graph of any polar equation in the form of.

 

 

 

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