C. Trigonometric Functions · Sine Functions

Periodic Functions: Cycle, Period, Axis and Amplitude

C2.1 C3.2About 70 minutes

Learning goals

I can decide whether a graph or table shows a periodic function.
I can identify a cycle and find the period of a periodic function.
I can find the maximum, minimum, equation of the axis and amplitude.
I can use the period to find values of a periodic function at other inputs.

Words to know

periodic function
A function whose graph repeats a pattern at regular intervals of the independent variable.

cycle
One complete repeating section of a periodic graph.

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

maximum / minimum
The greatest and least output values of the function.

axis of the curve
The horizontal line halfway between the maximum and minimum: y = (max + min) ÷ 2.

amplitude
The vertical distance from the axis to the maximum or minimum: (max − min) ÷ 2. Always positive.

sinusoidal function
A periodic function whose graph is a smooth wave with the shape of the sine curve.

Things that come around again

Think about a day in January in Ontario. The sun rises, climbs, sets, and the next day it does it all again. A ceiling fan blade passes the same spot again and again. Your heart beats in a pattern. The tide at a beach in Nova Scotia rises and falls about twice a day.

Each of these is a pattern that repeats at regular intervals. In math, a relationship like this is called a periodic function. In this unit you will learn to describe periodic functions, build the most important one (the sine function), move it around with transformations, and use it to solve real problems.

Remember from Grade 10 and the Quadratic Functions unit

In Grade 10 trigonometry you learned the primary trigonometric ratios for right triangles, including sin θ = opposite ÷ hypotenuse. In this course's unit Trigonometry of Acute Triangles you used them again, along with the sine law and cosine law. In the Quadratic Functions unit you learned that a function gives exactly one output for each input, and you used domain and range. All of these ideas come back in this unit.

What makes a function periodic?

A function is periodic if its graph repeats the same shape over and over at regular intervals of the independent variable (usually time or angle).

One complete repeating piece of the graph is a cycle.
The horizontal length of one cycle is the period. You can measure it from any point to the next point where the pattern starts again, for example from one peak to the next peak.
The maximum is the greatest output value. The minimum is the least output value.
The axis of the curve (also called the equation of the axis) is the horizontal line halfway between the maximum and minimum:

axis: y = (max + min) ÷ 2

The amplitude is the vertical distance from the axis to the maximum (or to the minimum). It is half the distance from the minimum to the maximum:

amplitude = (max − min) ÷ 2

Amplitude is a distance, so it is never negative.

The height of a mark on a spinning wheel. The pattern repeats every 4 s, so the period is 4 s. The axis is halfway between max 8 and min 2, at h = 5, and the amplitude is 3 m.

Worked example

Example 1: Reading a periodic graph

Use the graph above of the height h, in metres, of a mark on a wheel after t seconds. Find (a) the maximum and minimum, (b) the period, (c) the equation of the axis, (d) the amplitude, (e) the height at t = 13 s.

(a) The highest points are at h = 8 and the lowest at h = 2. Max = 8 m, min = 2 m.
(b) Peaks occur at t = 1 and t = 5. Period = 5 − 1 = 4 s. (Check another pair: minimums at t = 3 and t = 7 also give 4 s.)
(c) Axis: h = (8 + 2) ÷ 2 = 5, so the axis is h = 5.
(d) Amplitude = (8 − 2) ÷ 2 = 3 m.
(e) The pattern repeats every 4 s, so h(13) = h(13 − 4 − 4 − 4) = h(1). At t = 1 the graph is at a peak, so h(13) = 8 m.

Max 8 m, min 2 m, period 4 s, axis h = 5, amplitude 3 m, and h(13) = 8 m.

Periodic and sinusoidal

Some periodic graphs have sharp corners or flat parts, like a heartbeat trace or the distance of a swimmer from the end of a pool. Others are smooth, symmetrical waves. A smooth wave that has the same shape as the sine graph (which you will build in Lesson 4) is called sinusoidal. Every sinusoidal function is periodic, but not every periodic function is sinusoidal.

A graph is not periodic if no piece of it repeats exactly at regular intervals, even if it goes up and down.

Left: a repeating pattern with flat parts and corners, periodic but not sinusoidal. Middle: a smooth repeating wave, sinusoidal. Right: it goes up and down, but the shape changes, so it is not periodic.

Worked example

Example 2: Periodic behaviour in a table

The depth d, in metres, of water at the end of a dock is recorded every 3 hours.

t (h): 0, 3, 6, 9, 12, 15, 18, 21, 24
d (m): 5, 8, 5, 2, 5, 8, 5, 2, 5

Is the relationship periodic? If so, give the period, maximum, minimum, axis and amplitude.

Look for a block of values that repeats. From t = 0 to t = 12 the depths are 5, 8, 5, 2, 5. From t = 12 to t = 24 they are 5, 8, 5, 2, 5 again. The pattern repeats, so it is periodic.
Period: the pattern starts again every 12 h (for example the maximum 8 occurs at t = 3 and at t = 15, and 15 − 3 = 12).
Max = 8 m, min = 2 m.
Axis: d = (8 + 2) ÷ 2 = 5, so d = 5. Amplitude = (8 − 2) ÷ 2 = 3 m.

Periodic, period 12 h, max 8 m, min 2 m, axis d = 5, amplitude 3 m.

The dock data plotted as points. A smooth curve through them (dashed) is a sinusoidal model. Notice that one cycle (12 h) contains one high tide and one low tide.

Watch out

The period is a horizontal distance (time or angle). The amplitude is a vertical distance.
Amplitude is half the max-to-min distance, not the whole distance.
To find a period, use two matching points: peak to peak, trough to trough, or the same height going the same direction. Two points at the same height going opposite directions are usually not one period apart.

Worked example

Example 3: Same height, different direction

On the wheel graph, the height is 5 m at t = 0, t = 2 and t = 4. Is the period 2 s?

At t = 0 the graph is at h = 5 and rising.
At t = 2 it is at h = 5 but falling. This point is not at the same place in the cycle.
At t = 4 it is at h = 5 and rising again. This is the next matching point.
So the period is 4 − 0 = 4 s, not 2 s.

No. The period is 4 s. Matching points must have the same height and be moving in the same direction.

Just for fun

Why did the periodic function never worry about missing the bus?

It knew everything comes around again, right on schedule.

Hands-on activity

Periodic pattern hunt

You need: paper, a pencil, a ruler, and a clock or phone timer.

Walk around your home and list at least six things that change in a repeating way. Ideas: the height of the tip of a clock's minute hand, a ceiling fan blade, a washing machine drum, the light outside over several days, a swing, the hands of a wall clock.
Pick two of them. For each, sketch a graph of one measurable quantity against time for at least two cycles. Label both axes with units.
On each sketch, mark one cycle, the period, the maximum, the minimum and the axis. Estimate each value (for example, period of the minute hand = 60 min).
Decide whether each graph is sinusoidal (smooth wave) or only periodic (corners or flat parts).

What to look for: the minute-hand height has period 60 min and is sinusoidal. The amount of outdoor light over several days has a period of about 24 h but has flat parts at night, so it is periodic but not sinusoidal. A swing slowly loses height unless someone keeps pushing, so it is only roughly periodic.

Your turn

A periodic function has a maximum of 14 and a minimum of 4. Find the equation of the axis and the amplitude.

The peaks of a periodic graph are at x = 2, x = 9 and x = 16. What is the period?

A function is periodic with period 10, and f(3) = 6. Find f(13), f(23) and f(−7).

Is the pattern 2, 5, 2, 5, 2, 5, ... (one value per second) periodic? If so, what is its period?

The axis of a periodic graph is y = 3 and its amplitude is 4. Find the maximum and minimum.

Explain the difference between a periodic function and a sinusoidal function.

More practice: warm-up

What is one complete repeating piece of a periodic graph called?

Max = 10, min = 2. Find the amplitude.

Max = 10, min = 2. Find the equation of the axis.

Troughs (lowest points) of a graph are at t = 5 s and t = 17 s, with none in between. What is the period?

True or false: the amplitude can be negative.

More practice: on your own

A Ferris wheel seat is 1 m above the ground at its lowest point and 31 m above the ground at its highest point. One full turn takes 50 s. Find the period, amplitude and equation of the axis of the height graph.

Is this table periodic? x: 0, 1, 2, 3, 4, 5, 6, 7, 8; y: 0, 3, 0, −3, 0, 3, 0, −3, 0. Give the period, axis and amplitude.

Is this table periodic? x: 0, 1, 2, 3, 4, 5, 6; y: 1, 4, 1, 5, 1, 6, 1. Explain.

A periodic function has period 6 and g(1) = −2, g(2) = 0, g(4) = 5. Find g(7), g(16) and g(−2).

The height of a buoy bobbing on waves goes from a high of 1.6 m to a low of 0.4 m. Find its amplitude and axis.

Challenge

A periodic function has axis y = −2 and amplitude 3.5. What is its range? Explain.

A function is periodic with period 8. One cycle starts at x = 0 with a maximum. Name four different x-values between 0 and 40 where it must have a maximum, and explain how you know.

Max-to-min distance is 12 cm and the maximum is 20 cm. A student says the amplitude is 12 cm and the axis is at 14 cm. Find and fix both errors.

Check your understanding

Quiz

Which quantity is a horizontal distance on the graph of a periodic function?

A periodic graph has max 9 and min −3. What is the amplitude?

A periodic graph has max 9 and min −3. What is the equation of the axis?

f is periodic with period 5 and f(2) = 7. Which value must equal 7?

Which situation is not periodic?

Which statement is true?

For parents and teachers

Big idea: This lesson starts strand C2 (describing periodic functions from graphs and tables, expectation C2.1) and the skill of identifying periodic and sinusoidal functions (C3.2).

How to help: Point out periodic patterns in daily life: the phases of the moon, the clock, seasonal temperatures, the spin cycle. Ask: "What is one cycle? How long does it take? What are the highest and lowest values?"

Common difficulties:

Using the whole max-to-min distance as the amplitude. Amplitude is half of it.
Measuring the period between two points at the same height that move in opposite directions. Have your teen check "same height, same direction."
Mixing up the axis (a horizontal line, written y = ...) with the amplitude (a distance).