Case 3:


As an example, let's consider the equation

The graph of this equation is the following:

We see that this graph has k = 4 leaves, and the graph does not go all the way to the origin.  The points farthest from the origin are | a | + | b | = 8 units from the origin, and the points closest to the origin are | a | - | b | = 2 units from the origin.


As before, when a and b are both negative, when k is odd, the graph is rotated /k radians from the case when a and b are both positive.  When k is even, the graph is exactly the same as the case when a and b are both positive.  Here are some examples:

This graph looks the same as the above graph of 

Let's look at 


Also as before, when a is negative and b is positive, when k is odd, the graph is the same as the case when a and b are both positive.  When k is even, the graph is exactly the same as the case when a and b are both positive, but it is rotated /k radians.  Let's look at an example where a is negative, b is positive, and k is even.


Also as before, when a is positive and b is negative, he graph is the graph of the case when a and b are both positive, but it is rotated /k radians, regardless of whether k is odd or even.


 

Conclusions about the graph of the polar equation

when

The rose has k leaves, equally spaced about the origin.

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

The distance from the origin to the points closest to the origin is | a | - | b |. 

When a and b are both positive, the end of one leaf 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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