The n-leaf Rose
On your TI-83 (you can also do this investigation in Graphing Calculator, but with the TI-83 it's easy to see the graph "trace out"),
Hit the Y= button.
Note that instead of Y1=, Y2=, etc., you see r1=, r2=, etc.
This is because you just put your calculator into Polar Mode.
Instead of using the familiar parameters x and y, the graph of a
θ, the angle between the positive-x axis and
r, the distance from the origin to the point.
In r1=, type sin(θ)
Now hit GRAPH.
You might see a full circle, or part of one (or something crazy, if the step size is too big; or nothing, if your window is wrong). The amount of the circle you see depends on the limits and step size you have set for θ.
Hit the WINDOW key. Where you're used to seeing limits on just x and y, you can now set limits on θ as well.
Let θmin = 0 and θstep = 0.1.
Now experiment with different values of θmax until you see a whole circle when you hit GRAPH. You may also need to adjust the limits on x and y, as you normally would.
What is the smallest value for θmax (keeping θmin = 0)
The smallest value is π. Is this what you found?
We will call π the cycle of this graph--the smallest number you can choose for θmax so that you can see the whole graph.
Note that when you let θmax > π,
Because the graph eventually traces over itself, we will call it cyclic.
Now let's manipulate our equation.
Try r1 = sin(nθ) for various nonzero values of n--
Because of the limitations of the calculator, and our value for θstep,
Again, play with θmax to find the cycle of these various graphs.
Which values of n yield graphs that look cyclic? Noncyclic?
What else do you notice?