Assignment 11

This activity leads you through some investigations of the polar equation

Start with the equation .  Let's begin our investigation by observing what the function of this equation looks like for different values of a, b, and k respectively while holding the other two values constant.

Let r = a + cos (2ø).  The following graph shows what happens when a = -3, -2, -1, 0, 1, 2, 3.

Notice that when a < 0, the graph is stretched in the vertical direction, along the y axis. Also, notice when a < -1, the curve intersects the x axis at |a| - 1 and the y axis at |a| +1.   When a > 0, the graph is stretched in the horizontal direction, along the x axis.  When a >1, the curve intersects the y axis at |a| - 1 and the x axis at |a| +1.  These graphs are all centered on the origin.  Notice also, that when |a| is less than or equal to 1, the graph of the curve intersects the origin.

Now let r = 2 + b cos (2ø).  What happens for different values of b?

When b = 0, the graph is a circle.  When b does not equal 0, the graph has three leaves.   When b is negative the graph is symmetric about the line y = x.  When b is positive, the graph is symmetric about the line y = -x.

Compare this to the graph of the equation, r = b cos (3ø), where a = 0.

The same properties of symmetry still apply from the previous example.   This graph is a bit more simplistic then the last graph in that each equation only has one set of leaves.  Looking back at the previous picture, several of the equations have two sets of similar leaves.

Finally, look at the graphs of the equation r = 3 + 3 cos (kø), where k = -3, -2, -1, 0, 1, 2, 3.

When k = 0, the graph is a circle.  When k = 1, the graph is a circle like, and intersects the origin.  When k > 1, the graph of the equation is an n-leaf rose.  Notice, when k is even, the graph of the equation is centered on the x axis.

This graph brings us to another important observation - When a = b and k is an integer, the graph of r = a + b cos (kø) is an n-leaf rose centered on the origin.

Going back to the previous graph, compare this graph to the graphs of the equation r = 3 cos (kø), where k = -3, -2, -1, 0, 1, 2, 3.

Notice when k = 0 and k = 1, the graph is a circle.  When k > 1, the graph is an n-leaf rose.  Again, when k is even, the graph of the equation is centered on the x axis.