Transformations of trigonometric functions

Apply reflections, translations and dilations to

y=\sin x

y=\cos x

and

y=\tan x

; determine amplitude, period, phase shift and vertical shift; solve equations and sketch graphs within a specified domain.

Worked examples

Identifying key features from the equation

Straightforward

Problem

For

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

, state the amplitude, period, phase shift and vertical shift. Then state the range.

Solving a trigonometric equation in a given domain

Moderate

Problem

Solve

2\sin(2x) = \sqrt{3}

for

x \in [0,\, 2\pi]

Finding the equation from a description

Challenging

Problem

A trigonometric function

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

has maximum value

5

, minimum value

-3

, period

\pi

, and first maximum at

x = \pi/4

(with

a > 0

b > 0

). Find

a

b

c

d

and write the equation.

Practise

Q1·Straightforward

State the amplitude of

y = 3\sin(2x)

Q2·Straightforward

Find the period of

y = \cos\!\left(\dfrac{x}{3}\right)

in radians. Give your answer as a multiple of

\pi

, i.e. enter the coefficient

k

where the period is

k\pi

Q3·Straightforward

What is the period of

y = \sin(3x - \pi)

in radians? Express your answer as a fraction of

\pi

, i.e. enter the value of

k

where the period is

\dfrac{k\pi}{3}

...

Actually: give the period as a decimal rounded to 4 decimal places.

Q4·Straightforward

State the amplitude of

y = -4\cos(x/2) + 1

Q5·Moderate

The function

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

has amplitude 3, period

4\pi

, phase shift

\pi/4

to the right, and vertical shift

-1

.

Find the value of

b

Q6·Moderate

Find all solutions of

2\sin(x) - 1 = 0

in the interval

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q7·Moderate

Find all solutions of

2\sin(x) - 1 = 0

[0,\,2\pi]

. Enter the **larger** solution in radians, rounded to 4 decimal places.

Q8·Moderate

Find all solutions of

2\cos(x) - 1 = 0

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q9·Moderate

How many solutions does

\sin(2x) = \dfrac{\sqrt{3}}{2}

have in the interval

0 \leq x \leq 2\pi

Q10·Challenging

The graph of

y = A\sin(Bx + C)

passes through the origin, has amplitude 4 and period

\pi

. The graph is increasing at

x=0

.

Given

A > 0

B > 0

and

-\pi < C \leq \pi

, find the value of

C

Q11·Challenging

Find the maximum value of

y = 3\sin\!\left(2x - \dfrac{\pi}{3}\right) + 2

Q12·Challenging

Solve

\sqrt{3}\tan(x) + 1 = 0

for

x \in [0,\,\pi)

. Give your answer in radians, rounded to 4 decimal places.

Q13·Challenging

The function

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

has:
- maximum value

7

- minimum value

-1

- period

3\pi

- a maximum at

x = \pi

Find

a

(the amplitude).

Open Math

Transformations of trigonometric functions

Functions · MAV-12-01

Name:

Date:

Q1Straightforward

State the amplitude of

y = 3\sin(2x)

Q2Straightforward

Find the period of

y = \cos\!\left(\dfrac{x}{3}\right)

in radians. Give your answer as a multiple of

\pi

, i.e. enter the coefficient

k

where the period is

k\pi

Q3Straightforward

What is the period of

y = \sin(3x - \pi)

in radians? Express your answer as a fraction of

\pi

, i.e. enter the value of

k

where the period is

\dfrac{k\pi}{3}

...

Actually: give the period as a decimal rounded to 4 decimal places.

Q4Straightforward

State the amplitude of

y = -4\cos(x/2) + 1

Q5Moderate

The function

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

has amplitude 3, period

4\pi

, phase shift

\pi/4

to the right, and vertical shift

-1

.

Find the value of

b

Q6Moderate

Find all solutions of

2\sin(x) - 1 = 0

in the interval

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q7Moderate

Find all solutions of

2\sin(x) - 1 = 0

[0,\,2\pi]

. Enter the **larger** solution in radians, rounded to 4 decimal places.

Q8Moderate

Find all solutions of

2\cos(x) - 1 = 0

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q9Moderate

How many solutions does

\sin(2x) = \dfrac{\sqrt{3}}{2}

have in the interval

0 \leq x \leq 2\pi

Q10Challenging

The graph of

y = A\sin(Bx + C)

passes through the origin, has amplitude 4 and period

\pi

. The graph is increasing at

x=0

.

Given

A > 0

B > 0

and

-\pi < C \leq \pi

, find the value of

C

Q11Challenging

Find the maximum value of

y = 3\sin\!\left(2x - \dfrac{\pi}{3}\right) + 2

Q12Challenging

Solve

\sqrt{3}\tan(x) + 1 = 0

for

x \in [0,\,\pi)

. Give your answer in radians, rounded to 4 decimal places.

Q13Challenging

The function

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

has:
- maximum value

7

- minimum value

-1

- period

3\pi

- a maximum at

x = \pi

Find

a

(the amplitude).

Worked solutions and answers at openmath.au/year-12/advanced/further-graph-transformations-and-modelling/transformations-of-trigonometric-functions