This write up is for problems #2 in Assignment 11.

Investigating Polar Equations

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Investigation of

After we investigated the polar equation by varying "a" and "k", were given a similar polar equation except for the addition of b. At this point in our investigation, we will vary the "b" using the mentioned values for "a" and "k". Again the sine equation will remain symmetric about the y-axis.

When all values of a, b, and k are 1, we have the following graph.

The graph now has a distinct shape. There appears to be a smaller curve inside the larger curve, which occurs at the origin. Will the curve behave like the previous curves if we change the value of "b" to 2? Let's see.

The smaller curve appears to disappear. However, there is still a noticeable bend at the origin of the graph.

Now, let us vary both "k" and "b".

When a =1, k = 2, and b = 2, we get the following graph.

Here, we can observe two pedal-like curves that are connected at the origin. Now, let's vary "a" to be 2, also.

When a = 2, we get 4 pedal-like curves that join each other at the origin. Two of these pedals are smaller than the original two pedals. Now let's see what occurs when we change a = 5.

The pedals have expanded as with the previous changes of "a". Therefore, if we change "k", then the number of pedals should change if the same principles for "k" apply in this polar equation.

Let's try k = 5 while a = 2 and b = 2.

By observation one can see that there are 5 large pedals and 5 small pedals which are inside the larger ones.

Finally, we will vary all the values of a, b, and k.

Here a = 2, b = 1, and k = 5. What conclusions can you draw from this?

Improvement on the Theory of the Number of Pedals

The number of pedals is based upon "k". After careful investigation and research, we found the following information.

• If k is even, then the number of pedals is 2k.
• If k is odd, then the number of pedals is also 2k. This is the case although it appears that only one set of pedals exist in some cases. These pedals are actually laying one on top of the other. Furthermore, this is more apparent when you add and vary "b" in both the sine and cosine equations. One set of pedals shows up smaller than the other set of pedals.

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