C2.1 Describe properties of periodic functions in applications

Ontario Grade 11 Mathematics

Trigonometric Functions

Recognizing repeating patterns, their cycles, and their useful properties

Many situations repeat in a regular way. A rotating wheel returns to the same position. The height of a tide rises and falls in a repeating pattern. A graph can represent these changes over time. In this lesson, you will describe the repeating cycle and explain what important features mean in the situation. You will use familiar ideas about graphs and sine, but the focus is on interpreting the model, not on advanced algebra.

What you will learn

Describe what makes a function periodic.
Identify a period, maximum, minimum, and range from an application.
Connect a repeating situation to a graph, a table, and a simple sine model.
Explain what a function’s properties mean in the context of a real situation.

1. A bridge from graphs to repeating patterns

A function connects an input to an output. In an application, the input might be time and the output might be a height or temperature. A graph shows how the output changes as the input changes.

The range is the set of output values a function can have. The maximum is the greatest output in a cycle, and the minimum is the least output in a cycle. These ideas help describe how high, low, or wide a repeating pattern is.

A periodic function repeats its outputs in a regular pattern. The period is the length of one complete repeat. If the input is time in seconds, the period is measured in seconds. After one period, the function starts the same pattern again.

The period describes how long one complete cycle takes.
Maximum and minimum describe the highest and lowest outputs in a cycle.
The range describes the outputs that occur.

2. Read the pattern in more than one way

Start with a rotating light that moves up and down in a repeating path. Suppose it reaches a height of 6 metres at its highest point and 2 metres at its lowest point. It takes 8 seconds to complete one full cycle.

In words, the light repeats every 8 seconds. Its height stays between 2 metres and 6 metres. The maximum height is 6 metres, and the minimum height is 2 metres.

A table can show selected points in a cycle. The output values at 0 and 8 seconds match because those times are one full period apart. Other matching values occur at equal positions in later cycles. A graph would show the same rise and fall repeating every 8 seconds.

The range for this situation is all heights from 2 to 6 metres, including both endpoints. The endpoints matter because the light does reach its highest and lowest points.

Use the units of the input when stating a period.
Use the units of the output when stating a maximum, minimum, or range.
Values one period apart repeat, and the graph’s shape repeats as well.

3. Describe a simple sine model

The sine function is useful for modelling smooth, repeating changes. A basic sine graph repeats after its angle has increased by 360°. When the angle is linked to time, the time needed to reach the next repeat is the period.

For example, a simple model for a moving object’s height could be written as

h=4+2\sin(30t)

, with the angle measured in degrees and

t

measured in seconds. This model has a middle height of 4 metres. The sine part ranges from

-1

to 1, so multiplying it by 2 gives a change from

-2

to 2. Adding 4 gives heights from 2 to 6 metres.

The angle increases by 30° each second. It takes 12 seconds to increase by 360°, so the model repeats every 12 seconds. In this example, the period is found from the rate of angle change. The model is only useful if its units and repeating behaviour fit the situation.

A model gives more than a formula. Its properties have meanings. The period describes the timing of the cycle; the maximum and minimum describe the output limits; and the range describes every output value the model can produce.

h=4+2\sin(30t)

A sine model can describe smooth, repeating change.
Check that the model’s period and output range make sense in the application.
Always state what the variables and units represent.

4. Use properties to explain an application

When describing a periodic situation, connect each mathematical property to the real event. Instead of saying only that the period is 8, say that the event completes a cycle every 8 seconds. Instead of listing a range, explain what the lower and upper outputs represent.

A graph or table may show only part of a cycle. Look for a complete repeat before deciding on the period. For example, the time from one maximum to the next maximum is one full period. The time from a maximum to the next minimum is only half of a cycle when the pattern is a regular sine wave.

A periodic model is an approximation of a real situation. Real measurements may not match the model perfectly. Describe what the model predicts, and use the application’s units and context when interpreting its properties.

Find matching positions in successive cycles to identify a period.
Distinguish a full cycle from part of a cycle.
Interpret mathematical values in context, with units.

Selected heights in one cycle

Time (seconds)	Height (metres)
$0$	$4$
$2$	$6$
$4$	$4$
$6$	$2$
$8$	$4$

Worked example

Describe a rotating platform’s height

A sensor records the height of a point on a rotating platform. Its height is modelled by

h=5+1.5\sin(45t)

, where

h

is in metres,

t

is in seconds, and the sine angle is in degrees. Describe the period, maximum, minimum, and range. Explain what the results mean.

Interpret the output
The sine part can range from $-1$ to 1. Multiplying by 1.5 makes its values range from $-1.5$ to 1.5. The added 5 shifts these outputs so the height ranges from 3.5 to 6.5 metres.
$5-1.5=3.5,\quad 5+1.5=6.5$
Find the period
The angle increases by 45° each second. One full sine cycle is 360°, so divide 360° by the increase of 45° per second to find the time for one cycle.
$360\div45=8$
State the properties in context
The model predicts that the platform point reaches a maximum height of 6.5 metres and a minimum height of 3.5 metres. It repeats its height pattern every 8 seconds, so the range is all heights from 3.5 to 6.5 metres, inclusive.
$3.5\le h\le6.5$

Answer: The period is 8 seconds. The maximum height is 6.5 metres, the minimum height is 3.5 metres, and the range is 3.5 to 6.5 metres inclusive.

Check: The angle advances by 45° for each second. In 8 seconds it advances by 360°, which is one full sine cycle.

Common mistakes and how to avoid them

Calling the time from a maximum to the next minimum one full period.

Correction: That interval is half a cycle for a regular sine pattern. Find the time between matching positions, such as one maximum and the next maximum.

Giving a period without its units.

Correction: State the input units. If the input is time in seconds, report the period in seconds.

Confusing the range with the period.

Correction: The period measures the input interval for one repeat. The range lists output values.

Reporting a model’s maximum or minimum without explaining what it represents.

Correction: Name the output and its units. For example, say that the model predicts a maximum height of 6.5 metres.

Lesson summary

A periodic function repeats its outputs in a regular pattern.
The period is the input interval for one complete repeat.
Maximum and minimum are the greatest and least outputs in a cycle.
The range contains all possible output values.
In applications, describe each property with its meaning and units.

Check your understanding

Question 1

A machine’s repeating height pattern takes 10 seconds to return from one maximum to the next. What is its period?

5 seconds
10 seconds
20 seconds
correctIndex daga

Show answer and explanation

10 seconds

The interval from one maximum to the next maximum is one complete repeat, so the period is 10 seconds.

Question 2

A periodic temperature model has a minimum of 12°C and a maximum of 20°C. Which statement gives its range?

From 12°C to 20°C, inclusive
12 hours to 20 hours
Every 8°C
correctIndex daga

Show answer and explanation

From 12°C to 20°C, inclusive

The range describes output values. The temperatures in the model go from 12°C to 20°C, including the minimum and maximum.

Question 3

A sine model’s angle increases by 60° each minute. How long does it take for the angle to increase by 360°?

6 minutes
60 minutes
300 minutes
correctIndex daga

Show answer and explanation

6 minutes

A full sine cycle is 360°. At 60° per minute, it takes 360 divided by 60, or 6 minutes.

Key terms

Periodic function: A function whose outputs repeat in a regular pattern.
Period: The input interval needed for one complete repeat.
Maximum: The greatest output value in a cycle.
Minimum: The least output value in a cycle.
Range: The set of output values a function can have.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

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C2.1 · Describe properties of periodic functions in applications

Ontario Grade 11 Mathematics

Recognizing repeating patterns, their cycles, and their useful properties

What you will learn

1. A bridge from graphs to repeating patterns

2. Read the pattern in more than one way

3. Describe a simple sine model

4. Use properties to explain an application

Selected heights in one cycle

Describe a rotating platform’s height

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson