Investigate the general form of a polar equation:
We will begin by examining
where a = 0 and b = 1.
Effects of a, b, and k on the graphs:
a determines how the leaf unfolds.
By keeping b =1 and k=4, and varying a we see
what happens as a is increased.
here for a movie of how "a"
changes this graph.
b determines where the leaves intersect the x and
Keeping k constant at k=2, we can study the
effects of changing the value of b.
= 1 or -1, the leaves intersect at 1 and
-1 on both axes.
= 2 or -2, the leaves intersect at 2 and
-2 on both axes.
= 3 or -3, the leaves intersect at 3 and
-3 on both axes.
= 4 or -4, the leaves intersect at 4 and
-4 on both axes.
It does not matter if b > 0 or b < 0,
unless b is odd, the place of the intersection is still the same.
here for a movie of how b effects
k determines the number of leaves
When k=1, there is a one leaf rose
When k=2, there is a four-leaf rose
When k=3, there is a three-leaf rose
When k=4, there is an eight-leaf rose
We obtain the following conclusion:
When k is odd, the number of leaves is k.
When k is even, the number of leaves is
When a and b are equal, keeping k = 1, the
a = b = 1
a = b = 2
a = b = 3
a = b = 4
When a and b are equal, keeping k = 2, the
When a and b are equal, keeping k = 3, the
Finally, when a and b are equal, keeping k
= 4, the rose
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