Polar Equations


Summer Brown

On this page, you will find investigations of the graph of:

I will be exploring the graphs of the above equation when a and b are equal and k is any integer.

Click here to view examples where a and b are not equal.


Let's begin with a=b=1. Below are graphs when k = 0, 1, 2.

When k = 0, the graph is a circle with a radius of 2.

When k = 1, the graph forms one "leaf." I will denote the length of a leaf by the distance from the origin to the farthest point from the origin on the leaf. In this example, the length of the leaf is 2 units.

When k = 2, the graph forms two leaves, each with a length of 2 units.

Let's look at some more examples when k = 3, 4, 5.

When k = 3, the graph forms 3 leaves, each with a length of 2 units.

When k = 4, the graph forms 4 leaves, each with a length of 2 units.

When k = 5, the graph forms 5 leaves, each with a length of 2 units.


We can begin to make some conjectures about the graphs of the polar equation .


What about negative values of k? Take a look at the examples below to conclude that negative values of k produce the same graph as positive values of k. Below are examples when k = 7 and -7. View an animation of the graph of , where a,b =1 and k varies between -10 and 10. Negative values of k produce the same graph as their positive counterparts.


What happens if we change the values of a and b?

Below are examples of graphs when k = 3 and a,b = 1, 2, and 3.

As a and b increase, the leaves of the rose stretch our further. In each case, the length of each leaf is equal to a + b. If we compare the above graphs to the graphs when a,b = -1, -2, and -3 ; we see that the orientation of the graph is reversed (at least when k=3.)

One leaf now lies on the negative side of the x-axis instead of the positive side. In other words, the graph has been rotated by radians. Yet another way to describe the difference is to say that the graph has been reflected over the y-axis. Also, we need to modify our conjecture that the length of a leaf is a + b. The length is in fact, .

If we look at more examples of the graphs comparing a,b with -a,-b, we discover two cases. The first is when k is odd. The graph is reflected over the y-axis. An example of this case was shown above where k=3.

The second case is when k is even. The graph does not change when the values of a and b change signs. The negative values produce the same graphs as the positive values. An example of this is illustrated below where k=6. You see only one graph, because the negative values of a and b produce the same graph as the positives. This makes sense, because, if we tried to reflect the graph over the y-axis, we would obtain the same graph. This will hold true for all cases where k is even, because there are an even number of leaves, thus making the graph symmetric across the y-axis.

To summarize my findings:

Investigate the graphs of

Investigate the graphs of

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