D2.1 Describe properties of periodic functions in applications

Ontario Grade 11 Mathematics

Trigonometric Functions

Recognizing and Describing Repeating Patterns in Real-World Contexts

Have you ever noticed that the height of a Ferris wheel seat, the depth of ocean tides, or the loudness of a musical note all follow patterns that repeat over and over? These are real-life examples of periodic behaviour. In this lesson you will learn to spot those repeating patterns, name their key features precisely, and describe what those features mean in context. No new formula is harder than what you already met in Grade 10 — the focus here is on reading graphs carefully and using the right mathematical language to describe what you see.

What you will learn

Identify whether a real-world graph or data set represents a periodic function.
Define and calculate the period, amplitude, equation of the axis, and range of a periodic function.
Describe the properties of a periodic function using correct vocabulary in the context of an application.
Distinguish between periodic and non-periodic functions using key features of their graphs.

Prerequisite Bridge: Functions and Their Graphs

Before Grade 11, you learned that a function is a rule that assigns exactly one output to each input. You also learned to read key features of a graph: where it rises and falls, its highest and lowest points, and how the y-values behave as x increases. Those skills are the foundation of today's lesson.

Recall that the domain of a function is the set of all allowed inputs, and the range is the set of all possible outputs. When the graph has a clear maximum (highest point) and minimum (lowest point), the range runs from the minimum to the maximum. Keep these ideas in mind as you meet periodic functions, because every property defined below connects directly to these familiar graph-reading skills.

A function assigns exactly one output to every input.
The range is the set of all output values a function actually reaches.
Identifying maxima and minima on a graph is a Grade 10 skill used throughout this lesson.

What Makes a Function Periodic?

A function is called periodic if its graph repeats the same pattern over and over at regular intervals along the horizontal axis. Think of a swing: after one complete back-and-forth motion, it is in exactly the same position moving in the same direction. The graph of its height versus time would show identical humps repeating endlessly.

Formally, a function

f

is periodic if there is a positive number

p

such that

f(x + p) = f(x)

for every value of

x

in the domain. In plain language: shifting the graph to the right by exactly

p

units places it perfectly on top of itself. The smallest positive value of

p

that does this is called the period.

Not every graph that goes up and down is periodic. The key test is regularity: the pattern must repeat in chunks of equal width. A graph that bounces around unpredictably, or whose humps get wider or taller over time, is not periodic.

f(x + p) = f(x)

A periodic function repeats the same pattern at equal horizontal intervals.
The period is the length of one complete repeating cycle, measured along the horizontal axis.
Use the rule $f(x + p) = f(x)$ to confirm periodicity: the output is the same at $x$ and at $x + p$ .
Irregular or growing oscillations are not periodic.

Key Properties: Amplitude, Axis, and Range

Once you confirm a function is periodic, you describe it using four main properties: period, amplitude, equation of the axis, and range.

The period

p

is the horizontal distance for one full cycle. To measure it from a graph, find two consecutive points that look identical — same height, same direction of movement — and subtract their x-coordinates:

p = x_2 - x_1

.

The maximum value is the greatest output the function reaches, and the minimum value is the least. The amplitude

a

measures half the total vertical spread of the function. It tells you how far the graph reaches above or below the central horizontal line. The formula is

a = \frac{\text{max} - \text{min}}{2}

. Amplitude is always a positive number.

The equation of the axis is the equation of the horizontal line that runs exactly halfway between the maximum and the minimum. It acts like a 'rest position' for the function. Its equation is

y = \frac{\text{max} + \text{min}}{2}

.

Finally, the range of the function is the set of all y-values from the minimum to the maximum, written as

\text{min} \leq y \leq \text{max}

. Together, these four properties give a complete numerical portrait of any periodic function.

a = \frac{\mathrm{max} - \mathrm{min}}{2}

Period: horizontal length of one complete cycle, $p = x_2 - x_1$ between two matching points.
Amplitude: $a = \frac{\text{max} - \text{min}}{2}$ , always positive.
Equation of the axis: $y = \frac{\text{max} + \text{min}}{2}$ .
Range: all y-values from minimum to maximum, inclusive.
Each property has a meaning in the real-world context, not just a number.

Reading Properties Directly from an Application Graph

In an application problem, the graph axes carry units and labels. The period is not just a number — it is a number with a unit such as seconds, hours, or days. Always state the unit when describing the period in context.

For example, a graph of water depth (in metres) at a harbour over 24 hours might show the depth rising and falling twice in that time. That means each cycle takes 12 hours, so the period is 12 hours. If the maximum depth is 5.4 m and the minimum depth is 1.2 m, then the amplitude is

\frac{5.4 - 1.2}{2} = \frac{4.2}{2} = 2.1

m, the equation of the axis is

y = \frac{5.4 + 1.2}{2} = \frac{6.6}{2} = 3.3

, and the range is

1.2 \leq y \leq 5.4

.

Notice that the equation of the axis,

y = 3.3

, represents the average water depth. The amplitude, 2.1 m, tells you the water rises 2.1 m above and falls 2.1 m below that average. Linking numbers to their contextual meaning is exactly what 'describing properties in applications' requires.

When deciding whether a graph is periodic, ask: Does the overall shape repeat? Are the cycles the same width? Are the peaks and troughs equally high and low each time? If yes to all three, the function is periodic and you can measure its properties.

y = \frac{\mathrm{max} + \mathrm{min}}{2}

Always attach the correct unit to the period when working in context.
The equation of the axis represents the average or 'middle' value in many real situations.
The amplitude represents how far the quantity varies above and below its average.
Check that cycles are equal in width and equal in height before calling a function periodic.

Periodic vs. Non-Periodic: Making the Distinction

A common task is deciding whether a given graph or table of values describes a periodic function. Look for three signs of periodicity: equal cycle width, equal maximum values, and equal minimum values across all visible cycles. If any of these is missing, the function is not periodic.

For example, a graph of a bouncing ball loses height with each bounce because of energy loss. Although it goes up and down repeatedly, the peaks get shorter and the cycle widths change. This graph is not periodic. By contrast, an ideal pendulum swinging without friction produces identical cycles indefinitely and is periodic.

A straight line, a parabola, or an exponential curve never repeats its pattern, so none of these is periodic. A flat horizontal line does technically repeat (it looks the same at every shift), but by convention a constant function's period is undefined because there is no smallest positive period.

Practise the distinction by sketching several graphs and asking yourself: if I covered the left half and showed only the right half, could a reader reconstruct the left half exactly? If yes, the graph is likely periodic.

Periodic graphs have equal cycle widths and equal peak and trough heights throughout.
A decreasing oscillation (like a real bouncing ball) is not periodic.
Linear, quadratic, and exponential functions are not periodic.
A constant function does not have a well-defined period.

Summary of Periodic Function Properties

Property	Formula / How to Find It	What It Tells You in Context
Period ( $p$ )	$p = x_2 - x_1$ between matching points	How long one complete cycle takes
Amplitude ( $a$ )	$a = \frac{\text{max} - \text{min}}{2}$	How far the quantity swings above or below the centre
Equation of the axis	$y = \frac{\text{max} + \text{min}}{2}$	The average or 'rest' value of the quantity
Range	$\text{min} \leq y \leq \text{max}$	All possible output values the quantity can reach

Worked example

Example 1 — Ferris Wheel Height

A Ferris wheel completes one full rotation every 40 seconds. A rider's height above the ground, in metres, is recorded over time. The maximum height reached is 22 m and the minimum height is 2 m. The graph of height versus time is periodic. Determine the period, amplitude, equation of the axis, and range. Then describe what the amplitude means in this context.

Identify the period
The problem states the wheel completes one full rotation every 40 seconds. One full rotation is one complete cycle of the height graph, so the period is 40 seconds.
$p = 40 \text{ s}$
Calculate the amplitude
The amplitude is half the vertical distance between the maximum and minimum heights. Substitute max $= 22$ and min $= 2$ into the amplitude formula.
$a = \frac{22 - 2}{2} = \frac{20}{2} = 10$
Find the equation of the axis
The equation of the axis is the average of the maximum and minimum heights. This gives the horizontal line that runs through the middle of the graph.
$y = \frac{22 + 2}{2} = \frac{24}{2} = 12$
State the range
The rider's height goes no lower than 2 m and no higher than 22 m, so the range includes every value between those two extremes.
$2 ≤ y ≤ 22$
Interpret the amplitude in context
The amplitude is 10 m. This means the rider's height rises 10 m above the axis and falls 10 m below it during each cycle. In other words, the rider travels 10 m above the centre height and 10 m below it with every rotation.

Answer: Period: 40 s; Amplitude: 10 m; Equation of the axis:

y = 12

; Range:

2 \leq y \leq 22

. The amplitude of 10 m means the rider's height varies 10 m above and below the centre height of 12 m.

Check: Check the axis:

12 - 10 = 2

(matches the minimum) and

12 + 10 = 22

(matches the maximum). Both are correct, so the amplitude and axis equation are consistent.

Worked example

Example 2 — Tide Depth at a Coastal Harbour

A graph of water depth (in metres) at a coastal harbour is recorded over 72 hours. The graph shows three identical cycles in that time. The highest depth recorded is 6.8 m and the lowest is 0.4 m. Determine all four properties of this periodic function and explain what the equation of the axis represents for a harbour master planning boat arrivals.

Find the period
Three complete cycles occur in 72 hours. Divide the total time by the number of cycles to find the length of one cycle.
$p = \frac{72}{3} = 24 \text{ h}$
Calculate the amplitude
Use the amplitude formula with max $= 6.8$ m and min $= 0.4$ m.
$a = \frac{6.8 - 0.4}{2} = \frac{6.4}{2} = 3.2$
Find the equation of the axis
Average the maximum and minimum depths to locate the central horizontal line of the graph.
$y = \frac{6.8 + 0.4}{2} = \frac{7.2}{2} = 3.6$
State the range
The water depth takes every value from the minimum of 0.4 m up to the maximum of 6.8 m throughout the day.
$0.4 ≤ y ≤ 6.8$
Interpret the equation of the axis in context
The equation of the axis is $y = 3.6$ , meaning the average water depth is 3.6 m. A harbour master could use this as an estimate of the 'typical' depth. Boats requiring more than 3.6 m of water can only enter during the upper half of each tidal cycle, which lasts roughly half of 24 hours, or about 12 hours per day.

Answer: Period: 24 h; Amplitude: 3.2 m; Equation of the axis:

y = 3.6

; Range:

0.4 \leq y \leq 6.8

. The axis value of 3.6 m is the average depth, useful for scheduling boat traffic.

Check: Check:

3.6 + 3.2 = 6.8

(matches the maximum) and

3.6 - 3.2 = 0.4

(matches the minimum). The values are consistent.

Common mistakes and how to avoid them

Measuring the period as the distance from a minimum to the next maximum, which is only half a cycle.

Correction: Always measure the period from one point to the next point that is identical in both height and direction of movement — that is one full cycle.

Stating the amplitude as the full vertical distance from the minimum to the maximum (e.g., writing 20 instead of 10 in Example 1).

Correction: The amplitude is half the vertical spread:

a = \frac{\text{max} - \text{min}}{2}

. Divide by 2 after subtracting.

Forgetting to include units when describing the period or amplitude in a real-world problem.

Correction: Always attach the unit from the graph axis — seconds, hours, metres, etc. — when stating any property in context.

Calling a graph periodic just because it goes up and down, even when the peaks get shorter each time.

Correction: Check that all peaks reach the same maximum and all troughs reach the same minimum before labelling the function periodic.

Confusing the equation of the axis with the amplitude, or reporting only a number instead of writing

y = \text{value}

.

Correction: The equation of the axis is a horizontal line, so it must be written as an equation: for example,

y = 3.6

, not just 3.6.

Lesson summary

A periodic function is one whose graph repeats an identical pattern at equal horizontal intervals called cycles.
The period is the horizontal length of one complete cycle and must be reported with its unit in context.
The amplitude is half the vertical distance between the maximum and minimum values: $a = \frac{\text{max} - \text{min}}{2}$ .
The equation of the axis, $y = \frac{\text{max} + \text{min}}{2}$ , is the horizontal line halfway between the maximum and minimum and often represents an average value in context.
The range of a periodic function spans from its minimum to its maximum output value.
A function is periodic only if its cycles are equal in width and equal in height; decreasing oscillations and non-repeating curves such as parabolas are not periodic.

Check your understanding

Question 1

A periodic graph has a maximum value of 14 and a minimum value of 2. What is the amplitude?

16
12
6
7

Show answer and explanation

6

Amplitude = (max − min) ÷ 2 = (14 − 2) ÷ 2 = 12 ÷ 2 = 6. Option A adds instead of subtracts; option B forgets to divide by 2; option D uses the wrong formula.

Question 2

A graph shows 4 complete cycles over 48 seconds. What is the period?

48 s
4 s
192 s
12 s

Show answer and explanation

12 s

Period = total time ÷ number of cycles = 48 ÷ 4 = 12 seconds. Option A is the total time, not one cycle; option B is the number of cycles; option C multiplies instead of divides.

Question 3

Which of the following graphs is NOT periodic?

The height of a Ferris wheel seat over time
The depth of ocean tides over several days
The height of a ball dropped from a building, which bounces lower each time
The position of a pendulum swinging at a constant rate

Show answer and explanation

The height of a ball dropped from a building, which bounces lower each time

A ball that bounces lower on each bounce produces cycles of decreasing height, so the pattern is not truly repeating. The Ferris wheel, ocean tides, and constant-rate pendulum all repeat identical cycles and are periodic.

Question 4

A periodic function has a maximum of 9 and a minimum of 3. What is the equation of the axis?

$y = 6$
$y = 3$
$y = 9$
$y = 4.5$

Show answer and explanation

y = 6

Equation of the axis = (max + min) ÷ 2 = (9 + 3) ÷ 2 = 12 ÷ 2 = 6, so

y = 6

. Option B is the minimum; option C is the maximum; option D incorrectly halves only the minimum.

Key terms

Periodic function: A function whose graph repeats the same pattern at equal horizontal intervals.
Period: The horizontal length of one complete repeating cycle of a periodic function.
Cycle: One complete repetition of the repeating pattern in a periodic function.
Amplitude: Half the vertical distance between the maximum and minimum values of a periodic function; always a positive number.
Equation of the axis: The equation of the horizontal line located exactly halfway between the maximum and minimum values, written in the form $y = k$ .
Maximum value: The greatest output value reached by a function on its graph.
Minimum value: The least output value reached by a function on its graph.
Range: The complete set of output values a function can produce, from its minimum to its maximum.

Continue through MCR3U

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D2.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

D2.1 · Describe properties of periodic functions in applications

Ontario Grade 11 Mathematics

Recognizing and Describing Repeating Patterns in Real-World Contexts

What you will learn

Prerequisite Bridge: Functions and Their Graphs

What Makes a Function Periodic?

Key Properties: Amplitude, Axis, and Range

Reading Properties Directly from an Application Graph

Periodic vs. Non-Periodic: Making the Distinction

Summary of Periodic Function Properties

Example 1 — Ferris Wheel Height

Example 2 — Tide Depth at a Coastal Harbour

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson