Success! The petal has a radius of 44, as we suspected.

Success! Our hypothesis seems to hold.

Our results are as we hypothesized. You can click here for an animation showing a similar phenomenon as seen for the sine equation.

We can
immediately see that changing k = 4 to k = -4 and k = 5 to k = -5 flips
the graph about the x-axis. This is consistant with our knowledge
of the fact that sin starts increasing and cosine starts decreasing
when moving positively away from 0, however both sine and cosine are
decreasing when moving negatively away from zero, so the picture would
be shifted counter-clockwise by a positive k value and clockwise by a
negative k value.

In evaluating the above pictures, it also seems that k has similar effects on this equation that it had on our previous equations. It seems that when k is an odd integer, there is a k point star in the middle, but when k is an even integer there is a 2k point star in the middle. Let's explore this phenomenon and see what happens when k is not an integer via the animation below.

In evaluating the above pictures, it also seems that k has similar effects on this equation that it had on our previous equations. It seems that when k is an odd integer, there is a k point star in the middle, but when k is an even integer there is a 2k point star in the middle. Let's explore this phenomenon and see what happens when k is not an integer via the animation below.

We can observe a few different
things when watching the animation. First, our assumption about
the number of points of the star as k varries is correct for non-zero
and non-unit values of k. Second, we can see that when k is not
between -1 and 1, as k varries, the graph seems to spin and create new
stars on the odd integers that double immediately after the odd
integers. Finally we notice that there is a fixed point of (1,0)
for all of the graphs. We also notice that the stars completely
fall apart when the value of k is between -1 and 1. Probably
because the functions 1/(sin(x)) and 1/(cos(x)) vary quickly between -1
and 1.

We continue our exploration by fixing b = c = 1, k = 5, and varying a through the animation below.

We continue our exploration by fixing b = c = 1, k = 5, and varying a through the animation below.

We can see that a affects the size
of the star and how the star is twisted. As the value of a moves
away from 0, the star gets smaller. So the star is biggest when a
= 0. We also notice that as the value of a increases, the star
turns clockwise.

We continue with our exploration by holding a = c = 1, k = 5, and varying b.

We continue with our exploration by holding a = c = 1, k = 5, and varying b.

We can see in this animation, that
like a, varying b affects the size and the orientation of the
star. As b moves away from 0, the star gets smaller and
smaller. We also notice that when b = 0 the graph is oriented
with a point on the positive x-axis, but as b moves bigger positively
or negatively, the graph moves so a point is centered either on the
positive y-axis or the negative y-axis respectively. We note
again that the graph moves in a clockwise direction as the value of b
increases.

We conclude our exploration by fixing a = b = 1, k = 5 and varying c.

We conclude our exploration by fixing a = b = 1, k = 5 and varying c.

We can immediately see that varying c
has different effects than a and b. For instance, the difference
between c and -c is a precise rotation about the origin of 180
degrees. We also notice that when c = 0, the graph is
empty. Which makes sense, since 0/n = 0 for any n. We also
notice that the value of c mainly affects the size of the star.
As c moves away from 0, the star gets bigger and bigger, regardless of
whether c is moving positively away from 0 or negatively away from 0.

This concludes our exploration.

This concludes our exploration.