Varying a

Even Variation

For this part of our investigation, we will vary the value of a by even numbers while keeping b and k constant.

Note: K will be kept constant at 4. So, we know we will always have a four-petal symmetrical rose.

We see that as a increases on the even numbers, our rose becomes more spread out from the origin, but it keeps its symmetry.

So now let's see if we get the same expected results if b is larger than a.

What in the world have we gotten here?

It looks like instead of spreading our rose out, that it has pulled it in tighter to the origin and doubled our number of petals for each flower.

It is worth testing a few larger values of b to see if our observation is correct.

We will do so with an odd and even value of b.

It looks like we are right about the rose when b is larger than a.

But do we get something different if a is odd?

Odd Variation

For this part of our investigation, we will vary the value of a by odd numbers while keeping b and k constant.

Note: K will be kept constant at 4. So, we know we will always have a four-petal symmetrical rose.

It looks just like the even graph, except that rose crosses the x and y axes at the positive and negative even values immediately after the odd value of a.

We see that our rose is once again spreading out.

What if b is larger than a?

Once again, we see that same pattern for odd values of a.

We will assume that the rose will result the same for any values of b which are larger than a,

because it seems to be following the same pattern as the even case.

Feel free to test it on your own.