The graph of
with a = 1 and k = 1. Notice this defines a radius 1 circle
centered at (1,pi/2). 


The graph of
with a = 1 and k = 2. This is a 4 petal flower centered at
the origin with petals that appear to be 2 in length along their line of
symmetry. 


The graph of
with a = 1 and k = +/ 3. This is a 3 petal flower centered
at the origin with petals that appear to be 2 in length along their line
of symmetry. Notice the negative inverts the original flower. 


Continuing to change the values for k will just continue
to add petals to the flowers observed. When k is an even number,
there are always 2k as many petals. The first petal counterclockwise
from the positive xaxis is at (2pi / 4k) radians and the
next petals are (2pi / 2k) radians counterclockwise from there. When
k is an odd number, there are k petals. One petal will be
on the negative y axis, with the rest (2pi / k) radians counterclockwise
from there. A negative value for k only has an observable effect
on odd values of k, as seen in the final example above. This flips
the image of the graph with positive k over the horizontal axis.

Next to look at the graphs of the form .
First consider when a = 1 and k = 1. Notice this defines a
radius 1 circle centered at (1, 0). As you may expect, there is a clockwise
rotations of pi/2 compared with the same function that replaces the cosine
with sine (see the green circle above). 


The graph of
with a = 1 and k = 2. This is a 4 petal flower centered at
the origin with petals that appear to be 2 in length along their line of
symmetry. Again, there is a clockwise rotations of pi/2 compared with the
same function that replaces the cosine with sine (see the yellow flower
above). 


The graph of
with a = 1 and k = +/ 3. This is a 3 petal flower centered
at the origin with petals that appear to be 2 in length along their line
of symmetry. Again, there is a clockwise rotations of pi/2 compared with
the same function that replaces the cosine with sine. However, notice that
both graphs do not appear. In fact, both are present and the same inversion
about the xaxis occurs as seen in the sine function above. However,
because of the pi/2 rotation, this inversion maps the flower right back
to itself. 


Continuing to change the values for k will just continue
to add petals to the flowers. This behavior observed in the cosine functions
summarily acts the same as the sine examples above. When k is an
even number, there are always 2k as many petals. The first petal
counterclockwise from the positive xaxis is at 0 radians of counterclockwise
rotation and the next petals are (2pi / 2k) radians counterclockwise
from there. When k is an odd number, there are k petals. One
petal will be on the positive x axis, with the rest (2pi / k)
radians counterclockwise from there. A negative value for k only
has no observable effect on values of k, as seen in the final example
above. 