1.1 - Periodic Functions
Objectives:[edit]
- Decide whether a function could be periodic.
- Identify the midline, period, and amplitude of a periodic function.
- Interpret the graph of a periodic function in context.
- Definitions
- periodic function, period, midline, amplitude
Lesson Guide[edit]
Begin with an example similar to the one below to illustrate periodic functions. "Ferris wheel" examples recur through the trigonometry sections, so make sure students understand the set-up.
A Ferris wheel is $30$ meters in diameter and is boarded at ground level. The wheel completes one full revolution every $4$ minutes. At time $t=0$, an individual is at the 3:00 position and ascending. The Ferris wheel makes $2$ complete revolutions, and then returns the individual to the boarding platform. Sketch a graph of $h=f(t)$, where $h$ is the height above ground (in meters) after $t$ minutes.
Next, we can make a table. You can ask the students to estimate the height of the individual at each of the ``-.5" times (since students do not yet have a way of calculating the heights more accurately). This will be good enough for graphing.
| Time, $t$ (minutes) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| Height, $h=f(t)$ (meters) | 15 | 25.61 | 30 | 25.61 | 15 | 4.39 | 0 | 4.39 | 15 |
We can then use the table to create a graph:
A \underline{periodic function} is any function satisfying $f(x+c)=f(x)$ for some number $c$ and all values $x$ in the domain.
*It will be more intuitive to students to think of a periodic function as a function whose values repeat on a regular interval.
Given a periodic function $f(x)$, the \underline{period} of $f$ is the smallest possible choice of $c$, where $c>0$,
for which $f(x+c)=f(x)$ for all values $x$ in the domain.
*The period is effectively how long it takes for a function's values to begin repeating.
The \underline{midline} of a periodic function is the horizontal line midway between the function's minimum and maximum values.
The midline often represents the \underline{average value} of a wave-like periodic function, like the one in the example.
*The midline of a periodic function, being a line, should always be written as an equation.
The \underline{amplitude} of a periodic function is the distance between the function's maximum (or minimum) value and the midline.
*The amplitude represents how ``large" the function's oscillations are.
**Not all periodic functions have midlines or amplitudes: consider showing a graph of $\tan(x)$ as an example of this.
Point out that the Ferris wheel example is a periodic function. Ask students to identify the period, midline, and amplitude in the example. This can also be done as you introduce each of the definitions above.
Comments[edit]
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.
It should be noted that the midline may not always represent the average value of a periodic function. Instead of including this as part of the definition of the midline, you may want to let students explore this idea while working on problems in the course packet.
