Sketch transformed sine and cosine functions

Ontario Grade 12 Mathematics

Trigonometric Functions

Use key points to show changes in height, period, and horizontal position

Sine and cosine graphs repeat in smooth cycles. Transformations change a cycle’s height, width, or position. To sketch a transformed graph, first identify those changes. Then mark five key points across one cycle and join them with a smooth curve. This lesson uses radians. In radians, the basic sine and cosine functions each have period

2\pi

.

What you will learn

Identify the amplitude, period, horizontal shift, and vertical shift of a transformed sine or cosine function.
Use five key points over one cycle to sketch a transformed graph.
Check that a sketch matches the function’s equation and repeats at the correct interval.

1. Prerequisite bridge: the basic cycles

A cycle is one complete repeating pattern. For

y=\sin x

, one cycle runs from

x=0

to

x=2\pi

. The graph starts at

0

, rises to

1

, returns to

0

, falls to

-1

, and returns to

0

.

For

y=\cos x

, one cycle also runs from

x=0

to

x=2\pi

. It starts at

1

, falls to

0

, reaches

-1

, rises to

0

, and returns to

1

. These familiar points are useful anchors for sketching.

The amplitude is the distance from the graph’s middle level to its highest or lowest value. The period is the horizontal length of one cycle. The midline is the horizontal line halfway between the maximum and minimum values.

\text{period}=2\pi

Sine key points over one cycle: $0, 1, 0, -1, 0$ .
Cosine key points over one cycle: $1, 0, -1, 0, 1$ .
Both basic graphs have amplitude $1$ and period $2\pi$ .

2. Read the transformations

A useful form for a transformed sine or cosine function is

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

or

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

. The same features can be read from either form.

The number

a

changes the vertical size. The amplitude is |a|, where |a| means the positive distance represented by

a

. If

a

is negative, the graph is reflected across its midline: its usual high and low points switch roles.

The number

k

changes the period. For

k>0

, the period is

\frac{2\pi}{k}

. A larger

k

makes a cycle shorter; a positive

k

less than

1

makes it longer.

The number

d

gives the horizontal shift. The expression

x-d

shifts the graph right by

d

when

d

is positive. If

d

is negative, it shifts the graph left. The number

c

gives the vertical shift and the midline is

y=c

.

The maximum and minimum values can help check the vertical placement. They are

c+|a|

and

c-|a|

. The curve stays between these values.

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

Amplitude: |a|; midline: $y=c$ .
Period: $\frac{2\pi}{k}$ for positive $k$ .
Horizontal shift: $d$ in $x-d$ ; vertical shift: $c$ .
A negative $a$ reverses the graph’s high and low points around the midline.

3. Build a sketch from five points

A reliable sketch starts by finding one full cycle. For a sine function in the stated form, begin at

x=d

. Divide the period into four equal intervals. The five

x

-coordinates are

d

,

d+\frac{P}{4}

,

d+\frac{P}{2}

,

d+\frac{3P}{4}

, and

d+P

, where

P

is the period.

At those points, use the basic sine pattern

0,1,0,-1,0

, multiplied by

a

and shifted vertically by

c

. If

a

is negative, multiplication reverses the pattern. For cosine, begin with the pattern

1,0,-1,0,1

instead.

Mark the midline and the maximum and minimum levels before drawing. Plot the five points in order. Join them with one smooth wave; do not connect them using straight line segments. To show more than one cycle, repeat the pattern at intervals of one period.

For example, when

P=4\pi

, each quarter-cycle interval is

\pi

. This is why dividing the period by four gives the spacing between the five key points.

\text{quarter-cycle interval}=\frac{P}{4}

Find the period, then divide it by four.
Use the sine or cosine starting pattern, adjusted by $a$ and $c$ .
Plot points in order and draw a smooth repeating curve.

4. Application: use the graph’s features to check the model

A sketch can be checked without relying on appearance alone. Confirm that the midline is at

y=c

, that the graph stays between

c-|a|

and

c+|a|

, and that one cycle spans the calculated period.

Also check the starting point and direction. A sine graph with positive

a

begins on its midline at the shifted starting position and moves upward. With negative

a

, it begins on the midline and moves downward. A cosine graph begins at an extreme value at its shifted starting position.

These checks are especially helpful when several transformations occur together. A correct amplitude does not guarantee a correct sketch if the period or horizontal shift is wrong.

y_{\max}=c+|a|,\quad y_{\min}=c-|a|

Check the midline, maximum, and minimum.
Check the horizontal length of one cycle.
Check the starting point and direction of the curve.

Five key points for the worked cycle

Position in cycle	x-coordinate	y-coordinate
Start	$\pi$	$1$
Quarter	$2\pi$	$-1$
Halfway	$3\pi$	$1$
Three-quarter	$4\pi$	$3$
End	$5\pi$	$1$

Worked example

Sketch one transformed sine cycle

Sketch one cycle of

y=-2\sin\left(\frac{x-\pi}{2}\right)+1

and identify its main features.

Identify the transformations
Compare the function with $y=a\sin(k(x-d))+c$ . Here, $a=-2$ , $k=\frac{1}{2}$ , $d=\pi$ , and $c=1$ . The negative value of $a$ means the sine pattern is reflected across the midline.
$a=-2,\quad k=\frac{1}{2},\quad d=\pi,\quad c=1$
Find the period and vertical levels
The period is $\frac{2\pi}{k}=4\pi$ . The amplitude is $2$ , and the midline is $y=1$ . The maximum is $3$ and the minimum is $-1$ .
$P=4\pi,\quad y_{\max}=3,\quad y_{\min}=-1$
Set the five horizontal positions
The cycle begins at $x=d=\pi$ . Each interval is one quarter of $4\pi$ , or $\pi$ . Add this interval repeatedly to get all five positions.
$\pi,\quad 2\pi,\quad 3\pi,\quad 4\pi,\quad 5\pi$
Find the five heights
Use the sine pattern $0,1,0,-1,0$ . Multiplying by $-2$ gives $0,-2,0,2,0$ , and adding $1$ gives the graph’s heights. Plot the resulting points, then draw a smooth curve. The curve starts on its midline and moves downward because $a$ is negative.
$(\pi,1),\quad (2\pi,-1),\quad (3\pi,1),\quad (4\pi,3),\quad (5\pi,1)$

Answer: The graph has amplitude

2

, period

4\pi

, midline

y=1

, and a shift right by

\pi

. One cycle runs from

x=\pi

to

x=5\pi

through the five listed points.

Check: The lowest and highest points are

-1

and

3

, each two units from the midline

y=1

. The cycle length is

5\pi-\pi=4\pi

, as required.

Common mistakes and how to avoid them

Using

2\pi k

as the period.

Correction: For a function with

k

multiplying the shifted input, calculate the period as

\frac{2\pi}{k}

.

Reading the horizontal shift from the sign alone without checking the brackets.

Correction: In

x-d

, a positive

d

shifts right. For example,

x-\pi

gives a shift right by

\pi

.

Treating

a

as the midline or the maximum value.

Correction: The amplitude is |a| and the midline is

y=c

. Use

c+|a|

and

c-|a|

for the maximum and minimum.

Joining key points with straight segments.

Correction: Sine and cosine graphs are smooth waves. Draw a smooth curve through the points and repeat it after one period.

Lesson summary

Read the amplitude and midline from $a$ and $c$ .
Calculate the period using $\frac{2\pi}{k}$ .
Use $d$ to locate the start of a cycle and divide the period into four equal intervals.
Apply the basic sine or cosine pattern, plot five points, and draw a smooth curve.
Check the midline, vertical range, period, and starting direction.

Check your understanding

Question 1

For

y=3\cos(2(x-\frac{\pi}{4}))-2

, what are the amplitude, period, midline, and horizontal shift?

Amplitude $3$ , period $\pi$ , midline $y=-2$ , shift right by $\frac{\pi}{4}$
Amplitude $3$ , period $4\pi$ , midline $y=-2$ , shift left by $\frac{\pi}{4}$
Amplitude $2$ , period $\pi$ , midline $y=3$ , shift right by $\frac{\pi}{4}$
Amplitude $3$ , period $\pi$ , midline $y=2$ , shift left by $\frac{\pi}{4}$

Show answer and explanation

Amplitude

3

, period

\pi

, midline

y=-2

, shift right by

\frac{\pi}{4}

Here

a=3

,

k=2

,

c=-2

, and

d=\frac{\pi}{4}

. Thus the amplitude is

3

, the period is

\frac{2\pi}{2}=\pi

, the midline is

y=-2

, and the graph shifts right.

Question 2

For

y=-\sin(x)+4

, what is the value at the start of a cycle,

x=0

?

$4$
$5$
$3$
$-4$

Show answer and explanation

4

Since

\sin(0)=0

, the function value is

-0+4=4

. The negative coefficient changes the direction of the wave, but not its value at this starting point.

Question 3

A transformed sine graph has period

6\pi

. What is the horizontal spacing between its five key points?

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

Show answer and explanation

\frac{3\pi}{2}

The five key points divide one cycle into four equal intervals. Therefore the spacing is

\frac{6\pi}{4}=\frac{3\pi}{2}

.

Key terms

Amplitude: The positive distance from the midline to a maximum or minimum value.
Cycle: One complete repeating pattern of a graph.
Midline: The horizontal line halfway between the graph’s maximum and minimum values.
Period: The horizontal length of one complete cycle.
Horizontal shift: A movement of the graph left or right.
Vertical shift: A movement of the graph up or down.

Continue through MHF4U

View the complete MHF4U Ontario Grade 12 Mathematics curriculum and lessons

Tutor support for B2.5 · Sketch transformed sine and cosine functions

Working on B2.5 · Sketch transformed sine and cosine functions in MHF4U? Bring the question and your own attempt. In a live session, you can label the axes and connect each feature of the graph to the equation or situation. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث B2.5 · رسم تابع‌های تبدیل‌شده سینوس و کسینوس در درس MHF4U، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید محورها را برچسب بزنید و هر ویژگی نمودار را به معادله یا موقعیت مسئله ربط دهید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.5. 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.

Official curriculum reference

Report a correction or ask a question