Assignment 11
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Compare the graph of r = a + bcos(k theta) to the graph of r = bcos(k theta) for various values of k.
Since the equation r = bcos(k theta) has fewer variables, let's begin by looking at several graphs of the function for various values of b and k.
b = 1, k = 1 b = 1, k = 2
b = 1, k = 3 b = 1, k = 4
We are beginning to see a couple patterns with this equation.
1) When k is an odd integer, 2n + 1, where n is an integer, the equation has k number of 'petals.'
2) When k is an even integer, 2n, where n is an integer, the equation has 2k number of 'petals.'
Let's look at what happens when we vary b instead of k.
b = 2, k = 1 b = 3, k = 2
b = 2, k = 3 b = 3, k = 4
We can clearly see that b simply effects the length of the petals. The larger the value of b, the larger the graph of theequation.
Now, let's compare r = a + bcos(k theta) to the graph of r = bcos(k theta). Since b only affects the size of the graph, let b =1 and compare the graphs for various values of a and k.
a = 1, k = 1 a = 2, k = 1 a = 3, k = 1
As a increases, the graph of r = a +bcos(k theta) is beginning to look more like a circle.
a = 1, k = 2 a = 2, k = 2
a = 2, k = 3 a = 3, k = 2
When k increases, the graph of r = a +bcos(k theta) is much more elongated, similar to the infinity symbols. As a increases, its graph begins to round out again and take on the appearance of an oval.