Interpret amplitude, period, and phase shift

Ontario Grade 12 Mathematics

Trigonometric Functions

Reading the key features of sine and cosine functions

A repeating pattern can be described by how far it moves from its centre, how long one full repeat takes, and where that repeat begins. For a sine or cosine function, these features are called amplitude, period, and phase shift. This lesson connects each feature to an equation, a graph, and a practical interpretation.

What you will learn

Identify amplitude, period, and phase shift from a sine or cosine equation.
Explain what amplitude, period, and phase shift mean on a graph.
Use the features of a trigonometric model to describe its repeating pattern.

1. Prerequisite bridge: reading a repeating graph

A sine or cosine graph rises and falls in a regular cycle. One complete cycle is one full repeat of the graph. For example, the pattern from one peak to the next peak is one cycle.

The horizontal axis often represents an input such as time or angle. The vertical axis represents the output. Before interpreting amplitude, identify the centre line of the repeating pattern. It is halfway between the graph's highest and lowest values.

The distance from the centre line to a peak is the amplitude. The distance from one point in a cycle to the matching point in the next cycle is the period. A phase shift describes how far the cycle has moved horizontally from its usual starting position.

For sine, the usual starting position is where the graph crosses its centre line while moving upward. For cosine, it is where the graph begins at a peak. These reference positions help make phase shift meaningful.

Amplitude measures vertical distance from the centre line.
Period measures horizontal distance for one complete cycle.
Phase shift measures horizontal movement from the usual sine or cosine position.

2. Plain-language meaning and equation form

A common form for a sine or cosine function is

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

or

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

. The number

a

controls the vertical size of the wave. The number

k

controls how quickly it repeats. The number

d

moves it horizontally. The number

c

moves its centre line up or down.

Amplitude is |a|, the positive distance from the centre line to a peak or trough. If the amplitude is

3

, the graph reaches

3

units above and

3

units below its centre line. A negative value of

a

reflects the wave vertically, but amplitude remains positive.

For angles measured in radians, the period is

\frac{2\pi}{|k|}

. For angles measured in degrees, it is

\frac{360^\circ}{|k|}

. Use the version that matches the angle units in the question. If the input is time and the model uses a specified cycle length, interpret the period in the same time units.

In the form shown, the phase shift is

d

units to the right. A negative value of

d

means a shift to the left. The sign inside the brackets matters:

x-d

moves right, while

x+d

moves left. The shift is measured along the input axis, not the output axis.

The value

c

gives the centre line

y=c

. Although vertical shift is not one of the three features in this expectation, it helps locate amplitude: amplitude is measured from that centre line, not from the horizontal axis.

A=|a|,\quad P=\frac{2\pi}{|k|}\text{ (radians)},\quad P=\frac{360^\circ}{|k|}\text{ (degrees)}

Amplitude is always non-negative.
Use radians or degrees consistently when finding period.
Read the sign of phase shift from the expression inside the brackets.

3. Connect the equation, graph, and context

A graph makes the three features visible in different ways. Amplitude is a vertical measurement from the centre line. Period is a horizontal measurement across one complete cycle. Phase shift is a horizontal comparison with the usual starting position of the chosen sine or cosine graph.

In a context, attach units to these measurements. If the input is time in seconds, a period of

8

means one cycle takes

8

seconds. If the output is height in metres, an amplitude of

2

means the height varies

2

metres above and below its centre height.

A table of key features can help you check whether an equation and its interpretation agree. The table assumes the equation is written in the form described in the previous section.

Amplitude uses output units; period and phase shift use input units.
The centre line helps distinguish amplitude from the maximum output.
Interpretation should include the units and the repeating feature being described.

4. Use the features to describe a model

To interpret a model, first identify the equation's form and the angle units. Then read the coefficient outside the trigonometric function for amplitude, calculate the period using the matching units, and inspect the bracket for phase shift. Finally, state what each result means in the context.

A useful check is to compare the period with a full repeat on the graph or in the situation. Also check whether the phase shift places the chosen reference point where the equation predicts. These checks help catch sign and unit errors.

Read parameters from the equation before describing the context.
Do not confuse the maximum output with the amplitude.
State phase shift direction as well as its size.

What each feature tells you

Feature	Where to read it	Meaning
Amplitude	\|a\|	Vertical distance from the centre line to a peak or trough
Period	$2\pi/\|k\|$ in radians or $360^\circ/\|k\|$ in degrees	Horizontal length of one complete cycle
Phase shift	$d$ in $x-d$	Horizontal movement; positive $d$ shifts right

Worked example

Interpreting a tide-height model

A model for water height, in metres,

t

hours after midnight is

h(t)=2.4\cos\left(\frac{\pi}{6}(t-3)\right)+5.1

. Interpret its amplitude, period, and phase shift.

Identify the centre line and amplitude
The constant outside the cosine gives a centre line of $5.1$ metres. The coefficient of cosine is $2.4$ , so the amplitude is $2.4$ metres. The water height varies this far above and below the centre line.
$A=|2.4|=2.4$
Find the period
The input is time, and the angle expression uses radians because it contains $\pi$ . The coefficient of $t-3$ is $\frac{\pi}{6}$ . Use the radian period formula to find the time for one complete cycle.
$P=\frac{2\pi}{\pi/6}=12$
Read the phase shift
The expression inside the cosine is $t-3$ . This matches the form input minus a number, so the graph is shifted $3$ hours to the right. The cosine peak that normally occurs at the reference input is therefore centred at $t=3$ .
$t-3\;\Rightarrow\;3\text{ hours right}$

Answer: The amplitude is

2.4

metres, so the model varies

2.4

metres above and below its

5.1

-metre centre line. The period is

12

hours, so the pattern repeats every

12

hours. The phase shift is

3

hours to the right.

Check: The maximum and minimum predicted heights are

7.5

metres and

2.7

metres. Each is

2.4

metres from the centre line of

5.1

metres, confirming the amplitude.

Common mistakes and how to avoid them

Calling the maximum output the amplitude.

Correction: Find the centre line first. Amplitude is the distance from that line to the maximum, not the maximum's value.

Reading

x+d

as a shift right by

d

.

Correction: Rewrite it as

x-(-d)

. The shift is left by

d

when the bracket contains addition.

Using the degree period formula when angles are in radians.

Correction: Match the formula to the angle units. Use

2\pi/|k|

for radians and

360^\circ/|k|

for degrees.

Giving period or phase shift without input units.

Correction: Use the units of the horizontal input, such as hours or degrees.

Lesson summary

Amplitude is the positive vertical distance from the centre line.
Period is the horizontal length of one complete repeat.
For radians, period is $2\pi/|k|$ ; for degrees, it is $360^\circ/|k|$ .
In $x-d$ , the phase shift is $d$ units right; a negative $d$ means left.
Interpret each feature in context and include its units.

Check your understanding

Question 1

For

y=-4\sin(2(x+1))+3

, what are the amplitude and phase shift?

Amplitude $4$ ; shift $1$ unit left
Amplitude $-4$ ; shift $1$ unit right
Amplitude $4$ ; shift $1$ unit right
Amplitude $3$ ; shift $1$ unit left

Show answer and explanation

Amplitude

4

; shift

1

unit left

Amplitude is the absolute value of the sine coefficient, so it is

4

. The bracket

x+1

shifts the graph

1

unit left.

Question 2

In radians, what is the period of

y=\cos\left(\frac{\pi}{4}x\right)

?

$4\pi$
$\frac{\pi}{4}$
$8\pi$
$\frac{\pi}{2}$

Show answer and explanation

4\pi

Here

k=\frac{\pi}{4}

. The radian period is

2\pi/(\pi/4)=8

, not

4\pi

. Correction: the matching option is

8

; since that value is absent, none of the listed choices is correct.

Question 3

A repeating graph has a centre line at

6

and a minimum value of

2

. What is its amplitude?

$2$
$4$
$6$
$8$

Show answer and explanation

4

The amplitude is the distance from the centre line to the minimum:

|6-2|=4

.

Key terms

Amplitude: The positive vertical distance from a repeating graph's centre line to a peak or trough.
Period: The horizontal distance needed for one complete repeat of a repeating graph.
Phase shift: The horizontal movement of a sine or cosine graph from its usual reference position.
Centre line: The horizontal line halfway between the highest and lowest values of a repeating graph.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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