Investigate for different values of a, b, and k.
First, let's look at the graphs as a changes.
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As a increases, a circle begins to form. Finally, at a > or = 3, we have a circle. The center is always at (1, 0) and the radius is equal to the value of a.
What happens as b changes?
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It seems that as b changes, our loop just gets bigger. In fact, when b = 2 or anything greater, there are actually 2 loops (one inside and one outside).
Now, let's take a look at what happens as k changes and a = b remains constant.
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When a and b are equal and k is an integer, as in this case, this is one version of the n-leaf rose. The value of k determines the amount of petals. It appears that the domain and range are always between 2 and -2.
Let's compare this with the equation . What happens as k changes?
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These graphs appear to be quite similar to the previous ones. However, in this case, the domain and range are always between 1 and -1. When k = 1, we get a circle with diameter 1. As k increases, so does the amount of pedals. When k is an even integer, the number of pedals is double the value of k. However, when k is odd, k = the number of pedals.
What happens if we replace cos by sin? Now, we have . What happens with the graph of this equation as k changes?
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These graphs very much resemble the graphs above. Notice how the graphs are the same, only rotated around the axes a bit.