Let us first look at integer values of k. The following shows the general case when a,b, and k=1.
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Now let us look at various integer values for k less than 6.28318530718.
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We can see that we seem
to be drawing a flower each time, adding one more flower petal
or leaf. The value of k gives us the number of petals or
leaves. This is called the "n-leaf rose".
Next
let us look at when a<b. For this part of then
investigation we will let k = 6. (This is so we can see
what happens to the pretty flower.)
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We can see from the table
above that first for different values of a and b,
where a<b, gives us two distinct curves. It is
an appearance of a flower with two sets of petals. Also notice
the difference in the size of the flowers when the values of a
and b are changed.
We will now use the same
method as above to look at when a<b.
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Here we can see that we keep a six petal flower. The only difference is that the flower petals do not intersect at the origin as seen in the previous cases.
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