C2.7 Sketch transformed sine functions and state domain and range

Ontario Grade 11 Mathematics

Trigonometric Functions

Use key points to draw a graph and state its domain and range

A sine graph repeats a wave shape as its input continues. Transformations change the wave’s height or position. To sketch one, start with familiar points on the basic sine graph, adjust their outputs, and connect them with a smooth curve. The domain is the set of allowed inputs. The range is the set of possible outputs.

What you will learn

Recall the shape and key points of the basic sine graph.
Describe how vertical stretches, reflections, and vertical shifts change a sine graph.
Sketch a transformed sine graph using key points.
State the domain and range of a sine function.

1. Prerequisite bridge: the basic sine graph

A function pairs each input with one output. On a graph, the input is read along the horizontal axis and the output along the vertical axis. The basic sine function is written

y=\sin x

. Here,

x

is an angle. One complete cycle can be shown from

0

to

2\pi

.

The sine graph rises from zero to a high point, falls through zero to a low point, and returns to zero. A highest point is called a maximum; a lowest point is called a minimum. The basic graph’s maximum output is

1

and its minimum output is

-1

. Its range is therefore

[-1,1]

, including both endpoints.

The graph continues for every real input, in both directions. Its domain is all real numbers. Real numbers include values such as

-2

,

0

, and

1.5

.

y=\sin x

Five points show one basic cycle.
The basic sine graph has domain all real numbers and range $[-1,1]$ .

2. How transformations change the graph

A transformation is a change to a graph’s position, size, or orientation. Start with key points on the basic sine graph. Then change their outputs according to the equation.

Multiplying the sine output changes the graph’s height. For example,

y=2\sin x

doubles every output, so the high and low points become

2

and

-2

. This is a vertical stretch. A negative multiplier also reflects the graph across the horizontal axis: high points become low points, and low points become high points.

Adding a number outside the sine function moves every output up or down. In

y=\sin x+3

, each output increases by

3

, so the range changes from

[-1,1]

to

[2,4]

. This is a vertical shift.

Changing the input can change the horizontal spacing of the wave. For example,

y=\sin(2x)

completes a cycle over a shorter horizontal distance than the basic graph. In

y=\sin(x/2)

, a cycle takes a longer horizontal distance. In either case, any real input is allowed, so the domain remains all real numbers.

y=a\sin x+c

A multiplier outside the sine changes output heights; a negative multiplier also reflects the graph.
Adding outside the sine shifts all outputs up or down.
Changing the input changes the horizontal spacing of the wave, not its domain.

3. From key points to a sketch

For one basic cycle, use inputs

0

,

\frac{\pi}{2}

,

\pi

,

\frac{3\pi}{2}

, and

2\pi

. Their outputs are

0

,

1

,

0

,

-1

, and

0

.

For an equation with an outside multiplier and a vertical shift, keep these input positions and transform each output. If a basic output is

s

, the new output is

as+c

. Plot the new points in order and connect them with a smooth wave. Continue the pattern in both directions because the sine graph repeats.

To find the range, identify the lowest and highest outputs. In

y=a\sin x+c

, when

a

is not zero, the endpoints of the range are

c-|a|

and

c+|a|

. The notation |a| means the non-negative size of

a

. If

a=0

, the graph is a horizontal line at

c

.

Use brackets in an interval such as

[-1,1]

because the endpoint values are included. If the sine input has no restriction, state the domain as all real numbers.

\text{Range}=[c-|a|,\,c+|a|]

Transform the basic outputs, plot the points in order, and draw a smooth repeating curve.
For a nonzero outside multiplier, the range runs from $c-|a|$ to $c+|a|$ .
An unrestricted sine input has domain all real numbers.

4. Check a sketch

A transformed sine graph can show a quantity that rises and falls repeatedly. The sketch shows its outputs, while the domain and range describe its possible inputs and outputs.

Check the direction of the graph. A negative outside multiplier reverses the basic graph’s high and low points. Check its vertical position too: a vertical shift moves every point by the same amount. The graph should still repeat, and an unrestricted sine input should still have domain all real numbers.

You can check the range by starting with the basic outputs from

-1

to

1

, applying the multiplier, and then applying the vertical shift. The resulting lowest and highest values should match the sketch.

Check orientation and vertical position against the equation.
Check that the range matches the graph’s lowest and highest outputs.

Key points for the example

Input $x$	Basic output $\sin x$	Transformed output $-2\sin x+1$
$0$	$0$	$1$
$\frac{\pi}{2}$	$1$	$-1$
$\pi$	$0$	$1$
$\frac{3\pi}{2}$	$-1$	$3$
$2\pi$	$0$	$1$

Worked example

Sketch a reflected and shifted sine graph

Sketch

y=-2\sin x+1

for one cycle from

0

to

2\pi

. State its domain and range.

List the basic points
Use the five standard inputs for one cycle. The basic sine outputs at these inputs are $0$ , $1$ , $0$ , $-1$ , and $0$ . They mark the middle, high, middle, low, and middle of the wave.
$(0,0),\left(\frac{\pi}{2},1\right),(\pi,0),\left(\frac{3\pi}{2},-1\right),(2\pi,0)$
Transform the outputs
The multiplier $-2$ doubles each output and reverses its sign. Adding $1$ then moves each result up by one. Keep the inputs unchanged and apply both changes to each output.
$-2(0)+1=1, -2(1)+1=-1, -2(0)+1=1, -2(-1)+1=3, -2(0)+1=1$
Plot and connect
Plot the new points in order and join them with a smooth sine-shaped curve. The curve starts at $1$ , falls to $-1$ , rises to $1$ , rises further to $3$ , and returns to $1$ . It repeats beyond this cycle.
$(0,1),\left(\frac{\pi}{2},-1\right),(\pi,1),\left(\frac{3\pi}{2},3\right),(2\pi,1)$
State domain and range
The input can be any real number, so the domain is all real numbers. The lowest output is $-1$ and the highest is $3$ . Both endpoints are included in the range.
$\text{Domain}=\mathbb{R}, \text{Range}=[-1,3]$

Answer: The sketch passes smoothly through the five listed points and repeats. Its domain is all real numbers, and its range is

[-1,3]

.

Check: The basic outputs range from

-1

to

1

. Multiplying by

-2

gives outputs from

-2

to

2

, and adding

1

gives outputs from

-1

to

3

. This matches the plotted minimum and maximum.

Common mistakes and how to avoid them

Using the basic sine outputs as the transformed outputs.

Correction: Apply the multiplier and vertical shift to every output before plotting.

Forgetting that a negative multiplier flips the graph.

Correction: Check the transformed high and low points. A negative multiplier makes the basic maximum a transformed low point before any vertical shift.

Giving the range as all real numbers.

Correction: A sine graph continues for all real inputs, but its outputs stay between a lowest and highest value. State the domain and range separately.

Drawing straight line segments between the key points.

Correction: Use the points to guide a smooth wave. The sine graph bends between them.

Lesson summary

The basic sine graph has five key points in one cycle.
Apply an outside multiplier and vertical shift to the outputs, then draw a smooth repeating curve.
For $y=a\sin x+c$ with $a\ne 0$ , the range is $[c-|a|,c+|a|]$ .
The domain is all real numbers when the sine input is unrestricted.

Check your understanding

Question 1

For

y=3\sin x

, what is the range?

$[-3,3]$
$[-2,2]$
$[0,3]$
$\mathbb{R}$

Show answer and explanation

[-3,3]

The basic sine outputs range from

-1

to

1

. Multiplying by

3

gives outputs from

-3

to

3

.

Question 2

For

y=\sin x-2

, what is the range?

$[-3,-1]$
$[-1,1]$
$[-2,2]$
$\mathbb{R}$

Show answer and explanation

[-3,-1]

Subtracting

2

moves both basic endpoints down by

2

. The range becomes

[-1-2,1-2]=[-3,-1]

.

Question 3

What is the domain of

y=-\sin x+4

?

$[-1,1]$
$[3,5]$
$\mathbb{R}$
$[0,2\pi]$

Show answer and explanation

\mathbb{R}

The sine input is unrestricted, so any real number can be used. The domain is all real numbers.

Key terms

Domain: The set of input values a function can use.
Range: The set of output values a function can produce.
Transformation: A change to a graph’s position, size, or orientation.
Vertical shift: A movement of every graph point up or down by the same amount.
Reflection: A flip of a graph across a line; a negative outside multiplier flips a sine graph across the horizontal axis.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C2.7. It is a study resource, not an official curriculum publication.

Official curriculum reference

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C2.7 · Sketch transformed sine functions and state domain and range

Ontario Grade 11 Mathematics

Use key points to draw a graph and state its domain and range

What you will learn

1. Prerequisite bridge: the basic sine graph

2. How transformations change the graph

3. From key points to a sketch

4. Check a sketch

Key points for the example

Sketch a reflected and shifted sine graph

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