Polar Equations

by Morgan Guest

In this exploration, we will see what happens to the graph when a, b, and k are changed.

First, let's look at the graph when a = 1, b =1, and k = 1 and compare it to the graph when a = 1, b =1, k = 3 and when a = 1, b = 1, k = 8. We can see from the two graphs below that k determines how many pedals the graph has. The first graph has only 1 pedal while the second graph is a flower shape with 3 pedals. Finally, the third graph is a flower shape with 8 pedals. Notice that the y-intercepts are equal to (0,1) and (0,-1) so the y-intercept is equal to ±a/b.

r = 1 + cos (θ)

r = 1 + cos(3θ)

r = 1 + cos(8θ)

Notice that the y-intercepts are equal to (0,1) and (0,-1) so the y-intercept is equal to ±a/b. Let's see if this holds true when a = 2, b = 2, k =3, when a = 2, b = 2, k =8, and when a = 2, b = 2, k =6. You can see in the graphs below that the graphs have a flower shape with 3, 8, and 6 pedals, respectively. When k = 3, the y-intercepts still equal ±a/b where a = b = 2. If you explore further, you will see that this will hold true when k is odd. However, notice that when k = 8, the y-intercept is the tip of the pedal at ±a/b. If you explore further, you can see that this holds true when k is a multple of 4. When k = 6, the y-intercept does not equal ±a/b. As a and b increase, the shape of the graph stays the same, but the size of the graph increases along with a and b.

r = 2 + 2cos(3θ)

r = 2 + 2cos(8θ)

r = 2 + 2cos(6θ)

Now let's turn our attention to the equation r = bcos(kθ). Below are the graphs when b = 1 and k = 3 and k=4. As you can see the number of pedals each graph has is 3 and 8 respectively. If you do more investigation, you will see that when k is odd, the number of pedals is equal to k. When k is even, the number of pedals in the graph is equal to 2k.

r = cos(3θ)

r = cos(4θ)

Now, let's say b =2. Notice in the following two graphs, the same rule about the number of pedals has remained constant. When k = 3, there are 3 pedals and when k = 6, there are 12 pedals. The size of the graph has increased along with b.

r = 2cos(3θ)

r = 2cos(6θ)

Finally, let's compare the last two graphs with the graphs of r = 2sin(3θ) and r = 2sin(6θ). The number of pedals remains the same as above but the location of the pedals has rotated about the origin. Now, in the graph r = 2sin(3θ), the y intercept is at -2 instead of at (0,0). The y-intercept in the graph r = 2sin(6θ) is now (0,0) instead of at ±b where b = 2.

r = 2sin(3θ)

r = 2sin(6θ)

The overall shape of the graphs is the same as in the graph with cosine, but the pedals are rotated about the origin by 90 degrees.