Investigate vertical and horizontal sine transformations

Ontario Grade 11 Mathematics

Trigonometric Functions

Use graphs, tables, and equations to see how a sine pattern moves or changes width

A sine graph repeats a wave-like pattern. A transformation changes where the pattern appears or how much horizontal distance it takes to repeat. Before investigating these changes, recall that a function pairs each input with one output. Adding the same number to every output moves a graph up or down. Changing the input can affect where the graph’s features occur. In this lesson, you will investigate these vertical and horizontal changes using the basic sine graph.

What you will learn

Describe how adding a number outside a sine function shifts its graph vertically.
Describe how changes inside a sine function shift the graph horizontally or change its period.
Use a graph or table to investigate and check a prediction about a sine transformation.

1. Prerequisite bridge: reading the basic sine graph

The basic sine function is written as

y=\sin x

. Its input is an angle. In a graphing tool, use the angle setting specified for the graph and keep that setting consistent when comparing functions.

The basic sine graph repeats after one complete cycle. The horizontal length of that cycle is called the period. The graph’s highest output is

1

, and its lowest output is

-1

. These features give us a starting point for noticing what a transformation changes.

A vertical shift changes outputs. For example, adding

2

to every output moves every point up

2

units. A horizontal shift changes where points occur along the input axis. A horizontal shift alone does not change the highest or lowest output.

y=\sin x

The period is the horizontal length of one complete repeating cycle.
The basic sine graph has maximum $1$ , minimum $-1$ , and period $2\pi$ .
Vertical changes affect outputs; horizontal changes affect input locations.

2. Vertical transformations: changing outputs

Compare the basic graph with

y=\sin x+2

. The

+2

is outside the sine function, so it is added to each sine output. Every point moves up

2

units. The new maximum is

3

, the new minimum is

1

, and the period stays the same.

In general, adding a constant outside the sine function shifts the graph vertically. A positive constant shifts it up; a negative constant shifts it down. The maximum and minimum both change by the same amount, while the distance between them stays the same.

A table can help you check this change. Keep an input the same, find its basic sine output, and then add the vertical shift. On a graph, the wave keeps its shape and cycle length but appears at a new height.

y=\sin x+k

A constant outside $\sin$ changes every output by the same amount.
A vertical shift does not change the period or the distance between maximum and minimum.
Compare matching inputs to check a vertical shift.

3. Horizontal transformations: changing input locations

A change inside the sine function affects the input before the sine value is found. For

y=\sin(2x)

, one full cycle occurs when the inside input reaches

2\pi

. This happens when

x=\pi

. The cycle is therefore shorter than the basic cycle.

For

y=\sin(x-\frac{\pi}{3})

, the sine pattern appears later along the input axis. The graph shifts right by

\frac{\pi}{3}

. In contrast,

y=\sin(x+\frac{\pi}{3})

shifts left by

\frac{\pi}{3}

. To check the direction, find where the inside input equals zero.

For a positive multiplier

b

on the input, the period is found by asking how far

x

must go for the inside input to complete one cycle. A multiplier greater than

1

compresses the graph horizontally. A multiplier between

0

and

1

stretches it horizontally. Compare the distance between matching features, such as consecutive peaks, to investigate the period.

y=\sin(bx),\quad T=\frac{2\pi}{b},\quad b>0

Changes inside the sine function affect horizontal locations or the period.
A positive input multiplier greater than $1$ compresses the graph horizontally.
In $\sin(x-h)$ , the graph shifts right by $h$ .

4. Investigating with graphs and tables

To investigate, compare the basic sine graph with a graph that changes one part of the equation. For a vertical shift, compare the maximum and minimum. For an input change, compare where peaks or other matching features occur, and measure the distance between them.

A table makes it easier to separate inputs from outputs. With a vertical shift, keep the input values the same and change each output by the same amount. With a horizontal shift, familiar features appear at different input values. With a changed input multiplier, matching features appear closer together or farther apart.

A graphing tool can display the original and changed functions together. Choose a viewing window that shows at least one complete cycle. Use the same angle setting and viewing window for both graphs so the comparison is fair.

Change one part of the equation at a time to identify its effect.
Compare output heights and the locations of matching features.
Use a graph window that shows at least one complete cycle.

Comparing basic and transformed sine features

Function	Vertical shift	Period	Maximum and minimum
$y=\sin x$	None	$2\pi$	$1$ and $-1$
$y=\sin x+2$	Up $2$	$2\pi$	$3$ and $1$
$y=\sin(2x)$	None	$\pi$	$1$ and $-1$
$y=\sin(x-\frac{\pi}{3})$	None	$2\pi$	$1$ and $-1$

Worked example

Predicting and checking a transformed sine graph

For

y=\sin(2x)+1

, describe the vertical shift, period, maximum, and minimum. Then check the predictions using key points from one cycle.

Read the vertical change
The $+1$ is outside the sine function, so it adds $1$ to every output. The basic sine outputs range from $-1$ to $1$ . After adding $1$ , the outputs range from $0$ to $2$ .
$y=\sin(2x)+1$
Find the cycle length
One complete cycle occurs when the inside input reaches $2\pi$ . For this function, that means $2x=2\pi$ . Dividing both sides by $2$ gives a period of $\pi$ .
$2x=2\pi,\quad x=\pi$
Check key points
Use the inputs $0$ , $\frac{\pi}{4}$ , $\frac{\pi}{2}$ , $\frac{3\pi}{4}$ , and $\pi$ . Their inside inputs are $0$ , $\frac{\pi}{2}$ , $\pi$ , $\frac{3\pi}{2}$ , and $2\pi$ . The corresponding sine outputs are $0$ , $1$ , $0$ , $-1$ , and $0$ . Adding $1$ gives the points shown.
$(0,1),\left(\frac{\pi}{4},2\right),\left(\frac{\pi}{2},1\right),\left(\frac{3\pi}{4},0\right),(\pi,1)$

Answer: The graph shifts up

1

unit. Its period is

\pi

, its maximum is

2

, and its minimum is

0

. The listed points show one complete cycle from

x=0

to

x=\pi

.

Check: The greatest listed output is

2

and the least is

0

, as predicted. The cycle begins and ends at output

1

, with a horizontal distance of

\pi

between those matching points.

Common mistakes and how to avoid them

Thinking

\sin x+2

changes the period.

Correction: The

+2

is outside the sine function. It changes the outputs but not the horizontal length of a cycle.

Thinking

\sin(x-\frac{\pi}{3})

shifts left.

Correction: It shifts right by

\frac{\pi}{3}

. Set the inside input equal to zero to check: this occurs when

x=\frac{\pi}{3}

.

Using the input multiplier as the period.

Correction: For

y=\sin(bx)

with positive

b

, find how far

x

must go for the inside input to reach

2\pi

. The period is

\frac{2\pi}{b}

.

Lesson summary

A constant outside the sine function shifts all outputs up or down.
A change inside the sine function can shift the graph horizontally or change its period.
For $y=\sin(bx)$ with $b>0$ , the period is $\frac{2\pi}{b}$ .
Use tables and graphs to test predictions about transformed sine functions.

Check your understanding

Question 1

What is the period of

y=\sin(3x)

?

$6\pi$
$3$
$\frac{2\pi}{3}$
$2\pi$

Show answer and explanation

\frac{2\pi}{3}

One full cycle occurs when

3x=2\pi

. Dividing by

3

gives a period of

\frac{2\pi}{3}

.

Question 2

How does

y=\sin x-2

compare with

y=\sin x

?

It shifts down $2$ units.
It shifts right $2$ units.
Its period becomes $2$ .
It shifts up $2$ units.

Show answer and explanation

It shifts down

2

units.

The

-2

is outside the sine function, so it subtracts

2

from every output. The graph moves down

2

units.

Question 3

Which function shifts the basic sine graph right by

\frac{\pi}{4}

without changing its period?

$y=\sin(x+\frac{\pi}{4})$
$y=\sin(x-\frac{\pi}{4})$
$y=\sin(4x)$
$y=\sin x+\frac{\pi}{4}$

Show answer and explanation

y=\sin(x-\frac{\pi}{4})

Subtracting

\frac{\pi}{4}

inside the input shifts the graph right by that amount. The input multiplier is still

1

, so the period remains

2\pi

.

Key terms

Function: A rule that pairs each input with exactly one output.
Period: The horizontal length of one complete repeating cycle.
Vertical shift: A movement of a graph up or down that changes its output values.
Horizontal shift: A movement of a graph left or right that changes where its features occur along the input axis.
Horizontal compression: A change that makes a cycle take less horizontal distance.
Horizontal stretch: A change that makes a cycle take more horizontal distance.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.6. 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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