D2.6 Determine amplitude, period, phase shift, domain, and range

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U – D2.6 | Transformations of Trigonometric Functions

You already know that the sine and cosine curves repeat in a wave-like pattern. In Grade 10, you explored what these basic curves look like. Now, in Grade 11, you will learn how to read an equation and immediately know everything important about its graph: how tall the wave is, how long one full cycle takes, whether the wave has slid left or right, and what set of inputs and outputs the function uses. These five features, amplitude, period, phase shift, domain, and range, are the complete identity card of any sinusoidal function. Mastering them lets you sketch, interpret, and compare waves without plotting dozens of points.

What you will learn

Identify the values of a, k, d, and c in the general sinusoidal form and explain what each one does to the graph.
Calculate the amplitude, period, and phase shift of a sinusoidal function from its equation.
State the domain and range of a transformed sine or cosine function.
Read key features directly from a graph and connect them back to the equation.

Prerequisite Bridge: The Basic Sine and Cosine Curves

Before transformations, recall what the parent functions

y = \sin x

and

y = \cos x

look like. Both curves oscillate smoothly between a minimum value of

-1

and a maximum value of

1

. One complete wave, called a cycle, takes CAD 360° to finish. The sine curve starts at

(0°, 0)

, rises to a peak, falls through zero, dips to a trough, and returns to zero. The cosine curve starts at

(0°, 1)

, which is already at its peak.

Both parent functions are defined for every real-number input, so their domain is all real numbers. Because the output never goes below

-1

or above

1

, their range is

-1 \leq y \leq 1

. Keep these benchmarks in mind, since every transformation you study today is a change made to these starting values.

Parent functions: $y = \sin x$ and $y = \cos x$ .
Both oscillate between $-1$ and $1$ with a period of CAD 360°.
Domain of each parent function: all real numbers.
Range of each parent function: $-1 \leq y \leq 1$ .

The General Form and What Each Parameter Does

A transformed sinusoidal function is written in the general form

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

or

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

. There are four parameters:

a

,

k

,

d

, and

c

. Each one controls a different feature of the graph, and you can read all four directly from the equation.

The parameter

a

is the coefficient in front of the sine or cosine. It stretches or compresses the wave vertically and controls the amplitude. The parameter

k

is the coefficient inside the brackets, beside

x

. It stretches or compresses the wave horizontally and controls the period. The parameter

d

is the value being subtracted from

x

inside the brackets. It slides the entire graph left or right and controls the phase shift. The parameter

c

is the constant added on the outside. It shifts the graph up or down and controls the vertical shift, which in turn affects the range.

Notice that the form uses

k(x - d)

, not

kx - d

. This distinction is critical. If you see

y = \sin(2x - 90°)

, you must factor the inside first, giving

y = \sin(2(x - 45°))

. Only then can you read

d = 45°

correctly. Skipping this factoring step is the single most common error in this unit.

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

General form: $y = a\sin(k(x - d)) + c$ or $y = a\cos(k(x - d)) + c$ .
$a$ controls the amplitude (vertical stretch).
$k$ controls the period (horizontal stretch or compression).
$d$ controls the phase shift (horizontal slide).
$c$ controls the vertical shift (which changes the range).

Calculating Each Feature from the Equation

Amplitude is the distance from the midline of the wave to its maximum (or minimum) value. It is always a positive number. Calculate it using the absolute value of

a

. For example, if

a = -3

, the amplitude is

3

. The negative sign on

a

means the wave is reflected (flipped upside down), but the amplitude, the height, is still

3

.

Period is the horizontal length of one complete cycle, measured in degrees. It tells you how far the wave travels along the

x

-axis before repeating. Divide CAD 360°

by

|k| to find it. If

|k| > 1

, the wave is horizontally compressed, so cycles are shorter. If

0 < |k| < 1

, the wave is horizontally stretched, so cycles are longer.

Phase shift is the horizontal distance the graph has moved compared to the parent function. It equals

d

. If

d > 0

, the graph shifts to the right. If

d < 0

, the graph shifts to the left. Remember to always factor first so that the inside of the brackets matches the form

(x - d)

exactly.

Domain refers to all allowable input values, the

x

-values. Because the sine and cosine functions accept any angle, the domain of every sinusoidal function studied in this course is all real numbers, written as x ∈

\mathbb{R}

. Transformations of type

a

,

k

,

d

, and

c

do not restrict which

x

-values can be used.

Range refers to all possible output values, the

y

-values. The midline sits at

y = c

. The wave rises |a| units above the midline and falls |a| units below it. Therefore the minimum output value is

c - |a|

and the maximum is

c + |a|

.

\text{Period} = \frac{360°}{|k|}

Amplitude equals |a|; it is always positive.
Period equals CAD 360° divided by |k|.
Phase shift equals $d$ ; factor the inside of the brackets first.
Domain: all real numbers, x ∈ $\mathbb{R}$ , for every sinusoidal function in this course.
Range: $c - |a| \leq y \leq c + |a|$ .

Reading Features from a Graph

Sometimes you are given a graph instead of an equation, and you need to identify the features visually. Start by finding the maximum and minimum

y

-values on the graph. The amplitude is half the distance between them. The midline, the value of

c

, is the average of the maximum and minimum.

To find the period, locate two consecutive points where the wave is in the exact same position and moving in the same direction, for example, two successive peaks. Measure the horizontal distance between them; that distance is the period. To find the phase shift, compare where the first peak (or zero crossing) of the transformed graph occurs versus where it would occur on the parent function. The horizontal shift between those two positions is the phase shift.

Once you have the amplitude, period, and midline from the graph, you can work backwards to find

a

,

k

,

d

, and

c

, and then write the equation. This skill connects graphical and algebraic representations, which is a key goal of the course.

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

From a graph, amplitude equals half the distance between the maximum and minimum.
Midline (value of $c$ ) equals the average of the maximum and minimum.
Period equals the horizontal distance between two identical successive points.
Phase shift equals the horizontal distance the graph moved from the parent function.

Putting It All Together: A Summary Strategy

When you encounter any sinusoidal equation, follow these four steps in order. First, factor the expression inside the brackets to match the form

(x - d)

, and read off

k

and

d

. Second, read

a

and

c

directly from the equation. Third, calculate amplitude as |a|, period as CAD 360° divided by |k|, and phase shift as

d

. Fourth, state the domain as all real numbers and calculate the range using

c

and |a|.

Keeping this sequence consistent prevents you from mixing up parameters. For instance, it stops you from confusing the vertical shift

c

with the amplitude |a|, which is one of the most common errors. With practice, reading these five features from an equation will feel as natural as reading the slope and intercept from a linear function.

c - |a| \leq y \leq c + |a|

Step 1: Factor inside brackets to isolate $k$ and $d$ .
Step 2: Read $a$ and $c$ directly.
Step 3: Compute amplitude, period, and phase shift using the formulas.
Step 4: State domain (all reals) and compute range from $c$ and |a|.

Summary: Parameters in $y = a\sin(k(x-d)) + c$ and Their Effects

Parameter	Feature It Controls	Formula or Rule	Example Value	Result
$a$	Amplitude and reflection	\|a\|	$a = -3$	Amplitude equals $3$ ; graph reflected
$k$	Period	$\frac{360°}{\|k\|}$	$k = 2$	Period equals CAD 180°
$d$	Phase shift	Shift equals $d$ (factor first)	$d = 30°$	Graph moves CAD 30° right
$c$	Vertical shift and midline	Midline at $y = c$	$c = -2$	Midline at $y = -2$
$a$ and $c$	Range	$c - \|a\| \leq y \leq c + \|a\|$	$a = 3,\ c = 1$	$-2 \leq y \leq 4$

Worked example

Example 1 — Reading Features from an Equation

For the function

y = -4\sin(3(x - 20°)) + 1

, determine the amplitude, period, phase shift, domain, and range.

Identify the parameters
Match the equation to the general form $y = a\sin(k(x - d)) + c$ . The equation is already factored correctly inside the brackets, so you can read the parameters directly.
$a = -4,\ k = 3,\ d = 20°,\ c = 1$
Calculate the amplitude
Amplitude equals the absolute value of $a$ . The negative sign on $a$ tells you the graph is reflected over the midline (the wave goes down first instead of up), but it does not change the height of the wave.
$\text{Amplitude} = |-4| = 4$
Calculate the period
Divide CAD 360° $by$ |k|. Here $|k| = 3$ , so one full cycle is compressed, meaning the wave repeats three times in the space where the parent function repeats once.
$\text{Period} = \frac{360°}{3} = 120°$
State the phase shift
The phase shift equals $d$ . Because $d = 20°$ is positive, the entire graph has shifted to the right compared to $y = -4\sin(3x) + 1$ .
$\text{Phase shift} = 20°\ \text{right}$
State the domain
Sine accepts any real-number input, and none of the parameters restricts the $x$ -values. The domain is all real numbers, written x ∈ $\mathbb{R}$ .
Calculate the range
The midline is at $y = c = 1$ . The wave travels $|a| = 4$ units above and below that midline. So the minimum output is $1 - 4 = -3$ and the maximum is $1 + 4 = 5$ .
$-3 \leq y \leq 5$

Answer: Amplitude equals

4

; period equals CAD 120°; phase shift equals CAD 20° to the right; domain is x ∈

\mathbb{R}

; range is

-3 \leq y \leq 5

.

Check: Verify the range: midline at

y = 1

, amplitude

4

. Maximum equals

1 + 4 = 5

, correct. Minimum equals

1 - 4 = -3

, correct. Verify the period:

\frac{360°}{3} = 120°

, correct.

Worked example

Example 2 — Factoring First, Then Finding Features

For the function

y = 2\cos(4x - 60°) + 3

, determine the amplitude, period, phase shift, domain, and range.

Factor the inside of the brackets
The expression inside the cosine is $4x - 60°$ . This is not yet in the form $k(x - d)$ . Factor out the $4$ from both terms inside to get $4(x - 15°)$ . Now the equation can be rewritten in proper general form. This step is essential; if you skip it, you would incorrectly read the phase shift as CAD 60° instead of the correct CAD 15°.
$y = 2\cos(4(x - 15°)) + 3$
Identify the parameters
Now match to $y = a\cos(k(x - d)) + c$ . Read off each parameter from the rewritten equation.
$a = 2,\ k = 4,\ d = 15°,\ c = 3$
Calculate the amplitude
Amplitude equals |a|. The coefficient $a$ is positive, so the graph is not reflected; it starts at its maximum when $x = d$ .
$\text{Amplitude} = |2| = 2$
Calculate the period
Divide CAD 360° $by$ |k|, where $|k| = 4$ . The wave completes one full cycle every CAD 90°.
$\text{Period} = \frac{360°}{4} = 90°$
State the phase shift
The phase shift equals $d = 15°$ . Because $d$ is positive, the graph has moved to the right. If you had not factored and mistakenly used CAD 60°, your sketch would be off by a significant amount, which is a good reason to always factor first.
$\text{Phase shift} = 15°\ \text{right}$
State the domain
Cosine accepts every real-number input, and no parameter here restricts the $x$ -values. The domain is all real numbers, written x ∈ $\mathbb{R}$ .
Calculate the range
The midline is $y = c = 3$ . The amplitude is $2$ , so the wave reaches as high as $3 + 2 = 5$ and as low as $3 - 2 = 1$ .
$1 \leq y \leq 5$

Answer: Amplitude equals

2

; period equals CAD 90°; phase shift equals CAD 15° to the right; domain is x ∈

\mathbb{R}

; range is

1 \leq y \leq 5

.

Check: Verify factoring:

4(x - 15°) = 4x - 60°

, correct. Verify range:

3 - 2 = 1

and

3 + 2 = 5

, correct. Verify period:

\frac{360°}{4} = 90°

, correct.

Common mistakes and how to avoid them

Reading the phase shift directly from

kx - d

without factoring out

k

first, giving a phase shift that is

k

times too large.

Correction: Always factor the expression inside the brackets into the form

k(x - d)

before identifying

d

. For

4x - 60°

, factor to get

4(x - 15°)

, so the phase shift is CAD 15°, not CAD 60°.

Using a negative value for amplitude, for example writing amplitude equals

-4

when

a = -4

.

Correction: Amplitude is always a positive number, equal to |a|. The negative sign means the wave is reflected, but amplitude still equals

|-4| = 4

.

Confusing the vertical shift

c

with the amplitude |a|, especially when calculating the range.

Correction: The amplitude tells you how far the wave moves above and below the midline. The midline itself is at

y = c

. The range runs from

c - |a|

to

c + |a|

.

Stating the period as CAD 360° multiplied by |k| instead of divided.

Correction: A larger |k| compresses the wave, making the period shorter. The correct formula divides CAD 360°

by

|k|.

Thinking the domain changes when vertical or horizontal shifts are applied.

Correction: For sine and cosine functions, transformations of type

a

,

k

,

d

, and

c

never restrict the

x

-values. The domain remains all real numbers regardless of the transformation.

Lesson summary

The general form $y = a\sin(k(x - d)) + c$ (or cosine) contains four parameters that each control a different feature of the graph.
Amplitude equals |a|; it is always positive and measures the height from the midline to a peak.
Period equals CAD 360° divided by |k|; it gives the horizontal length of one complete cycle in degrees.
Phase shift equals $d$ ; always factor the inside of the brackets first to read $d$ correctly.
Domain is all real numbers, x ∈ $\mathbb{R}$ , for every sinusoidal function in this course; no transformation restricts the $x$ -values.
Range is $c - |a| \leq y \leq c + |a|$ , where $c$ is the midline and |a| is the amplitude.

Check your understanding

Question 1

What is the amplitude of

y = -5\sin(2(x - 10°)) + 4

?

-5
5
4
2

Show answer and explanation

5

Amplitude equals

|a| = |-5| = 5

. The negative sign reflects the graph but does not affect the amplitude, which is always a positive value.

Question 2

What is the period of

y = 3\cos\left(\frac{1}{2}(x + 30°)\right) - 1

?

180°
720°
360°
90°

Show answer and explanation

720°

Period equals CAD 360° divided by

|k| = \frac{1}{2}

, which gives

360° \times 2 = 720°

. Because

|k| < 1

, the period is longer than the parent function's CAD 360°.

Question 3

A student writes the phase shift of

y = \sin(3x - 90°)

as CAD 90° to the right. What went wrong?

The student forgot to take the absolute value of $k$ .
The student did not factor out $k = 3$ first, so the phase shift should be CAD 30° to the right.
The student confused sine and cosine.
The student used the wrong period formula.

Show answer and explanation

The student did not factor out

k = 3

first, so the phase shift should be CAD 30° to the right.

Factoring correctly:

3x - 90° = 3(x - 30°)

, so

d = 30°

. The phase shift is CAD 30° to the right, not CAD 90°. Skipping the factoring step produced an answer three times too large.

Question 4

What is the range of

y = 6\cos(x - 45°) - 2

?

$-6 \leq y \leq 6$
$-8 \leq y \leq 4$
$2 \leq y \leq 8$
$-2 \leq y \leq 6$

Show answer and explanation

-8 \leq y \leq 4

Here

|a| = 6

and

c = -2

. Minimum equals

c - |a| = -2 - 6 = -8

and maximum equals

c + |a| = -2 + 6 = 4

. Range:

-8 \leq y \leq 4

.

Key terms

Sinusoidal function: A function whose graph has the repeating wave shape of sine or cosine.
Amplitude: The positive distance from the midline of a sinusoidal wave to its maximum (or minimum) value; equal to |a|.
Period: The horizontal length, in degrees, of one complete cycle of a sinusoidal function; equal to CAD 360° divided by |k|.
Phase shift: The horizontal distance a sinusoidal graph has moved left or right compared to the parent function; equal to $d$ after factoring.
Midline: The horizontal line $y = c$ that runs through the vertical centre of a sinusoidal wave, halfway between its maximum and minimum.
Domain: The complete set of allowable input ( $x$ ) values for a function. For all sinusoidal functions in this course, the domain is all real numbers, written x ∈ $\mathbb{R}$ .
Range: The complete set of possible output ( $y$ ) values for a function. For a sinusoidal function, the range is $c - |a| \leq y \leq c + |a|$ .
Vertical shift: The value $c$ in the general form; it moves the entire graph up (if $c > 0$ ) or down (if $c < 0$ ) and sets the position of the midline.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D2.6. It is a study resource, not an official curriculum publication.

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D2.6 · Determine amplitude, period, phase shift, domain, and range

Ontario Grade 11 Mathematics

MCR3U – D2.6 | Transformations of Trigonometric Functions

What you will learn

Prerequisite Bridge: The Basic Sine and Cosine Curves

The General Form and What Each Parameter Does

Calculating Each Feature from the Equation

Reading Features from a Graph

Putting It All Together: A Summary Strategy

Summary: Parameters in $y = a\sin(k(x-d)) + c$ and Their Effects

Example 1 — Reading Features from an Equation

Example 2 — Factoring First, Then Finding Features

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