Assignment 11:

Every Rose has its Thorns

by

Kristina Little

*Polar equations* are equations using polar coordinates which are difined by a certain distance from a point and an angle from a fixed direction. These equations are used to develope more highly involved graphs than are easily accessible by our rectangular coordinate system of (x,y).

Let's look at the equation: .

We can see a little how each value of a, b, and k effect the graph. The purple graph and the blue graph seem to be similar in shape, just not size. Changing just the value of a in the red graph has effected the the shape that we recognize in both the purple and blue graph.

The green graph is completely different. It is the only one that has the value of k changed. So let's let k continue to change and see if we can begin to see a correlation between the values of k and the picture of the graph.

When a = b = c, a constant integer c, then we will see what happens as we let k vary.

You should be able to tell by now that the value of k directly effects the number of petals that are displayed by the graph.

Now consider, .

See how the number of petals have doubled since the value a was removed from the equation? If k=2 then the graph had 4 petals and so on.

The last graph in the table simply replaces the cos in our previous graphs with sin. What specific difference is there between this new graph and the corresponding graph with cos? Do you believe that this rotation will be true for all graphs of this form?

Look at the movie below of a polar equation. Considering what you have observed in the previous examples of polar equations, try to estimate what the equation for the graph below looks like. Which values do you believe are being varied? Is it more than one?

When think you have the correct equation, click the link below the graph to see the proper equation.

**Assignment 12: Seeing the Forest through the Trees**