Polar Equations

by:

Brandt Hacker

For this assignment we are asked to explore the polar equation .  A polar equation is one that uses the variable r as a distance from a given point in the plane, with theta giving direction in relation to the point.  We graph the equation above with a=1, b=1, and k=1.

LetÕs next explore several the same equation for different values of k.

When k=1, the graph shows one enclosed region.  When k=2, the graph shows two enclosed regions.  This pattern progresses as the value of k increases.  As we see in this series of images, the value of k represents the number of petals stemming from the origin.

Next weÕll look at what happens when the values of a and b are varied.

Above we see that as the values of a and b increase the length of the petals, our shape continue to get larger and larger as a and b increase.  But, what if the values of a and b are not equal to one another?  How might this effect the graph of this particular polar equation?

We see here that as the value of a increases, if b remains the same, the petals no longer come back to the origin, but instead become more and more rounded.