Exploring Polar Equations

By Princess Browne

 

Investigate

Note:

- 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 ( )?

r = b sin (kq)

 

We will start this investigation by graphing the function of

r = a + b cos (kq)

 

a=b=k=1

 

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.

 

 

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