An investigation of

Steve Messig

Stated in the assignment

is an "n leaf rose". If a=b=1, then the graph looks like:

To see the roses as k increase from one to ten click here.

Observing several equal values for a and b say 2, 3, 5 when k equals one the graph is similar to the one above. The graph expands in the horizontal direction twice the value of a, and in the vertical direction twice the value of b. As the value of k goes from 1 to n, the number of leafs increase and the length of each leaf from the origin is twice the value of a and b

Next let consider what happen when a and b are not equal. Let a = 1 and b=3, while use k = 4.

We see that there are four small leafs lying diagonally which are 2.12  units long and 4 leafs lying horizontally or verically which are 4 units long. We know from above that as when a and b are equal in value, then we have a k leaf rose. The farther apart a and b are the longer the diagonal leafs become. It appears as though the number of leafs of the rose becomes twice k.             

  

If a = 0 , then the equation becomes

.

If b = 1 and k = 1, then we get a circle with diameter 1.

If b is 3, and k = 1 we get a circle with diameter 3. However as we let k range from 0 to n some interesting observations occur. If k is odd we get a k leaf rose, if k is even we get a 2k leaf rose.

If we switch sin for cos the graphs appear to rotate 90/k degrees.