Investigate

When a and b are equal, the graph is like the following(a=b=2) according to k=1, k=2, k=3, k=4.

When a=b=2, the graph pass through (4,0) and (0,0) regardless of the value of k.

If k is odd then the graph meets at 2 and -2 with y-axis and if k is even then it does at 4 and -4 with x-axis.

The number of leaves is the value of k.

What if the values of a and b are different?

Case of a>b. Let's see the case of a=2 and b=1(k=1, 2, 3,4).

Case of a<b. Let's see the case of a=1 and b=2(k=1, 2, 3,4).

First observation

If a is bigger than b then the figure becomes to a circle as the difference is bigger regardless of k. To explore Click here

If b is bigger than a then new leaves appear and the size of new ones is difference of a and b. To explore Click here

If then the graph is like the following(b=2) according to k=1, k=2, k=3, k=4.

When b=2, the graph pass through (2,0) and (0,0) regardless of the value of k.

If k is even then it does at 2 and -2 with y-axis and x-axis.

If k is odd then the number of leaves is the value of k and if k is even then it is twice value of k.

What if cosine function is changed to sine function?

In fact, we can expect a rotation since sine function with addition of 90 degree becomes cosine function.

The followings are graphs ofwhen a=2, b=2, and k=1, 2, 3, 4.

Second observation

The graph of is some degree rotation of .

The rotation degree is decreasing as k is increasing.