Investigate the graphs of

Let a=b and k be an integer.

For k=1 and k=-1, the graph is

For k=2 and k=-2, the graph is

For k=3 and k=-3, the graph is

For k=4 and k=-4, the graph is

For k=5 and k=-5, the graph is

For k=6 and k=-6, the graph is

The graph of

when a and b are equal and k is an integer is one version of the "n-leaf" rose.

Let a=b and k is not an integer.

For k=.5 and

,

the graph is

For k=0.5 and

,

the graph is

The graph of

when a and b are equal and k is not an integer, is a one-leaf rose with a loop in the center.

For a=1 and b=2 and k is an integer.

For k=1, the graph is

For k=2, the graph is

For k=3, the graph is

For k=4, the graph is

The graph of

when a=1 and b=2 and k is an integer is similar to the "n-leaf" rose with additional "n" mini-leaves added

For a=2 and b=1 and k is an integer.

For k=1, the graph is

For k=2, the graph is

For k=3, the graph is

For k=4, the graph is

For k=5, the graph is

The graph of

when a=2 and b=1 and k is an integer is another form of an "n-leaf" rose.