We examine a very basic polar equation and pay attention to the arrangement of the constants,

variables, and powers. Below is the graph of the following function

This is the n leaf rose for k=n. Here, a,b, and k are set at 1 and n =3.

Click here to see modifications for the following equations as n varies. Notice that n makes k

change even if k is left alone at 1.

Lets examine a multitude of functions graphed together and make some generalizations.

In the case below n=3=k . The a,b,and c are left constant, and the constants at the end of the

equations are in place just to spread out the scenario.

For n very large in the above scenarios you should see something like this

Looking back at the first example, lets explore the intrinsic complexities of manipulating variables

and constants of even a simpler equation.

As a grows large the curve ( n -leaf) blows up, eventually minimizing the leafing characteristic

until the entire picture from the far out view becomes circle like

Lets look at the case where the constant b is manipulated and k=4 (the 4 leaf rose for the given

equation.

When k is odd and greater than1 the following happens . This specific case k=5 and nb=3, as nb gets bigger( n is a scaler here for graphing) thye picture simply blows up...

and new leaves appear inside the original leaves

When k is even (here k=6) the following happens...

new leaves grow spur outside the original, and again, increasing b expands the entire picture.

When all three manipulations happen together, they play their indiviudal role to a certain extent, but

still there is much to discuss. Exploration with middle of high school students with similar

situations are endless.

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