Assignment 11:  Polar Equations

By

Jonathan Sabo

Investigate

Note:

• When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."
• Compare with

for various k. What if . . . cos( ) is replaced with sin( )?

We will first look at when a and b are equal  and k is an integer.

Observe when a = b = k = 1,

We can see that the graph is symmetric across the x-axis.

Now lets observe when a = b = 1 and k = 2

Observe when a = b = 1 and k = 3

Observe when a = b = 1 and k = 5

We can see from each of the previous graphs that the equation is symmetric across the x-axis.  The number of leaves is also equal to k for each equation.

Now observe when a = b = 2 and k = 2,

We can see that as a and b both increase the graph gets much larger.  We can also see that the number of leaves is equal to k for each equation.

Now lets observe when a is greater than b.

Observe when a = 2, b = 1, and k = 5

In this graph we can see that the number of leaves is still equal to k.  In this example k = 5 and there are 5 leaves.  However, when a is greater then b, the leaves to not reach the origin of the graph.  The graph is still symmetric acrosst he x - axis.

Now lets observe when b is greater than a.

Observe when a = 1, b = 2, and k = 5,

In this example where b is greater than a we can see that there are five smaller leaves inside of the five larger leaves.  We can see that k is sill equal to the number of leves.  This graph is also symmetric across the x - axis.

Now we will replace cos() with sin().

Observe , where a = b = 1 and k = 5

When we replace cos( ) with sin( ), we can see that our graph looks very similar.  The sin( ) graph is different from the cos( ) graph becausethis graph is symmetric across the y - axis instead of being symmetric across the y - axis.