Kasey Nored

Polar Equations

We want to explore the following equations

Looking at as we vary a and k we see that changing a only increases the diameter while changing k is interesting

It appears that when k is even the number of leaves of the rose is equal to 2k and when k is odd the number of leaves equals k.

If we make k negative

The image reflects over the x-axis. The same effect occurs when a is negative, this reflection is masked when k is even due to the symmetry of the image.

When exploring we see that our image is rotated about the center slightly when k is odd.

When k is even the rotation is not discernable.

Looking at if b is even it appears that b just scaled the image.

BUT, if we have an odd b...

 a = 1, k = 5, b =1 a = 1, k = 5, b =3 a = 1, k = 6, b = 1

All kinds of things happen... Interestingly, if b = 2 we see the previous images for .

The same types of things happen for

 a = 1, k = 6, b =1 a =1, k=5, b=1 a = 1, k=5, b =3

One last equation series...

When c = 1, a =1, k =5 and b = 2 we see...

What a cool way to create a star.

As we vary c our star just gets bigger. Here c = 3 with all other above parameters being the same.

 a = 5 the graph shrank a = -5 the graph rotated b=5 the graph shrank and rotated b = -1 the graph rotated c = -1 the graph rotated k = 4 our graph is really different k = 6 same differences as k = 4 k = -2 the graph rotates and has the qualities as k = 4 and k =6