Assignment Eleven: Polar Equations

By: Melissa Wilson

Polar equations plot points using (r,), where r is the distance to a point that is located at from the positive x-axis.

In this assignment we will investigate the polar equations and vary some of the parameters. For these investigations, will always be on the interval [0, 2pi].

First we will look at the equation of the form:

Here we varied the a parameter and kept k=1. As you can see, as the value of a increases in magnitude, the size of the circle formed increases. Also, notice how when a is negative, then the graph is formed with negative x-values. The positive a values also correspond to positive x-values. The cosine function affects the x-values in the polar equation graphs.

Now we will vary the k parameter and keep a =1. The value for k changes the number of loops, or petals, that our graph has.

Now we will look at equations of the form:

In this first graph we varied a and kept k =1. Like we saw in the previous case, the magnitude of the a parameter changes the size of the loop. Here we can see that the negative values for a have negative y-values for the graph. This makes sense because sine function affects the y-values in the polar equations.

Now we kept a=1 and varied the value of the k parameter. As with the previous cosine function, the value for the k affects the number of loops in the graph. Also, notice that the negative coefficient also gives a similar looking graph.

Now we introduce a new parameter, b. The first equation we will look at is:

Here we has a = 2 and k = 1 for each graph. We varied the value of b. As you can see there is no difference between the positive and negative values. As the magnitude of the b value increases, the size of the graph increases. Notice that if you multiply the values of a and b together you get the range of v-values. Ex: for the blue graph, a*b = 2*3 = 6 and the range of x goes from -1 to 5, which is a spread of 6.

Now we will look at an analogous sine function:

For this equation we kept a =2 and k=1 and varied the value of b. The same types of changes occur in this graph as did in the previous example with the consine function. Notice how the graphs are now along the y-axis instead of the x-axis.

Here we varied the a parameter and kept k=1. As you can see, as the value of a increases in magnitude, the size of the circle formed increases. Also, notice how when a is negative, then the graph is formed with negative x-values. The positive a values also correspond to positive x-values. The cosine function affects the x-values in the polar equation graphs.
Now we will vary the k parameter and keep a =1. The value for k changes the number of loops, or petals, that our graph has.

In this first graph we varied a and kept k =1. Like we saw in the previous case, the magnitude of the a parameter changes the size of the loop. Here we can see that the negative values for a have negative y-values for the graph. This makes sense because sine function affects the y-values in the polar equations.
Now we kept a=1 and varied the value of the k parameter. As with the previous cosine function, the value for the k affects the number of loops in the graph. Also, notice that the negative coefficient also gives a similar looking graph.

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