Assignment 11
Polar Equations
by Debra Jackson
In this assignment, we are asked to discuss and compare the textbook version of the "n-leaf rose" when a and b are equal, and k is an integer when k is varied.
The equation is r = a + b cos (k theta) with k = 1, 2, 3, and 4 and is compared to r = a + b sin (k theta) in this first table.
 a,b,k = 1.PNG) |
 a,b,k = 1.PNG) |
|---|---|
 a,b,k = 2.PNG) |
 k = 2.PNG) |
 a,b,k = 3.PNG) |
 k = 3.PNG) |
 a,b,k = 4.PNG) |
 k = 4.PNG) |
In conclusion, when looking at the equations r = a + b cos (k theta) and r = a + b sin (k theta) with a nd b the same and k varying, the k parameter's value represents the number of petals on the "n-leaf rose". The graphs look very much alike except that the sin graph is rotated the following amounts:
At k = 1, it is rotated 90 degrees.
At k = 2, it is rotated 45 degrees.
At k = 3, it is rotated 30 degrees.
At k = 4, it is rotated 22.5 degrees.
The equation r = a + b cos (k theta) with k = 1, 2, 3, and 4 and is now compared to r = b cos (k theta) in this second table.
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 a,b,k = 1.PNG) |
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 a,b,k = 2.PNG) |
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 a,b,k = 3.PNG) |
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 a,b,k = 4.PNG) |
When r = a + b cos (k theta) is compared to r = b cos (k theta) the following observations are made:
When k = 1, the graph changes to a circle.
When k = 2, the domain and range of the graph reduces by 1 and the number of pedals doubles from 2 to 4.
When k = 3, the domain and range of the graph reduces by 1 but the number of pedal stays the same.
When k = 4, the domain and range of the graph reduces by 1 and the number of pedals doubles from 4 to 8.
In conclusion, if the k is an even number the number of pedals doubles and if it is an odd number they stay the same.
Here is an animation of r = cos n theta where n varies from -10 to +10
...and here is an animation of r = a + b cos (k theta + n) where a = 1, b = 2, k = 5 and n varying between -2 and +10