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.

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

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

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,

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

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,

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

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.

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