Investigation 11

Polar Equations

Charles Meyer



In Investigation 11, I will look at Polar Equations and how changes within the equation changes the look of the "n-leaf rose". I will begin by setting up my equation.

By setting a and b equal to one another and setting k to an integer, I find the following graphs.

k = 3



k = 4



k = 5

Each graph represents something unique about the rose. The variable k gives the rose the number of leaves it has. The variables a and b gives the rose its size. However, when the variable a is removed from the equation, changes occur within the rose in two areas. One is the size of the rose, the other is the number of leaves the rose contains.

The following examples are of different values of k. When k is an odd number, the quantity of leaves of the rose is equal to k, however when k is even, the number of rose leaves is equal to 2k.

k = 3

k = 5

k = 4, number of leaves = 2k = 8

k = 6, number of leaves = 2k = 12


While the cosine function plays a role in the original equation, it would be interesting to see how its counterpart, the sine function affects the rose.

k = 3

At first glance it would appear that no change has occurred to the rose, but pay attention to the placement of the rose on the axis. The sine function forces the rose to have a pedal that is symetric along the y-axis as opposed to the x-axis of the cosine function.


This investigation would be very beneficial to high school algebra 3 and trigonometry students. I believe the knowledge of the changes caused by the variables as well as the changing of the function would help them understand the power of small changes to a function.


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