Periodic functions

Certain naturally occurring phenomena eventually repeat themselves, especially when the phenomenon is somehow connected to a circle. For example, suppose that you are taking a ride on a Ferris wheel and we consider your height, , above the ground and how your height changes in tandem with the distance, , that you have traveled around the wheel. We can see a full animation of this situation at http://gvsu.edu/s/0Dt.

Because we have two quantities changing in tandem, it is natural to wonder if it is possible to represent one as a function of the other.

In the context of the ferris wheel mentioned above, assume that the height, , of the moving point (the cab in which you are riding), and the distance, , that the point has traveled around the circumference of the ferris wheel are both measured in meters.

Further, assume that the circumference of the ferris wheel is 24 meters (it’s a pretty short ferris wheel). In addition, suppose that after getting in your cab at the lowest point on the wheel, you traverse the full circle several times.

(a)
Recall that the circumference, , of a circle is connected to the circle’s radius, , by the formula . What is the radius of the ferris wheel? How high is the highest point on the ferris wheel?
(b)
How high is the cab after it has traveled of the circumference of the circle?
(c)
How much distance along the circle has the cab traversed at the moment it first reaches a height of meters?
(d)
Can be thought of as a function of ? Why or why not?
(e)
Can be thought of as a function of ? Why or why not?

Let’s consider a point traversing a circle of circumference 24 and examine how the point’s height, , changes as the distance traversed, , changes. Note particularly that each time the point traverses of the circumference of the circle, it travels a distance of units, as seen below, where each noted point lies 3 additional units along the circle beyond the preceding one.

Note that we know the exact heights of certain points. Since the circle has circumference , we know that and therefore . Hence, the point where (located of the way along the circle) is at a height of . Doubling this value, the point where has height . Other heights, such as those that correspond to and (identified on the figure by the green line segments) are not obvious from the circle’s radius, but can be estimated from the grid in the figure above as (for ) and (for ). Using all of these observations along with the symmetry of the circle, we can construct a table..

Moreover, if we now let the point continue traversing the circle, we observe that the -values will increase accordingly, but the -values will repeat according to the already-established pattern, resulting in the data in the table below.

It is apparent that each point on the circle corresponds to one and only one height, and thus we can view the height of a point as a function of the distance the point has traversed around the circle, say . Using the data from the two tables and connecting the points in an intuitive way, we get the graph shown below

Notice that the graph above resembles the graph of the sine function. As it turns out, the sine function exhibits some of the same oscillatory behavior as . This shared property turns out to be very important, especially when looking at functions that are related to circles.

For our ferris wheel example above, the period is the circumference of the circle that generates the curve. In the graph, we see how the curve has completed one full cycle of behavior every 24 units, regardless of where we start on the curve.

Two important periodic functions are the sine function, which you have seen, and the cosine function, which resembles the sine function. You will study these functions and learn about their relationship with circles in trigonometry.

As a reminder, here is a graph of the sine function along with a table listing some of its values.

-

-π--- 2−1
0 0 π___2 1 π 0

3π --- 2−1
2 π 0

Notice that and as well. In fact, the sine function is periodic with period .

Now, here is a graph of the cosine function along with a table listing some of its values.

-

-π2-- 0
0 1 π___2 0 π − 1

3π --- 2 0
2 π 1

Notice that and as well. In fact, the cosine function is also periodic with period .