Polar Equations

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

• The graph forms k leaves of a "rose."
• The length of each leaf is equal to a + b or 2a. Cases where a and b are not equal can be used to disprove that the length of each leaf is 2a. Click here to see these counterexamples, where a and b are not equal. Thus, the length of each leaf is a + b. (This will be modified later when negative values of a and b are considered.)
• The leaves are equally spaced apart, with each leaf rotated by radians.
• When k is even, the graph is symmetric with respect to both the x-axis and the y-axis.

To summarize my findings:

• k designates the number of leaves of the rose
• The length of each leaf is equal to
• The leaves are equally spaced apart around 2 radians, with an angle of radians between each leaf
• When k is even, the graph is symmetric with respect to the x-axis and to the y-axis
• When k is odd, the graph is symmetric with respect to the x-axis only
• When k is odd, negative values of a and b reflect the graph across the y-axis