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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.

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 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 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 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.