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

Investigating Polar Equations

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Investigation of

When a = 1 and k = 1, the polar equation will produce the following graph.

At this point we can observe that the circular shape has a center at 1 on the polar scale. The sine equation is symmetric about the y-axis. If we first allow "a" to vary we can observe some of the following situations.

When a = 3 and k = 1, the circular graph has expanded.

The circular graph is now located between 0 and 6. Since "a" is multiplied by 2, then the graph is expanding by 2 each time.

Now, we will vary "k" and hold "a" constant at 1.

When k > 1, the graph no longer has a circular shape. It now begins to appear to have pedals. Such as the following graph where k = 2.

We notice that the graph has 4 pedal-like curves. Can we conjecture that the number of pedals is k times 2? When k = 5, the graph does not have double the number of pedals, but it only has 5 pedal-like curves.

Therefore, one can observe that when k is an odd number, the number of pedals will be equal to k. However, when k is an even number, the number of pedals will be 2 times k.

Finally, we will vary both "a" and "k".

When a = 2 and k = 5, we would expect the graph to have 5 pedals and to be expanded more than the graph above. (See graph below)

We were correct. There are 5 pedals and the graph has been expanded from a magnitude of 2 to 4.

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