D3.2 Identify periodic and sinusoidal models and contextual restrictions

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U · Strand D · Expectation D3.2

Many real-world quantities repeat in a predictable pattern: the height of a Ferris wheel seat, the depth of water at a harbour, the hours of daylight through a year. Mathematics captures this repetition with periodic functions. Among all periodic functions, one family — sinusoidal functions — is especially common because its smooth, wave-like shape matches many natural phenomena closely. In this lesson you will learn to recognise both kinds of models, extract their key measurements, and apply the restrictions that a real-life context places on the domain and range.

What you will learn

Explain what makes a function periodic and identify its key features from a graph or table of values.
Distinguish between a general periodic function and a sinusoidal function.
Identify the amplitude, period, equation of the axis, and phase shift of a sinusoidal model.
Recognise contextual restrictions that limit the domain or range of a periodic or sinusoidal model.
Match a real-world situation to either a periodic or a sinusoidal model and justify the choice.

Prerequisite Bridge: What You Already Know

In Grade 10 you studied the sine and cosine ratios in right triangles and on the unit circle. You also worked with function notation, so you know that

f(x)

means the output of function

f

at input

x

. This lesson builds directly on both ideas.

You also know that the domain of a function is the set of allowed input values and the range is the set of possible output values. Keep those definitions in mind — context will shrink both sets in this lesson.

Sine and cosine produce values between $-1$ and $1$ for any angle input.
Domain = allowed inputs; Range = possible outputs.
Function notation $f(x)$ is used throughout this lesson.

Periodic Functions: The Idea of Repetition

A function is called periodic if its output values repeat at regular intervals. More precisely, a function

f

is periodic if there is a positive number

T

such that

f(x + T) = f(x)

for every value of

x

in the domain. The smallest such positive value of

T

is called the period — it is the length of one complete cycle.

Think of a clock's minute hand. After exactly 60 minutes it returns to the same position. If you graphed the height of the tip of the minute hand over time, the graph would look identical every 60 minutes. That repeated pattern is a period of 60 minutes.

You can spot a periodic function on a graph by checking that the wave pattern (hills and valleys) repeats at equal horizontal distances. The period

T

is measured from any point on the graph to the very next point with the same height moving in the same direction — for example, from one peak to the next peak.

Two other key features to identify are the maximum value (the highest output, called the crest or peak) and the minimum value (the lowest output, called the trough). From these two values you can calculate the equation of the axis and the amplitude.

f(x + T) = f(x)

A function is periodic when $f(x + T) = f(x)$ for all $x$ , where $T > 0$ .
The period $T$ is the length of one complete cycle, read as a horizontal distance on a graph.
Measure $T$ from peak to peak, or trough to trough, or any point to the next matching point.
Maximum value and minimum value are the highest and lowest outputs of the function.

Key Measurements of a Periodic Function

Once you know the maximum and minimum values of a periodic function, four measurements fully describe its shape and position.

The equation of the axis is the horizontal line exactly halfway between the maximum and the minimum. It represents the average (middle) value of the function. Its equation is

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

. On a graph it sits right through the centre of the wave.

The amplitude is the vertical distance from the equation of the axis up to the maximum (or equivalently, down to the minimum). It tells you how 'tall' the wave is:

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

. Amplitude is always a positive number.

Not every periodic function is sinusoidal. A square wave — one that jumps instantly between two levels and stays flat — is periodic but not sinusoidal. A sinusoidal function has a specific smooth, curved shape that comes from the sine or cosine curve. Recognising this smooth, continuous wave shape is the first step in deciding whether to use a sinusoidal model.

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

Equation of the axis: $y = \frac{\text{max} + \text{min}}{2}$
Amplitude: $a = \frac{\text{max} - \text{min}}{2}$ , always positive.
Period: horizontal distance from one peak to the next.
Sinusoidal functions are smooth, continuous, and wave-shaped — not jagged or stepped.

Sinusoidal Models: Form and Parameters

A sinusoidal function is any function whose graph has the same smooth, symmetric wave shape as

y = \sin x

or

y = \cos x

. In this course all angle inputs are measured in degrees. The general form of a sinusoidal model is

f(x) = a\sin(k(x - d)) + c

, where each letter controls a specific feature of the graph.

The parameter

a

is the amplitude. It stretches or compresses the wave vertically. If

a

is negative the wave is reflected (flipped upside down), but the amplitude is still |a|, which is always positive.

The parameter

k

is a dimensionless scale factor that controls the period. The relationship between

k

and the period

T

is

T = \frac{360^\circ}{k}

. A larger

k

squeezes the wave horizontally, producing a shorter period. You find

k

by rearranging:

k = \frac{360^\circ}{T}

.

The parameter

d

is the phase shift — a horizontal translation. A positive

d

shifts the graph to the right; a negative

d

shifts it to the left. The phase shift tells you where one cycle starts relative to the origin.

The parameter

c

is the vertical shift. It equals the value of the equation of the axis:

y = c

. It moves the entire wave up or down. The cosine form

f(x) = a\cos(k(x - d)) + c

describes the same family and all four parameters mean exactly the same things.

f(x) = a\sin(k(x - d)) + c

General sinusoidal form (degrees): $f(x) = a\sin(k(x-d)) + c$ .
|a| $= amplitude;$ c = equation of the axis, written $y = c$ .
Period $T = \frac{360^\circ}{k}$ ; equivalently $k = \frac{360^\circ}{T}$ . The value of $k$ is dimensionless.
$d$ = phase shift (positive $d$ shifts the graph right).
Cosine form $f(x) = a\cos(k(x-d)) + c$ belongs to the same sinusoidal family.

Contextual Restrictions on Domain and Range

A mathematical sinusoidal function is defined for all real number inputs and produces outputs between

c - |a|

and

c + |a|

forever in both directions. Real-world situations rarely allow this. A contextual restriction is a limitation on the domain or range that comes from the meaning of the situation, not from the mathematics itself.

Domain restrictions arise from time or physical limits. For example, a Ferris wheel model makes sense only while the wheel is actually running. If the ride lasts 5 minutes, the domain is

0 \leq t \leq 5

(with

t

in minutes) rather than all real numbers. You cannot have negative time in most physical situations.

Range restrictions arise from physical limits on the output. The height of a seat on a Ferris wheel cannot be below ground level and cannot exceed the top of the wheel. Even if the mathematical formula could produce values outside those bounds, the physical context caps the range.

When identifying a model you must state these restrictions explicitly. A complete model includes: the type (periodic or sinusoidal), the equation with all four parameters identified, the restricted domain, and the restricted range. Skipping the contextual restrictions gives an incomplete — and often physically impossible — answer.

Contextual restrictions limit domain and range to physically meaningful values.
Domain restrictions come from time limits, starting conditions, or physical boundaries.
Range restrictions come from physical maximum and minimum values.
A complete model states: equation, restricted domain, and restricted range.

Parameters of a Sinusoidal Model f(x) = a sin(k(x − d)) + c

Parameter	What It Controls	How to Find It from a Graph or Context
$a$ (amplitude)	Vertical stretch; height of wave above or below axis	$(\text{max} - \text{min}) \div 2$ ; always positive
$k$	Horizontal compression; affects period; dimensionless	$k = 360^\circ \div T$
$d$ (phase shift)	Horizontal translation of the wave	Read where the cycle starts; right shift is positive $d$
$c$ (vertical shift)	Moves the axis up or down; equals equation of axis	$(\text{max} + \text{min}) \div 2$
$T$ (period)	Length of one complete cycle	Horizontal distance from one peak to the next peak

Worked example

Example 1 — Reading a Periodic Model from a Table

A buoy in a harbour bobs up and down. Its height above the harbour floor (in metres) is recorded every 10 seconds. The data shows a maximum height of 4.6 m, a minimum height of 0.8 m, and one complete cycle takes 40 seconds. A technician claims the motion is sinusoidal. (a) Find the amplitude and the equation of the axis. (b) Write a sinusoidal equation of the form

f(t) = a\sin(kt) + c

assuming the buoy starts at its middle height and is moving upward at

t = 0

. (c) State a realistic restricted domain and range for the model if the buoy is monitored for exactly 3 minutes.

Identify the maximum and minimum
From the problem, the maximum height is $4.6$ m and the minimum height is $0.8$ m. These are the tallest and shortest positions of the buoy.
$\text{max} = 4.6, \quad \text{min} = 0.8$
Calculate the equation of the axis
The equation of the axis is the average of the maximum and minimum. Add them and divide by 2 to find the middle value $c$ .
$c = \frac{4.6 + 0.8}{2} = \frac{5.4}{2} = 2.7$
Calculate the amplitude
The amplitude is half the difference between the maximum and minimum. This tells you how far the buoy travels above or below its middle position.
$a = \frac{4.6 - 0.8}{2} = \frac{3.8}{2} = 1.9$
Find k from the period
The period is 40 seconds. Use the relationship $T = \frac{360^\circ}{k}$ and rearrange to solve for $k$ . Because $k$ is a dimensionless scale factor, you divide $360^\circ$ by the period value of 40.
$k = \frac{360^\circ}{40} = 9$
Determine the phase shift
The buoy starts at its middle height ( $y = 2.7$ m) and moves upward at $t = 0$ . This matches the standard sine curve, which also starts at its middle value and moves upward. No horizontal shift is needed, so $d = 0$ .
$d = 0$
Write the sinusoidal equation
Substitute $a = 1.9$ , $k = 9$ , $d = 0$ , and $c = 2.7$ into the general form $f(t) = a\sin(kt) + c$ . Because $d = 0$ , the phase-shift bracket simplifies and the equation is straightforward.
$f(t) = 1.9\sin(9t) + 2.7$
State the restricted domain
Three minutes equals 180 seconds. The buoy is monitored from $t = 0$ to $t = 180$ seconds. Negative time has no meaning here, so the domain is restricted to this interval.
$0 \leq t \leq 180$
State the restricted range
The buoy physically reaches no lower than 0.8 m and no higher than 4.6 m above the harbour floor. The range is restricted to those heights.
$0.8 \leq f(t) \leq 4.6$

Answer: Amplitude

= 1.9

m; equation of the axis

y = 2.7

m; sinusoidal model

f(t) = 1.9\sin(9t) + 2.7

; domain

0 \leq t \leq 180

s; range

0.8 \leq f(t) \leq 4.6

m.

Check: Verify at

t = 0

:

f(0) = 1.9\sin(0^\circ) + 2.7 = 0 + 2.7 = 2.7

✓ (middle height). At a peak,

\sin(9t) = 1

, giving

f = 1.9(1) + 2.7 = 4.6

✓. At a trough,

\sin(9t) = -1

, giving

f = 1.9(-1) + 2.7 = -1.9 + 2.7 = 0.8

✓.

Worked example

Example 2 — Choosing Between Periodic and Sinusoidal Models

Two machines in a factory produce signals that repeat. Machine A produces a signal whose graph shows a smooth, symmetric wave with a maximum output of 10 volts, a minimum output of 2 volts, and a period of 120°. Machine B produces a signal that instantly jumps to 10 volts, stays flat, then instantly drops to 2 volts and stays flat (a square wave), repeating every 120°. (a) Identify which machine produces a sinusoidal signal and which produces a non-sinusoidal periodic signal. Justify your choice. (b) For Machine A, identify the amplitude, the equation of the axis, and write its sinusoidal model using cosine with no phase shift. (c) State one contextual domain restriction that would apply to both machines if they only operate during an 8-hour shift, expressing the restriction in degrees where one full shift equals 3600°.

Identify the type of each signal
Machine A's graph is smooth and symmetric — this matches the shape of a cosine or sine curve, so it is sinusoidal. Machine B's graph has instant vertical jumps and flat sections — this is a square wave, which is periodic but not sinusoidal because it is not smooth or continuously curving.
Calculate the amplitude for Machine A
Use the amplitude formula with the maximum of 10 volts and the minimum of 2 volts.
$a = \frac{10 - 2}{2} = \frac{8}{2} = 4$
Calculate the equation of the axis for Machine A
Average the maximum and minimum outputs to find the middle value $c$ .
$c = \frac{10 + 2}{2} = \frac{12}{2} = 6$
Find k for Machine A
The period is $120^\circ$ . Rearrange $T = \frac{360^\circ}{k}$ to solve for $k$ by dividing $360^\circ$ by the period. The result is a dimensionless number.
$k = \frac{360^\circ}{120^\circ} = 3$
Write the cosine model for Machine A with no phase shift
With no phase shift, $d = 0$ . Substitute all values into $f(x) = a\cos(k(x - d)) + c$ . The cosine function starts at its maximum when $x = 0$ and no shift is needed, which fits a signal that begins at its peak.
$f(x) = 4\cos(3x) + 6$
Verify the model at key points
Check the maximum and minimum. At $x = 0^\circ$ : $\cos(0^\circ) = 1$ , so $f(0^\circ) = 4(1) + 6 = 10$ ✓. At $x = 60^\circ$ : the argument is $3 \times 60^\circ = 180^\circ$ and $\cos(180^\circ) = -1$ , so $f(60^\circ) = 4(-1) + 6 = 2$ ✓.
$f(0^\circ) = 10, \quad f(60^\circ) = 2$
State the contextual domain restriction
Both machines operate only during the shift: from the start ( $x = 0^\circ$ ) to the end of the shift ( $x = 3600^\circ$ ). Before and after the shift the machines are off, so inputs outside this interval have no physical meaning.
$0^\circ \leq x \leq 3600^\circ$

Answer: Machine A is sinusoidal; Machine B is periodic but not sinusoidal. Machine A: amplitude

= 4

V, equation of axis

y = 6

, model

f(x) = 4\cos(3x) + 6

. Both machines: domain restricted to

0^\circ \leq x \leq 3600^\circ

.

Check: Period check for Machine A:

T = \frac{360^\circ}{3} = 120^\circ

✓. Number of complete cycles in the shift:

\frac{3600^\circ}{120^\circ} = 30

cycles — a reasonable operating count for a machine over an 8-hour shift ✓.

Common mistakes and how to avoid them

Measuring the period as the horizontal distance from a peak to the nearest trough, giving half the true period.

Correction: Always measure from peak to peak (or trough to trough, or any point to the next identical point moving in the same direction). Peak to trough is only half a cycle.

Confusing amplitude with the maximum value. For example, writing

a = 10

when the max is 10 V and the min is 2 V.

Correction: Amplitude is half the total height of the wave:

a = (\text{max} - \text{min}) \div 2 = 4

, not the maximum itself.

Forgetting to state contextual domain and range restrictions, leaving a model that allows negative time or impossible physical heights.

Correction: Every real-world model must include a restricted domain (for example,

0 \leq t \leq 180

) and a restricted range (for example,

0.8 \leq f(t) \leq 4.6

) that reflect physical reality.

Labelling a square wave or any repeating graph as sinusoidal just because it is periodic.

Correction: Sinusoidal means the wave is smooth, continuous, and shaped exactly like a sine or cosine curve. Jagged, stepped, or angular repeating patterns are periodic but not sinusoidal.

Using the wrong sign for the phase shift — for example, shifting the graph right but writing a negative

d

.

Correction: In

f(x) = a\sin(k(x - d)) + c

, a rightward shift means

d

is positive. A leftward shift means

d

is negative. Check by substituting: the cycle starts at

x = d

.

Lesson summary

A periodic function repeats its output values at regular intervals; the period $T$ is the length of one complete cycle measured horizontally.
Key measurements — amplitude $a = (\text{max} - \text{min}) \div 2$ , equation of the axis $y = (\text{max} + \text{min}) \div 2$ , and period — fully describe the shape of a periodic wave.
A sinusoidal function has the specific smooth, symmetric wave shape of a sine or cosine curve; not every periodic function is sinusoidal.
The general sinusoidal model in degrees is $f(x) = a\sin(k(x-d)) + c$ , where $a$ is amplitude, $k = 360^\circ \div T$ is a dimensionless scale factor, $d$ is the phase shift, and $c$ is the equation of the axis.
Contextual restrictions limit the domain to physically meaningful input values (such as non-negative time) and the range to physically possible output values.
A complete identification of a sinusoidal model requires: the equation with all four parameters, the restricted domain, and the restricted range — each justified by the context.

Check your understanding

Question 1

A sinusoidal function has a maximum value of 14 and a minimum value of 2. What is its amplitude?

14
2
8
6

Show answer and explanation

6

Amplitude

= (\text{max} - \text{min}) \div 2 = (14 - 2) \div 2 = 12 \div 2 = 6

. The maximum value (14) is not the amplitude, and neither is the minimum (2) or the axis value (8).

Question 2

A periodic function has a period of

T = 90^\circ

. What is the value of

k

in the model

f(x) = a\sin(k(x-d)) + c

?

90
45
4
270

Show answer and explanation

4

k = 360^\circ \div T = 360^\circ \div 90^\circ = 4

. Dividing by 2 gives 45 (a common error of halving the period instead of dividing 360 by it), and the other options do not follow the formula.

Question 3

A machine operates for exactly 2 hours and produces a sinusoidal signal. Which of the following best describes a contextual domain restriction for this model, where

t

is in hours?

All real numbers
$t \geq 0$
$0 \leq t \leq 2$
$-2 \leq t \leq 2$

Show answer and explanation

0 \leq t \leq 2

The machine runs from the start (

t = 0

) to the end of its 2-hour operation (

t = 2

), so the domain is

0 \leq t \leq 2

. Allowing all real numbers or negative time ignores the physical context. The interval

-2 \leq t \leq 2

includes impossible negative time values.

Question 4

Which of the following graphs represents a periodic function that is NOT sinusoidal?

A smooth wave that rises and falls symmetrically like a cosine curve
A graph that repeats a sharp sawtooth shape — rising steadily then dropping instantly — every 60°
A graph described by $f(x) = 3\sin(2x) + 1$
A graph described by $f(x) = -\cos(x) + 4$

Show answer and explanation

A graph that repeats a sharp sawtooth shape — rising steadily then dropping instantly — every 60°

A sawtooth wave repeats at regular intervals (it is periodic) but its shape is not smooth or symmetric — it rises linearly then drops vertically, so it is not sinusoidal. The other three options all describe smooth, symmetric sine or cosine curves, which are sinusoidal.

Key terms

Periodic function: A function whose output values repeat exactly at regular intervals; satisfies $f(x + T) = f(x)$ for a fixed positive number $T$ .
Period ( $T$ ): The length of one complete cycle of a periodic function, measured as a horizontal distance on the graph.
Sinusoidal function: A periodic function whose graph has the same smooth, symmetric wave shape as a sine or cosine curve.
Amplitude ( $a$ ): The vertical distance from the equation of the axis to the maximum (or minimum) value; always positive. Calculated as $(\text{max} - \text{min}) \div 2$ .
Equation of the axis: The horizontal line exactly halfway between the maximum and minimum values of a periodic function; $y = (\text{max} + \text{min}) \div 2$ .
Phase shift ( $d$ ): A horizontal translation of a sinusoidal graph; in $f(x) = a\sin(k(x-d)) + c$ , a positive $d$ shifts the graph to the right.
Contextual restriction: A limitation on the domain or range of a mathematical model that comes from the real-world meaning of the situation, not from the algebra.
Vertical shift ( $c$ ): The value that moves the sinusoidal wave up or down; it equals the $y$ -value of the equation of the axis.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D3.2. It is a study resource, not an official curriculum publication.

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D3.2 · Identify periodic and sinusoidal models and contextual restrictions

Ontario Grade 11 Mathematics

MCR3U · Strand D · Expectation D3.2

What you will learn

Prerequisite Bridge: What You Already Know

Periodic Functions: The Idea of Repetition

Key Measurements of a Periodic Function

Sinusoidal Models: Form and Parameters

Contextual Restrictions on Domain and Range

Parameters of a Sinusoidal Model f(x) = a sin(k(x − d)) + c

Example 1 — Reading a Periodic Model from a Table

Example 2 — Choosing Between Periodic and Sinusoidal Models

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