Assignment Eleven

Here, we will compare and . Before doing so, we will explore r = bcos(k theta).

Predict how the graph of r =b cos(k theta) will look. Click here to check your assumption. Now, Let's examine the polar equation when b = 0, 1, 2, 3, 4, where k =1.

It seems as though that the values of b effect the diameter in each circle. In other words, b = diameter, when k = 1. When compare to r = a + b cos(ktheta), it seems, here, that the graph was translated over the x-axis.

Now, we should experiment with various values of k. Consider the equation when k = 2, 3, 4, 5, 6, 7 and b = 1. How do these graphs compare to where k was equal to the integers, 0 through 6?

From the graph above, it is hard to investigate what happens as the values of k change, where b = 1; however, it seems as though the domain and range goes from -1 to 1 for each value k. We need to separate the graphs to further examine the values of k. Let's see what happens when k is even and odd.

Notice: Separating the graphs based on the odd and even values of k has helped us to investigate the polar equations even more. By examining the graphs, what conclusions can you draw? There seems to be some difference between the number of n-leaf rose when k is even and odd and b remains equal to one. When k is even integer, the number of pedals double the value of k. In other words, if k= even number, then there will be 2k pedals or leaves on the rose, 2k-leaf rose. When k is an odd integer, the number of pedals equals the value of k. Thus, we will have a n-leaf rose. This is quite similar to polar equations that we investigated in the beginining.

Click here to see an animation of different values for k.