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Section 10.1 Periodic Functions

Subsection 10.1.1 Introduction to Periodic Functions

A function is called periodic if it repeats itself regularly. In this section, we will explore the basics of periodic functions by considering real-world examples.

In the last exercise, the function repeated itself every \(24\) hours. This is called the period of the function — the smallest value \(k\) such that

\begin{equation*} f(x + k) = f(x) \end{equation*}

for all \(x\) in the domain.

Note that the cell phone function also repeats itself every \(48\) hours, but that isn't the smallest number for which it repeats. The period is defined as the smallest such number.

The next exercise shows an important example of a periodic function — one we will return to many times. It involves a person riding a Ferris Wheel.

In the Ferris Wheel problem, there are a few other things to notice.

First, the height of the person as they go around the circle has nice symmetry — the graph on the way up is a reflection of the graph on the way down.

Next, the graph oscillates up and down around a “middle height”. For periodic functions, this height is called the midline. For example, when the radius was \(20\) meters and the loading platform was \(10\) meters, the midline was the height \(y = 30\) meters.

In fact, the midline is usually given in equation form, such as: \(y = b\)

Finally, the radius of the wheel is easily seen on the graph, as the distance between the midline and either the highest point, or the lowest point.

Because not every periodic function involves movement around a circle, we give this distance a more general term: The Amplitude

In general, the amplitude of a periodic function is the difference (along the output axis) between the maximum output and midline (or between the midline and the minimum output).

In the next exercise, you'll plot points on the graph of a periodic function.

As we see examples of periodic functions which describe motion around a circle (such as a Ferris Wheel), we often use the language of analog clocks to describe certain positions on the circle.

For example, if a person begins at the bottom of the circle, we may refer to this as the \(6\) O'clock position.

For reference, our naming of points on a circle will be as follows:

Figure 10.1.5.

Imagine riding a Ferris wheel where you board at the \(6\) O'clock position, and suppose it takes \(30\) seconds to make one complete revolution.

  1. At what times do you reach your maximum height above the ground?

  2. At what times will your height be the same as the center of the wheel?

  1. Notice that this question is the same as asking “When will you be at the \(12\) O'clock position?”

    This will occur at \(15\) seconds, \(30 + 15 = 45\) seconds, \(2(30) + 15 = 75\) seconds, etc.

    In general, this is \(15\) seconds after any multiple of the period (\(30\) seconds), or: \(15 + 30k\text{,}\) where \(k\) is any whole number.

  2. This question is just like asking “When will you be at the \(3\) O'clock or \(9\) O'clock positions?”

    You will be at the same height as the center of the circle after \(7.5\) seconds, \(15 + 7.5 = 22.5\) seconds, \(30 + 7.5 = 37.5\) seconds, etc.

    In general, this is \(7.5\) seconds after any multiple of half the period (\(15\) seconds), or: \(7.5 + 15k\text{,}\) where \(k\) is any whole number.