Assignment 11

Investigating Polar Equations

Drew Wilson

In this write up we will investigate the following equation above and we will determine how the parameters a, b, and k affect the graph of the polar function.

First lets consider the case when a=0 thus giving us the equation in the later half of the problem, r = b cos (k). Below are graphs of different scenerios of this equation that will help us get an idea of how the parameter b and k affect the graph of this equation. First we will notice that when the value of k is constant at 1 and the value of k varies, then the diamter of the circle varies as b varies. The parameter b determines the diameter of the circle, when b=2 the diameter is 2, when b=3 the diameter is 3and so on for all values of b. Now we have to look at when b is positive and when b is negative. When b is positive, then the graph will have diameter of b with the center of the circle at the point (, 0) because b is the length of the diameter and the equation passes through the origin (when r=0, =0). When b is negative, if we refer back to how a negative leading coefficient affects the graph of a typical equation we will know that a negative leading coefficient reflects the graph across the y-axis. This also occurs with polar equations, when b is negative, then the graph is reflected across the vertical axis. Below are some graphs when k=1 and the value of b varies.

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Now let's look at the case when b is a fixed value of 1 and the parameter k changes. We will notice that the parameter k determines the number of pedals in the graph of the equation. If k=1 there will be one pedal which is represented by the circle. When k=2 there will be 4 pedals, k=3 there will be 3 pedals, k=4 there will be 8 pedals, k=5 there will be 5 pedals. The common occurence here is between even and odd values of k. When k is even, then the number of pedals is two times the paramter k (pedals=2k) and when k is odd, then the number of pedals equals the parameter k (pedals=k). As we will see, it doesnt matter if the value of k is positive or negative, the same outcom will be observed. As mentioned in the previous scenario when the paramter b was negative, if the parameter b was fixed at a constant value of -1 we would see the same image reflected across the vertical axis. Below are some graphs that show you how the value of k affects the graph.

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Now let's look at graphs as the values of b and k are changing. We know the value of b determines the lengh of the diameter of the circle when k=1, and we can notice in the graphs above when b=1 the length of the pedals are 1. Therefore, we can conclude that the value of b will determine the lengths of the pedals, and if the value of b is negative, the lengths of the pedals will still have a length of b, but the graph will be reflected across the vertical axis. As we know that if the value of k is even then there are 2k number of pedals and these graphs are symmetric about the vertical and horizontal axese, so we know that a negative value for b will not affect the graph of the equation. A negative value for b will only affect the graph of an equation with a value of k that is odd. We also know the number of pedals is determined by the value of k and we know that the sign of k doesn't change the graph as you can observe above. Now that we know how the values of b and k affect the graph we can observe the equations below and see more examples.

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Now if the value b=0, then we will be left with the equation r=a which we know is a circle with center at the origin and radius equal to a. If the value of k=0 then we will have the equation r=a+b because cos(0)=1, this will give us a circle with center at the origin and radius equal to a+b.

Now let's look at how the value of a affects the graph. If the value of a varies and the value of b and k are left constant at a value of 1, then we will notice that the value of a affects the width of the graph. The value of a makes the graph wider no matter the sign of the value of a. The sign doesn't change how the value of a affects the graph. The value length of the pedals of the graph seem to lengthen by a value of a. We already know that the length of the pedals are b, but when a=b the length of the pedals seem to be a+b. When b < a the graph doesnt come back to the orgin after each pedal, instead each of the pedals comes back to a point on the circle r=a-b. When the value of b>a we can also look at the graphs and notice something interesting. When the value of k is even, then part of the graph lies between the pedals and when the value of k is odd, then there are smaller pedals within the larger pedals. The length of the pedals is still a+b in both cases when a<b and when a>b, the only difference is when a>b, the pedals meet the equation r=a-b. Below are some examples that show what is happening for different values of a, b and k.

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