Polar Equations

By: Kimberly Young and Mindy Swain


Investigate varying a,b, c, and k.


The affects of k on the function .

Let a=1 and k vary. The below animation captures the affect that k has on the function .

Conclusions about k-values:

When k is even, there are are a total of 2k petals with k petals above the x-axis and k petals below the x-axis. The graph is the same for all positive and negative even k values. Consider the following function .

When k is odd, there are total of k petals.

When k is an odd positive value, there one more petal above the x-axis than below. We could further conclude that the number of petals below the x-axis is [k/2], (where [ ] is the greatest integer function). And the number of petals above the x-axis is [k/2] + 1. Consider the following function .

 

When k is an odd negative value the opposite is true. The number of petals above the x-axis is[k/2]. And the number of petals below the x-axis is [k/2] + 1. Consider the following function .

When k = 0, the function is a single point, (0,0).


The affects of a on the function .

Let k=1 and a vary. The below animation captures the affect that a has on the function .

Conclusion about a values:

a determines the radius of the circle formed by the function.

When a is positive the circle is above the x-axis. Consider .

When a is negative the circle is below the x-axis. Consider .


The affects of b on the function .

Let k=1 and a = 1 and b vary. The below animation captures the affect that b has on the function .

 

When b=0, the graph of the function is a circle centered at (0,1) with radius of 1.

When -1<b<1, the graph of the function is a curve that loops inside itself.

When b<-2 or b>2, the graph of the function is a cardioid that approaches a circle as |b| increases.


The affects of a and k on

Recall that a affected the radius of a circle and k affected the number of petals. Consider the following graphs.

 

Again, k has affected the number of petals and a determines the amplitude, or size of the petal. The observations that we made earlier about k still hold. The amplitude of the petal can be determined by the a value and is equal to 2a.

 

The negative values for both a and k will cancel each other out. As seen in the following graph.

 


The affects of a, k and b on

 

Now lets consider changes in a, k, and b. Consider the following animation, where n = b.

 

Again, as |b| increases, the graph of the function approaches a circle.

When b=0, the obvious is true.

When -2a<b<2a, the graph of the function loops inside of itself.


A New Function:

We can now look at how these graphs compare to . We began by looking at how the a values compared. We noticed that a still determines the radius of a circle or the amplitude of the petals. However, for the cosine functions, when holding k constant, the graphs of the functions were rotated 90 degrees. We also noticed that for the k values, there was another shift. For even values of k, the graph of the sine function can be rotated 45 degrees to equal the graph of the cosine function. For odd values of k, the same is true with a 90 degree rotation. Similarly, the b value of the cosine function has the same effect as on the sine function except there is a 90 degree rotation. When a, k and b vary, the cosine graphs are equivalent to rotated sine graphs. The rotation is determined by the descriptions above. Consider the following graph.

 


Another Function:

When exploring this function, we found interesting characteristic in the graph. We found that the k value affects the number of lines that form asymptotes. For values of k that are even there are 2k asymptotes and for odd values of k there are k asymptotes. As |k| gets larger, the graph rotates and a k-sided regular polygon approaches the origin as well as shrinks in size. The a and b values determine the distance from the origin to the vertices of the hyperbola and cause rotations, a in a positive direction and b in the negative direction. For |c|>1, the k-sided polygon formed by the asymptotes has a larger area and as a result the graph approaches the origin at a slower pace. For |c|<1, the k-sided polygon formed by the asymptotes has a smaller area and as a result the graph approaches the origin at a faster pace.


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