Case 2:


As an example, let's consider the equation

Here is the graph:

In this case, there are two sets of leaves:  large leaves and small leaves.  The number of large leaves and the number of small leaves is k.  The length of the large leaves is | a | + | b |, and the length of the small leaves is | b | - | a |.  In this example, k is odd, and the small leaves are inside the large leaves.  


Let's see what happens when k is even.

The graph of the equation

is shown below

Again, there are k large leaves and k small leaves.  The length of the large leaves is | a | + | b |, and the length of the small leaves is | b | - | a |.  The difference between this example and the example when k is odd is that the small leaves are between the large leaves instead of inside the large leaves.


Let's consider an example when a and b are both negative, and k is odd, as in the following equation:

When we plot this equation on the same axes as the equation

we see that 

there are k large leaves and k small leaves.  The length of the large leaves is | a | + | b |, and the length of the small leaves is | b | - | a |.  The graph when a and b are both negative, though, is the graph when a and b are both positive rotated /k radians.


When a and b are both negative, and k is even, the graph is exactly the same as the graph when a and b are both positive.  The graph of

looks like the following:

which is the same as the graph of 

above.


Let's consider an example when a is negative, b is positive, and k is odd, as in the following equation:

The graph looks like the following:

This graph is exactly the same as the graph when a and b are both positive, as in the above equation


Now let's consider an example where a is negative, b is positive, and k is even, as in the following equation:

The graph is the following:

This graph is the same as the graph when a and b are both positive, but it is rotated /k radians.


Finally, let's look at an example where a is positive and b is negative.  Regardless of whether k is even or odd, the graph is the same as the graph when a and b are both positive, but it is rotated /k radians.

For example, look at the graphs of the following equations:

For an example when k is odd, look at the following:


Conclusions about the graph of the polar equation

when

The rose has k large leaves and k small leaves.

The leaves are equally spaced about the origin.

The length of each large leaf (the distance from the origin to the point farthest from the origin) is | a | + | b |.

The length of each small leaf is | b | - | a |. 

When a and b are both positive, the end of one small leaf and one large leave is on the positive x-axis.

When a and b are both negative and k is odd, the graph is the graph when a and b are both positive rotated /k radians.

When a and b are both negative and k is even, the graph is the same as the graph when a and b are both positive.

When a is negative, b is positive, and k is odd, the graph is the same as the graph when a and b are both positive.

When a is negative, b is positive, and k is even, the graph is the graph when a and b are both positive rotated /k radians.

When a is positive, b is negative, and k is odd, the graph is the graph when a and b are both positive rotated /k radians.

When a is positive, b is negative, and k is even, the graph is the graph when a and b are both positive rotated /k radians.



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