The Investigation we want to see is that when when a =1, b=1, and k =1, this is what we start with and we get the following graph:
So we can explore our options when a and b are set equal to each other we will be able to see how k changes the graph. When k=3 we get the following graph:
We can now exlpore when k is equal to 5 we might expect and five leaf rose.
This is what we get and can expect similar as n increases that we will get that many leaves, but at some point we will lose the capacity to graph because there will be too many leaves. Lets see what it will give us when k is equal to 2.5.
We get half of a leaf which we maybe could've predicted.
We can now compare this equation with but we will take the A out of the equation which will leave us with the following when b is equal to 10 and k is equal to 3.
So we can now see that this will give us close to the same graph. Lets try one more when k is equal to 10.
We can now see that it gave us approximately double what k was which was 10. So we still get leaves but as k grows bigger it will give us more leaves.
Lets explore one more equation to see the difference in the graphs if we replace cos with sin. So the equation will give us this graph when k =3.
So lets try when k is equal to 10. We get close to what we had before but two of the leaves do not lie on the x axis. Let's look at it:
We can determine that the number k we use for the first equation with a and b will give us the exact number of leaves as n is, but with the sin and cos equations without an A, when k =10, both give us 20 leaves.
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