Exploring Polar Equations
- When a and b are equal, and k is an integer, this is one textbook version of the “n-leaf rose."
- Compare with
For various k. What if . . . cos ( ) is replaced with sin ( )?
We will start this investigation by graphing the function of
r = a + b cos (kq)
The following function gives us a cardiod, which is known as the textbook version of the ‘n-leaf rose’. We want to graph the function leaving k as a constant and change the values of a and b. Next, we will graph the function leaving a and b constant and change the values of k. Finally, we will compare the equations by using the different graphs.
Purple r = a + b cos (kq), Red r = a + 2 cos (kq)
Purple: r = a + b cos (kq), Red: r = a - 2 cos (kq)
Purple: a = 1, Red: a = 2, Blue: a = 3
When a and b are constant and k varies
Red: k = 2, Purple: k = 3, Blue: k = 4, and Green: K = 5
Next, we will look at r = b cos (kq)
Purple: K = 2, Red: K = 3, Blue: k = 5, Green: k =4
Basally, k indicates the number of leaf rose the graph creates. When k is an odd integer the graph will create a rose with k petals. When k is an even integer, the graph will create a rose with 2k petals. For instance, if k = 3 the rose will produce a rose with 3 pedals and if k = 4 the rose will produce a rose with 8 pedals.
Finally, we will graph the equations using sine instead of Cosine!
Teal: r = a + b sin (kq)
Purple: r = a + b sin (2q)
Red: r = a + b sin (3q)
Blue: r = a + b sin (4q)
Green: r = a + b sin (5q)
Teal: r = b sin (kq)
Purple: r = b sin (2q)
Red: r = b sin (3q)
Green: r = b sin (4q)
Blue: r = b sin (5q)
The graphs rotate towards the y-axis. The value of k also determines the number of pedals created. We can see that the graphs with sine and cosine are the same with the exception of the rotation.