Topics
21. Trigonometric Functions
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iGCSE (2021 Edition)

21.02 Transformations of trigonometric functions

Lesson
Worksheet
Practice
Worksheet
Vertical translations
1

Consider the two graphs y = \sin x and y = \sin x - 2 below:

-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y

Describe the transformation required to obtain the graph of y = \sin x -2 from y = \sin x.

2

Consider the two graphs y = \cos x and y = \cos x + 2 below:

-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y

Describe the transformation required to obtain the graph of y = \cos x + 2 from y = \cos x.

3

Describe the transformation required to obtain the graph of y = \sin x +4 from y = \sin x.

4

State the equation of each of the functions graphed below given that they are of the form y=\sin x + k:

a
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
c
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
d
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
5

State the equation of each of the functions graphed below given that they are of the form y=\cos x + k:

a
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
y
b
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
y
c
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
y
d
-270
-180
-90
90
180
270
x
-5
-4
-3
-2
-1
1
2
3
4
y
6

State the equation of the line that the function y = \sin x + 2 oscillates about.

7

Consider the function y = \cos x + 4.

a

Describe the transformation required to obtain the graph of y = \cos x+ 4 from y = \cos x.

b
State the period of the function.
c

Find the maximum value of the function.

d

Find the minimum value of the function.

8

For each of the functions below:

i

State the period of the function.

ii

State the amplitude of the function.

iii

Find the maximum value of the function.

iv

Find the minimum value of the function.

v

Sketch a graph of the function for -360 \leq x \leq 360.

a
y = \sin x + 1
b
y = \cos x - 3
c
y = \sin x - 2
9

The function y = \cos x + 5 is translated 4 units up.

a

State the equation of the new function after the translation.

b

Find the maximum value of the new function.

Vertical dilations
10

Consider the expression \cos \theta.

a

Complete the table of values for different values of \theta.

\theta 06090 120180 240270300 360
\cos\theta
b

Sketch a graph of the function y = \cos \theta.

c

State the maximum value of \cos \theta.

d

State the minimum value of \cos \theta.

e

State the range of values of 4 \cos \theta.

11

Consider the functions y =\sin x and y =\cos x.

a

State the domain of both functions.

b

State the range of both functions.

12

Consider the function y = - 2 \cos x.

a

State the domain of the function.

b

State the range of the function.

13

Consider the given graph of a function of the form f \left( x \right) = A \sin x:

State the amplitude of the function.

-180
180
x
-8
-6
-4
-2
2
4
6
8
y
14

State the equation of each of the functions graphed below given that they are of the form y = a \sin x or y = a \cos x:

a
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
b
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
c
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
d
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
e
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
f
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
15

Consider the graph of the function y = \sin x for 0 \leq x < 360:

At which value of x in the given domain would y = - \sin x have a maximum value?

-270
-180
-90
90
180
270
x
-1
1
y
Vertical dilations (Extended)
16

Consider the function f \left( x \right) = - 8 \sin x, where 0 \leq x \leq 180.

a

State the amplitude of the function.

b

Find the value of f \left( 180 \right).

c

Find the minimum value of the function.

17

The function y = k \sin x has a maximum value of 5. Find the value of k, where k > 0.

18

A sine function has the form y = c \sin x, a range of \left[ - 2 , 2\right] and a minimum at 90. Find an expression for y.

19

For each of following functions:

i
State the amplitude.
ii
Sketch a graph of the function.
a

y = 3 \sin x.

b

y = 4 \cos x

c
y = \dfrac{4}{3} \cos x
d

y = - 5 \cos x

e

y = \dfrac{5}{4} \sin x

f
y = - 4 \sin x
20

Consider the function y = - 3 \cos x.

a

State the maximum value of the function.

b

State the minimum value of the function.

c

State the amplitude of the function.

d

Describe the two transformations required to obtain the graph of y = -3\cos x from the graph of y = \cos x.

21

Describe the two transformations required to turn the graph of y = \cos x into the graph of y = - \cos x + 3.

22

For each of the functions below:

i

State the period of the function.

ii

State the amplitude of the function.

iii

Find the maximum value of the function.

iv

Find the minimum value of the function.

v

Sketch a graph of the function for -360 \leq x \leq 360.

a
y = 3 \sin x + 2
b
y = 3 \cos x - 3
c
y = 2 \sin x - 3
23

For each of the functions below:

i

Find the value of y when x = 90.

ii

State the amplitude of of the function.

iii

Find the maximum value of the function.

iv

Find the minimum value of the function.

v

Describe the transformations required to obtain the graph of the function from y = \cos x.

vi

Sketch a graph of the function for 0 \leq x \leq 360.

a
y = 2 \cos x + 3
b
y = 2 \cos x - 3
c
y = - 4 \cos x + 2
d
y = \dfrac{1}{3} \sin x + 2
24

For each of the functions below:

i

Find the maximum value of the function.

ii

Find the minimum value of the function.

a
y = 2 - 3 \sin x
b
y = 4 - 3 \sin x
25

A sine function, y, has the form y = c \sin x + d and a range of \left[0, 4\right]. Find an expression for y, where c \gt 0.

26

A cosine function, y, has the form y = c \cos x - d and a range of \left[ - 10 , 6\right]. Find an expression for y, where c \gt 0.

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Outcomes

0607C3.8

Description and identification, using the language of transformations, of the changes to the graph of y = f(x) when y = f(x) + k, y = f(x + k).

0607E3.2G

Recognition of trigonometric, f(x) = a sin(bx); a cos(bx); tan x, function types from the shape of their graphs.

0607E3.8

Description and identification, using the language of transformations, of the changes to the graph of y = f(x) when y = f(x) + k, y = kf(x), y = f(x + k).

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