D. Trigonometric Functions · Sinusoidal Functions

Periodic Functions: Cycle, Period, Amplitude and Axis

D2.1 D3.2About 70 minutes

Learning goals

I can recognize a periodic function from a graph, a table or a description.
I can identify one cycle and find the period of a periodic function.
I can find the maximum, minimum, amplitude, equation of the axis and range of a periodic function.
I can tell a sinusoidal function apart from other periodic functions.

Words to know

periodic function
A function whose y-values repeat in a regular pattern. If the period is p, then f(x + p) = f(x) for every x.

cycle
One complete repeating pattern of a periodic function.

period
The horizontal length of one cycle (for example, in seconds, days or degrees).

axis of the curve
The horizontal line halfway between the maximum and minimum values: y = (max + min) ÷ 2. Also called the equation of the axis or the midline.

amplitude
Half the distance between the maximum and minimum values: (max − min) ÷ 2. It is the vertical distance from the axis to a maximum.

sinusoidal function
A periodic function whose graph is a smooth wave with the same shape as the graph of y = sin x (stretched or moved).

Things that go around come around

Picture a Ferris wheel at a summer fair. You climb on at the bottom, rise to the top, come back down, and then do it all again. If you graphed your height above the ground against time, the graph would go up and down in the same pattern over and over.

Lots of things behave like this: the tides on the Bay of Fundy, the hours of daylight from season to season, the electricity in the wall outlet, a swinging pendulum and the sound of a guitar string. This unit is about the functions that describe them, and especially about the most important family: the sinusoidal functions, built from sine and cosine.

Where this unit fits

In the unit Trigonometric Ratios and Laws you found sine and cosine for any angle from 0° to 360°. In this unit you turn those ratios into functions, graph them, transform them, and use them to model things that repeat. In Grade 10 you transformed parabolas with y = a(x − h)² + k; the same kind of thinking returns here with new parameters. In Grade 12 (Advanced Functions, MHF4U) you will meet these functions again with angles measured in radians.

What makes a function periodic?

A function is periodic if its y-values repeat in a regular pattern as x increases.

One complete pattern is called a cycle. A cycle can start anywhere, as long as it ends where the same pattern starts again.
The horizontal length of one cycle is the period. If the period is p, then f(x + p) = f(x) for every x.
The maximum and minimum are the greatest and least y-values.
The axis of the curve is the horizontal line halfway between them: y = (max + min) ÷ 2.
The amplitude is the distance from the axis up to a maximum (or down to a minimum): amplitude = (max − min) ÷ 2.
The range is every y-value from the minimum to the maximum: {y ∈ ℝ | min ≤ y ≤ max}.

Amplitude is always positive, because it is a distance.

A Ferris wheel with radius 20 m, axle 22 m above the ground, turning once every 60 s. The height repeats every 60 s (the period), swings 20 m above and below the axis h = 22, and stays between 2 m and 42 m.

Worked example

Example 1: Reading a Ferris wheel graph

Use the graph above. (a) Find the period, maximum, minimum, axis and amplitude. (b) State the range. (c) What does each value mean for the rider?

The graph is at a minimum (2 m) at t = 0, 60 and 120 s. The pattern repeats every 60 s, so the period is 60 s.
The maximum is 42 m (at t = 30 s, 90 s, …) and the minimum is 2 m.
Axis: h = (42 + 2) ÷ 2 = 22. Amplitude: (42 − 2) ÷ 2 = 20 m.
Range: {h ∈ ℝ | 2 ≤ h ≤ 42}.
Meaning: one full turn takes 60 s; the axle (the centre) is 22 m up; the radius of the wheel is 20 m, which equals the amplitude; you get on 2 m above the ground and reach 42 m at the top.

Period 60 s, max 42 m, min 2 m, axis h = 22, amplitude 20 m, range 2 ≤ h ≤ 42.

x	0	1	2	3	4	5	6	7	8
y	3	5	7	5	3	5	7	5	3

Data for Example 2.

Worked example

Example 2: Is a table periodic?

Decide whether the data in the table below suggest a periodic function. If so, find the period, amplitude and axis.

x: 0, 1, 2, 3, 4, 5, 6, 7, 8
y: 3, 5, 7, 5, 3, 5, 7, 5, 3

Look for a block of y-values that repeats: 3, 5, 7, 5 then 3, 5, 7, 5 again. So the data look periodic.
The pattern starts over every 4 units of x (y = 3 at x = 0, 4, 8). Period = 4.
Max = 7, min = 3. Amplitude = (7 − 3) ÷ 2 = 2. Axis: y = (7 + 3) ÷ 2, so y = 5.
A table only shows a few points. We are assuming the pattern continues the same way. Real data usually repeat only approximately.

Periodic with period 4, amplitude 2 and axis y = 5.

Periodic is not the same as sinusoidal

Every sinusoidal function is periodic, but not every periodic function is sinusoidal. A sinusoidal function has a smooth, rounded wave with exactly the shape of the sine graph (perhaps stretched, flipped or moved). Its graph is symmetric around each maximum and minimum, and it spends equal time above and below its axis.

A periodic function can have sharp corners, straight pieces or jumps and still repeat: for example, the height of water in a tank that fills slowly and then empties all at once, or the electrical signal of a heartbeat. The diagram below compares the three cases.

Left: a sinusoidal function (smooth wave). Middle: a periodic function that is not sinusoidal (it rises in straight lines and drops suddenly). Right: a function that is not periodic (no cycle repeats).

Worked example

Example 3: Periodic or not?

Which of these are periodic? For each periodic one, estimate the period.

(a) The height of the tip of a clock's minute hand above the floor, over a day. (b) The temperature of a cup of hot chocolate left on the counter. (c) The number of hours of daylight in Thunder Bay each day, over five years.

(a) The minute hand goes around once every hour, and each turn is identical. Periodic, period 60 min.
(b) The hot chocolate cools toward room temperature and stays there. Nothing repeats. Not periodic.
(c) Daylight is longest in June and shortest in December, every year. Periodic (approximately), period about 365 days (1 year).

(a) and (c) are periodic, with periods of about 60 min and 365 days.

Just for fun

I tried to tell a joke about periodic functions.

Then I tried to tell it again. And again. And again… It has a period of about one groan.

Hands-on activity

Build a paper-plate Ferris wheel

You need: a paper plate, a pushpin or thumbtack, a piece of cardboard, masking tape, a ruler, a pencil and graph paper. (Take care with the pin; a grown-up can help push it through.)

Tape the cardboard upright against a wall or the fridge door so that its bottom edge rests on the counter.
Draw a line through the centre of the plate and mark 8 equally spaced points around the rim like the points of a compass (every 45°). Number them 0 to 7. Put a big dot on point 0.
Pin the plate through its centre to the cardboard so it can turn. Turn it so the dot is at the bottom.
Turn the plate one-eighth of a turn at a time (to the next numbered point). Each time, measure the height of the dot above the counter. Do two full turns (17 measurements, counting the start).
Plot height (vertical) against the number of eighth-turns (horizontal). Join the points with a smooth curve.
Find the period, maximum, minimum, axis and amplitude of your graph.

What to look for: a smooth wave whose period is 8 eighth-turns (one full turn). The amplitude should equal the radius of the plate, and the axis should be the height of the pin above the counter. Small differences (a few millimetres) are measuring error.

Your turn

A periodic function repeats every 8 s. Its maximum is 10 and its minimum is 2. Find its period, amplitude, equation of the axis and range.

Is this function periodic? If so, give the period, amplitude and axis.

x: 0, 1, 2, 3, 4, 5, 6, 7, 8
y: 1, 4, 1, −2, 1, 4, 1, −2, 1

Is the bounce height of a dropped basketball (each bounce lower than the one before) a periodic function of time? Explain.

Is the height above the floor of the tip of a clock's second hand a periodic function of time? If so, what is its period?

A function has period 5 and f(2) = 7. Find f(12).

A wave goes from a maximum of 3 down to a minimum of −5. Find the amplitude and the axis.

Explain why the amplitude is (max − min) ÷ 2 and not max − min.

More practice: warm-up

In your own words, what is the period of a periodic function?

A periodic function has a maximum of 9 and a minimum of 1. Find the amplitude.

For the same function, write the equation of the axis.

One cycle of a graph starts at t = 3 s and ends at t = 11 s. What is the period?

Write the range of a periodic function with maximum 9 and minimum 1.

More practice: on your own

A Ferris wheel has a radius of 15 m, its axle is 17 m above the ground, and it turns once every 2 min. For a rider's height, find the maximum, minimum, amplitude, axis and period.

At one harbour, high tide is 11.2 m at 2:00 a.m. and the next low tide is 0.8 m at 8:12 a.m. Assume the tide is periodic. Find the amplitude, the axis and the period.

A periodic graph has a maximum at x = 40 and the next maximum at x = 130. Find the period and say where you expect the minimum between them (for a sinusoidal graph).

A function has period 6, f(1) = 3 and f(2) = −1. Find f(13) and f(−4).

The water level in a tank rises steadily from 0 cm to 50 cm over 4 min, then the tank empties almost instantly, and this repeats. Is this function periodic? Is it sinusoidal? Give the period and amplitude.

Challenge

A sinusoidal function has a maximum of 4 at x = 10 and the next minimum of −8 at x = 25. Find the amplitude, axis and period, and the x-values of the next two maxima after x = 10.

The tides at a harbour have a period of about 12.4 h. High tide is at 6:00 a.m. on a Monday. Predict the next three high tides.

Can a periodic function have an inverse that is also a function? Explain.

Check your understanding

Quiz

The period of a periodic function is…

A periodic function has a maximum of 12 and a minimum of 4. What is its amplitude?

Which of these is a periodic function of time?

A periodic function has a maximum of 3 and a minimum of −7. What is the equation of its axis?

A function has period 10 and f(4) = 6. What is f(34)?

Which statement is true?

For parents and teachers

Big idea: D2.1 asks students to describe the cycle, amplitude and period of periodic functions from real data, and D3.2 asks them to tell periodic and sinusoidal functions apart. This is the vocabulary for the whole unit, so it is worth getting it solid.

How to help: Point out periodic things at home: the dishwasher cycle, a ceiling fan, the seasons, the phases of the Moon. Ask "What is one cycle? How long is the period?" For a graph, ask your teen to point to the maximum, the minimum and the axis before calculating.

Common difficulties:

Using max − min for the amplitude. The amplitude is half of that.
Reading the period as the distance from a maximum to the next minimum. That is only half a period; the period goes from a point to the next point where the pattern is the same (for example, maximum to maximum).
Thinking any up-and-down graph is periodic. A bouncing ball goes up and down, but its bounces shrink, so it does not repeat.
Forgetting units: the period is a time (or an angle), and the amplitude has the units of the y-values.