Varying k

Even Variation

Let's take a look at the first case. Here we hae kept a and b constant at 1 and varied k by even increments.

What do you notice?

We can see that the number of petals on hte rose has changed as we vary k.

As a matter of fact, the number of petals corresponds exactly to the value of k.

The petals seem to be remaining symmetric.

Above we only took into account if a and b are less than k.

In this next graph we see if there is any difference if a and b are greater than k

We still result in a symmetrical rose with the number of petals equalling k, although the petals are much larger.

Generalization: When a and b remain constant, and k is varied evenly, we will get a symmetric rose with k petals

Odd Variation

Let's see what happens when a and b remain constant, and k varies by odd numbers.

Our first example keeps a and b constant at 1.

In the above graph we can see that the roses do not remain symmetric, however k is still equal to the number of petals.

If we take into consideration a and b values larger than k we get

The same thing happens with the odd values of k as with the even values.

If k is an odd number and a and b remain constant we will get a rose with k petals, but it will not be symmetric.

We get the same results for -k as we did above.