Section 1.1 Periodic Functions

Supplemental Videos.

The main topics of this section are also presented in the following videos:

The London Eye is a huge Ferris wheel located on the South Bank of the River Thames in London, England. It is 135 meters (443 feet) tall and completes one full rotation every 30 minutes. If we look at the behavior of one of the wheel’s passenger capsules, we see that it completes one cycle, or one revolution, and then repeats this revolution over and over again. The motion of this Ferris wheel is an example of a periodic function because its revolution or cycle is repeated every 30 minutes, and so we say that it has a period of 30 minutes.

described in detail following the image — Figure 1.2. London Eye Photo by Lippman & Rasmussen, 2010, CC-BY

In this section, we use the rotation of the London Eye as an example of a periodic function and will work to sketch a graph of a rider’s height above ground over time.

A periodic function is a function whose output values repeat on a regular interval. In mathematical terms, we say that a periodic function is a function for which a specific horizontal shift, \(P\text{,}\) results in the original function: \(f(x+P)=f(x)\) for all values of \(x\text{.}\)

The period of a periodic function is how long it takes for the output values to begin repeating, or the smallest horizontal shift with \(P>0\text{,}\) such that \(f(x+P)=f(x)\) for all values of \(x\text{.}\)

figure showing the period of a periodic function

Note 1.3.

In the figure above, we show the period as the interval from one maximum value to the next maximum value. We could just as easily have shown the period as the interval between minimum values or between any two points on the graph where the output values begin to repeat, as shown in the figure below.

figure showing two other periods of a periodic function

Example 1.4.

The London Eye is a huge Ferris wheel in London, England. It completes one full rotation every 30 minutes, and the diameter of its passenger capsules is 130 meters. Riders board the passenger capsules from a platform that is 5 meters above the ground. At time \(t=0\text{,}\) an individual boards the Ferris wheel.

Sketch a graph of \(h=f(t)\text{,}\) where \(h\) is the height of the individual above ground (in meters) after \(t\) minutes.

Solution.

Let’s start by drawing a picture of the Ferris wheel and labeling the known information.

We know that at time \(t=0\) a rider boards the Ferris wheel from the boarding platform, which is 5 meters above the ground. We also know that the Ferris wheel has a diameter of 130 meters and it takes 30 minutes to complete one full revolution around the Ferris wheel.

Since it takes 30 minutes to complete a trip around the Ferris wheel, a rider will reach the top of the Ferris wheel after 15 minutes (assuming that the wheel rotates at a constant speed). Similarly, the rider will reach the three o’clock and nine o’clock positions on the Ferris wheel at 7.5 minutes and 22.5 minutes.

sketch of the London Eye

Next, we can make a table of the rider’s height above ground vs. time.

At time \(t=0\text{,}\) the rider is at the boarding platform, which corresponds to a height of 5 meters above ground. After 15 minutes, the rider reaches the top of the Ferris wheel, which corresponds to a height of 135 meters above ground. We can find this height by taking the height of the boarding platform plus the diameter of the Ferris wheel.

We can also find an intermediate height corresponding to when the rider has been on the Ferris wheel for 7.5 minutes and 22.5 minutes. At these two positions, the rider is halfway between the top of the Ferris wheel and the boarding platform. We can find this height by adding the height of the boarding platform to the radius of the Ferris wheel. Therefore, we get a height of \(5+65 = 70\) meters above ground at these two times.

After 30 minutes, the rider is back at the boarding platform, which corresponds to a height of 5 meters above ground. If the individual were to continue riding the Ferris wheel, this pattern of heights would continue and we could extend our table with additional times and heights as shown below.

time, \(t\) (minutes)	0	7.5	15	22.5	30	37.5	45
height, \(h=f(t)\) (meters)	5	70	135	70	5	70	135

Now that we have a table of times and corresponding heights, we can plot these points and draw a smooth curve between them to create a height vs. time graph.

Since we are graphing height as a function of time, time is our independent variable and should be plotted along the horizontal axis while height is the dependent variable and should be plotted along the vertical axis.

London Eye Graph

Note that in this example, we assumed that the Ferris wheel was rotating counterclockwise, as shown in our sketch above. For this problem, it did not matter what direction the Ferris wheel was rotating because both clockwise and counterclockwise rotation would have produced the same height vs. time graph. However, this will not always be the case. If the rider were to start in the three o’clock or nine o’clock position, we would need to know whether the Ferris wheel was turning clockwise or counterclockwise in order to sketch a graph of the rider’s height above ground.

Subsection 1.1.1 Midline and Amplitude

In the previous example, we sketched a graph of a periodic function representing the height of a passenger on the London Eye over time. By looking at our graph, we can see that the periodic function we sketched has both a maximum value and a minimum value. We can use these maximum and minimum values to define special properties of periodic functions.

The midline of a periodic function is the horizontal line halfway, or midway, between the function’s maximum and minimum output values.

The amplitude of a periodic function is the distance between the function’s maximum (or minimum) output value and the midline.

figure showing midline and amplitude of a periodic function

The midline often represents the average value of a wave-like periodic function and should always be written as the equation of a horizontal line (this means that the midline will look like \(y=a\) for some constant \(a\text{,}\) as opposed to just the number \(a\)). The amplitude of a periodic function represents how "large" the function’s oscillations are.

Example 1.5.

Find the period, midline, and amplitude of the function shown below.

example for period, midline, and amplitude

Solution.

By definition, the period of the function is how long it takes for the function to start repeating. If we think about the function "starting" at \(x=0\text{,}\) or when it crosses the \(y\)-axis, we can see that it begins repeating when \(x=4\text{.}\) Therefore, the period of the function is 4.

The midline of the function is the horizontal line halfway between the function’s maximum and minimum values. Here, the maximum value of the function is 5 and the minimum value is 1. To find the number halfway between 5 and 1, we can take the average of the two numbers. The average of 5 and 1 is

\begin{equation*} \frac{5+1}{2} = 3 \end{equation*}

Thus, the midline of the function is the line \(y=3\text{.}\)

The amplitude of the function is the distance between the function’s maximum value and the midline. We can find this value by subtracting the midline from the maximum value. Therefore, the amplitude of this function is \(5 - 3 = 2\text{.}\)

example solution for period, midline, and amplitude

Note 1.6.

Not all periodic functions have a midline or amplitude. Shown below is an example of a periodic function that does not have a minimum or maximum value. Since there are no minimum or maximum values, we cannot find a midline or an amplitude for this periodic function.

However, all periodic functions must have a period. If a function does not have a period, then it cannot be a periodic function!

graph of tangent

Exercises 1.1.2 Exercises

1. Elk Population.

An elk population \(P(t)\) in a state park oscillates periodically as a function of time \(t\) in months after January (so January is \(t = 0\)). In January, the population is at its lowest, with only \(29\) members. The population is at its highest in July, with \(39\) members.

(a) What is the period of the population \(P(t)\text{?}\)

period = months.

(b) What is the midline of the function \(y=P(t)\text{?}\)

\(y =\) members.

amplitude = members.

(d) Consider the six possible graphs of \(P(t)\) below. Be sure to enlarge each graph and carefully read the labels on the axes in order distinguish the key features of each graph.

Which (if any) of the graphs A-F represents a possible graph of the population over two years?

Select
A
B
C
D
E
F
None of the graphs above


A	B	C

D	E	F

Hint.

a) Recall that the period is how long it takes for the function to repeat output values.

b) Recall that the midline is the equation of the line between the maximum and minimum of the output values.

c) Recall that the amplitude is the distance from the midline to the maximum (or minimum).

d) Combine the information found in parts a)-c) to construct a graph.

2. Periodic or Not? (Tables).

(a) Determine whether the function described by the following table is periodic. If it is periodic, determine its period. If not, enter "none".

\(x\)	1	2	3	4	5	6	7	8	9	10	11	12
\(f(x)\)	5	10	5	0	5	10	5	0	5	10	5	0

The function is...

periodic.
not periodic.

The period is .

(b) Determine whether the function described by the following table is periodic. If it is periodic, determine its period. If not, enter "none".

\(t\)	1	2	3	4	5	6	7	8	9	10	11	12
\(g(t)\)	-2	-0.267949	-0.267949	-2	-3.73205	-3.73205	-2	-0.267949	-0.267949	-2	-3.73205	-3.73205

The function is...

periodic.
not periodic.

The period is .

(c) Determine whether the function described by the following table is periodic. If it is periodic, determine its period. If not, enter "none".

\(\theta\)	1	2	3	4	5	6	7	8	9	10	11	12
\(h(\theta)\)	2	6	8	2	6	8	-1	2	6	8	-1	-5

The function is...

not periodic.
periodic.

The period is .

Hint.

Recall that a function is periodic if its output values repeat on a regular interval.

3. Periodic or Not? (Graphs).

Decide if the following graph represents a periodic function.

If the function is periodic, determine the period, midline, and amplitude (if possible). If the function has no well-defined period, midline, or amplitude, enter "none".

The period is .

The midline is \(y=\).

The amplitude is .

Hint.

Try to find an interval where the function repeats output values.

4. Properties of Periodic Functions.

Let \(y = f(x)\) be a periodic function whose values are given below. Find the period, amplitude, and midline.

x	4	19	34	49	64	79	94	109	124
f(x)	15	13	2	15	13	2	15	13	2

(a) The period of the graph is

(b) The midline of the graph is \(y =\)

5. Periodic Behavior.

A weight is suspended from the ceiling by a spring. Let \(d\) be the distance in centimeters from the ceiling to the weight. When the weight is motionless, \(d = 12 \, \mbox{cm}\text{.}\) If the weight is disturbed, it begins to bob up and down, or oscillate. Then \(d\) is a periodic function of \(t\text{,}\) the time in seconds, so \(d = f(t)\text{.}\) Consider the graph of \(d = f(t)\) below, which represents the distance of the weight from the ceiling at time \(t\text{.}\)

(a) Based on the graph of \(d =f(t)\) above, which of the statements below correctly describes the motion of the weight as it bobs up and down?

The weight starts closest to the floor and begins by bouncing up towards the ceiling.
The spring starts at its average distance between the ceiling and floor and begins by stretching the spring down towards the floor.
The weight starts closest to the ceiling and begins by stretching the spring down towards the floor.
None of the above

(b) How long does it take the weight to bounce completely up and down (or down and up) and return to its starting position?

(Include in your answer.)

(d) What is the furthest the weight gets from the ceiling?

(Include in your answer.)

(e) What is the amplitude of the graph of \(d = f(t)\text{?}\)

(Include in your answer.)

6. Ferris Wheels.

A ferris wheel 80 meters in diameter is boarded at its lowest point (6 O’Clock) from a platform that is 4 meters above ground. The wheel makes one full rotation every 6 minutes, and at time \(t = 0\) you are at the loading platform (6 O’Clock). Let \(h = f(t)\) denote your height above ground in meters after \(t\) minutes.

(a) What is the period of the function \(h = f(t)\text{?}\)

period = Include in your answer.

(b) What is the midline of the function \(h = f(t)\text{?}\)

\(h =\) Include in your answer.

amplitude = Include in your answer.

(d) Consider the six possible graphs of \(h = f(t)\) below. Be sure to enlarge each graph and carefully read the labels on the axes in order distinguish the key features of each graph.

Which (if any) of the graphs A-F represents two full revolutions of the ferris wheel described above?

Select
A
B
C
D
E
F
None of the graphs above


A	B	C

D	E	F

Prev Top Next