Exploration of Polar Equations
We are going to explore what happens when we change various values in the following equation:
First, we need to see what the most basic graph looks like.† Suppose a = b = k = 1.
Letís explore the values of a while b and k are constant.
As the absolute value of a increases, the radius of the circle grows.
However, something interesting occurs for rational values of a between -1 and 1.† The graph is transitioning into various sized pedals.
We can conclude the value of a affects the actual size and shape of the graph.
Now we can explore what happens when we change the k value in the equation
We will maintain b as a constant of 1 and allow 0£q£2p.
We see a circle of radius 0.5 and centered at (0.5,0).†
What if we try k=2?
We have a 4-pedaled flower centered at the origin with pedal unit length.
What if we try k=4?
Now we have an 8-pedaled flower centered at the origin with pedal unit length.
If we try other even-numbered integers, we will see similar results- that is 2k-pedaled flower centered at the origin with pedals unit length.
For instance, letís consider k=10.† We can predict that we will see 20 pedals of unit length centered at the origin.
What happens for odd-integer k-values?
It does not appear to follow the 2k pattern, but it maintains unit length pedals and centered at the origin.
Letís find out if the pattern holds true for k=5.†
So far our conclusions are correct.† Letís try k=13.
We can conclude for even-numbered integer values of k, we will yield a result of 2k pedals, all unit length and centered at the origin.† However, for odd-numbered integer values of k, we will see k pedals which are also unit length and centered at the origin.†
What if k is not an integer?†
Suppose k is only ďpartĒ of a whole number, or a rational number.
For values of k between 0 and 1, we see only part of the flower.† Instead we see various pedal shapes, but not in entirety.
However, if we set the values of k to be between 0 and 5 (that is, including rational and irrational numbers), we can watch the graph transition between the different k-pedaled flowers.
Letís consider what would happen if we used the sine function rather than cosine.† Let us use the following equation for the remainder of the exploration:
Just like with cosine, letís first look at b=k=1.† For comparison, cosine is in the color blue while the sine equation is in gray.
So far, these graphs appear to be very similar except for where the center is located.† For the sine graph, the center is on the y-axis at (0,0.5).
Letís try various values of k:
We again see the trend of 2k pedals, but the flower is simply rotated 45 degrees.†
We again see k-pedals with odd integer values of k.
Lastly, letís verify what will happen with rational values of k.
Not surprisingly, we see similar results as the cosine equation.†