The graphs of the equation are known as **roses**
with various numbers of leaves. Below are graphs of the above
equation with **a = 1** and varied values for **k**. It
appears that **k** affects the number of "petals"
of the graph. If **k is
odd**, then there are ** k petals**.
If

ODDk graphs (k = 1, 3, 5, 7)

EVENk graphs (k = 2, 4, 6)

Notice that when **k** is odd, there is
always one petal that "lies" on the y-axis in such a
way that the y-axis goes through the center of the leaf (as if
it were a vein through the center of the leaf). We will call this
petal the "axis petal."

When

k = 1, 5, 9, 13,..., the axis petal is on they-axis. Notice that these are the odd numbers which arepositivecongruent to 1 modulo 4.When

k = 3, 7, 11, 15,..., the axis petal is on thenegativey-axis. Notice that these are the odd numbers which arecongruent to 3 modulo 4.

Glance at the the graphs to verify these conclusions.
Then go on to "What
does **a** do?"