We are to investigate the following equation:
Changes in 'a'
We will keep both 'b' and 'k' equal to 1. Let's look at what happens when we vary "a" ...
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As we go from our inner circle (in purple) to our outer one (in light blue), our equations are as follows:
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So we get interesting behavior when a = 1 and when a = 2. The others produce circles always one more in diameter.
What happens if a is a fraction?
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Our curves will go inward. Starting
with the purple curve (a = 1/2) to the blue curve (a = 1/4).
Additional curves follow the pattern.   |
Changes in 'b'
Now let's keep 'a' and 'k' constant at 1 and see what happens when we vary 'b'.
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As we go from our inner circle (in red) to our outer one (in light blue), our equations are as follows:
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So a pattern begins to form after b = 2. Now let's look at fractions:
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Curves: Purple: b = 1/2 Red: b = 1/3 Blue: b = 1/4 |
Changes in 'k'
Here we make 'a' and 'b' both equal to 1 and we see how changing k effects our graph:
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The number seems to be related to the number of leaflets (after k > 1). We get the standard circle when k = 0 and a heart when k = 1.
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On the other hand, if we are to do away with the 'a' value altogether, here is what we come up with:
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The leaflets seem to double: Again, our last equation is in light blue.
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