Assignment 11: Polar Equations

by

Melissa Silverman
I will begin my investigation with a simple polar equation, where a, b and k are equal to 1.

As the absolute value of a increases, the shape becomes more circular. If -1<a<1, then the shape becomes smaller and an inner loop is formed.

What happens if the value of b is changed? If b is positive and increasing, then the graph increases in size and forms an inner loop. The graph is oriented towards the positive x-axis.

If b is negative and increasing, then the graph increases in size and forms an inner loop. The graph is oriented towards the negative x-axis.

If 0<b<1, then the graph becomes more circular and is oriented towards the positive x-axis.

If -1<b<0, then the graph becomes more circular and is oriented towards the negative x-axis.

The last variable to test is k. As the absolute value of k increases, the graph is flower with k number of petals, all centered around the origin.

If -1<b<1, then the graph is a spiral.

What if the cosine function is replaced by the sine function? How do a, b, and k affect the graph? We can begin by graphing a simple polar equation, where a, b, and k equal 1.

As the absolute value of a increases, the shape of the graph becomes more circular and the size of the enclosed figure increases.

If -1<a<1, then the graph forms an inner loop. The orientation of the graph does not change whether b is positive or negative.

If b is positive and increasing, the graph increases in size, forms an inner loop, and is oriented about the positive y-axis. Inversely, if b is negative and increasing, the graph increases in size, forms an inner loop, and in oriented about the negative y-axis.

The value of k determines how many petals form the flower graph. If k is positive, the petals are centered at (0,0) and are symmetric about the line y=x, without the line y=x bisecting the petal(s) in the first quadrant. If k is negative, the petals are centered at (0,0) and are symmetric about the line y=x, with the line y=x bisecting a petal in the first quadrant. The sign of k does affect the orientation of the flower.

It seems that the key to this polar equation is the k value. The k value determines the number of petals that the flower has. The a and b terms change the look and size of the flower, but cannot determine the flower. It is also interesting that both sine and cosine produce similar flowers, something which may not have easily been predicted without using a graphing program.

One frustration with this type of investigation is the myriad of possible scenarios. One can use sine or cosine. One can change the variables a, b, and k for different discoveries. Different combinations of a, b, and k values will also affect the graph and the flower shape. Playing with positives and negatives changes the graph's shape and/or its orientation. A graphing program makes this easier to accomplish because the results are instant. But there certainly seem to be infinite possibilities to this investigation!
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