The n-leaf Rose
by Emily Kennedy

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 MODE button, and choose Pol on the line that says Func Par Pol Seq.
Also, make sure you're in Radians Mode by choosing Radian on the line that says Radian Degree.

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
polar equation graphs points by two other parameters:

θ, the angle between the positive-x axis and
a segment connecting the point to the origin,

and

r, the distance from the origin to the point.

In r₁=, type sin(θ)
(The θ will show up when you hit the XTθn key).

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)
you can find that still shows the whole circle?
(Hint: since we're working in radians,
you might want to try multiples of π.)

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,
the graph was actually tracing over itself.

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--
use integers, fractions, even irrational numbers.

Because of the limitations of the calculator, and our value for θstep,
it is best to use small values of n (less than about 10).

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?

Play around with your calculator, or with this Graphing Calculator file, for a while before clicking here.

Next (Cycles) >>

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