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2.02 Period and phase shifts of trigonometric graphs

Worksheet
Horizontal and vertical dilations
1

Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos 4 x.

a

State the period of f \left( x \right) in radians.

b

Complete the table of values for g \left( x \right).

x0\dfrac{\pi}{8}\dfrac{\pi}{4}\dfrac{3\pi}{8}\dfrac{\pi}{2}\dfrac{5\pi}{8}\dfrac{3\pi}{4}\dfrac{7\pi}{8}\pi
g(x)
c

State the period of g \left( x \right) in radians.

d

Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).

e

Sketch the graph of g \left( x \right) for 0 \leq x \leq \pi.

2

Consider the function y = 2 \cos 3 x.

a

State the amplitude of the function.

b

Find the period of the function in radians.

c

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

3

For each of the following functions:

i

State the amplitude.

ii

Find the period in radians.

iii

Sketch a graph of the function for 0 \leq x \leq 2\pi.

a
y = \cos 3x
b
y = \sin \left( \dfrac{x}{2} \right)
c
y = \sin \left( \dfrac{3x}{4} \right)
d
y = \cos \left(\dfrac{x}{3} \right)
e
y = - \cos 4x
f
y = 3 \sin \left(0.25x\right)
g
y = 2 \sin 3x
h
y = -5 \sin 3x
4

Consider the function y = \sin \left( \dfrac{2x}{3} \right).

a

State the amplitude of the function.

b

Find the period of the function in degrees.

c

Sketch a graph of the function for 0\degree \leq x \leq 720 \degree.

5

A table of values for the the first period of the graph y=\sin x for x \geq 0 is given in the first table on the right:

a

Complete the second table given with equivalent values for x in the the first period of the graph y = \sin \left(\dfrac{x}{4}\right) for \\x \geq 0.

b

Hence, state the period of y = \sin \left(\dfrac{x}{4}\right).

x0\dfrac{\pi}{2}\pi\dfrac{3\pi}{2}2\pi
\sin x010-10
x
\sin\left(\dfrac{x}{4}\right)010-10
6

Consider functions of the form y=\tan bx.

a

Complete the given table identifying the periods of the functions.

b

State the period of y = \tan b x.

c

As the value of b in \tan b x increases, does the period become shorter or longer?

FunctionPeriod
\tan x\pi
\tan 2x
\tan 3x
\tan 4x
7

State the period of the following functions:

a
y = \cos 3 x
b
y=\sin \left( 2 \pi x\right)
c
y = \tan 2x
d
y = \tan \left(\dfrac{x}{4}\right)
e
y = \cos\left( \pi x\right)
f
y=\tan 0.75 x
g
y = \tan \left(\dfrac{2x}{3}\right)
h
y = \tan \left(\dfrac{\pi x}{2}\right)
8

The functions f \left( x \right) and g \left( x \right) = f \left( kx \right) have been graphed on the same set of axes below.

-\frac{3}{4}π
-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
x
-1
1
y
a

Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right).

b

Find the value of k.

9

Consider the function f\left(x\right)=\tan x and functions of the form g\left(x\right)=\tan bx.

a
Sketch the function y=f\left(x\right) for 0 \leq x \leq 2\pi.
b
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f\left(x\right).
c

Hence, sketch each of the following funtions for 0 \leq x \leq 2\pi:

i
g \left( x \right) = \tan 2 x
ii
g \left( x \right) = \tan 4 x
iii
g(x) = \tan \left(\dfrac{x}{3}\right)
iv
g \left( x \right) = \tan \left( \dfrac{x}{4} \right)
10

For each of the following functions:

i

Find the y-intercept.

ii

Find the period of the function in radians.

iii

Find the distance between the asymptotes of the function.

iv

State the first asymptote of the function for x \geq 0

v

State the first asymptote of the function for x \leq 0

vi

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

a
y = \tan 2 x
b
y = \tan 3 x
c
y = \tan \left(\dfrac{x}{2}\right)
d
y = \tan \left( - 4 x \right)
11

Consider the equation y = \tan 9 x.

a

State the period of the equation in radians.

b

Sketch the graph of the equation y = \tan 9 x for 0 \leq x \leq \pi

12

Consider the function y = \tan 7 x.

a

Complete the given table of values for the function.

b

Graph the function for -\dfrac{5\pi}{28}\leq x \leq \dfrac{5\pi}{28}.

x-\dfrac{\pi}{28}0\dfrac{\pi}{28}\dfrac{3\pi}{28}\dfrac{\pi}{7}\dfrac{5\pi}{28}
y
13

A function in the form f \left( x \right) = \tan b x has adjacent x-intercepts at x = \dfrac{13 \pi}{6} and x = \dfrac{9 \pi}{4}.

a

State the equation of the asymptote lying between the two x-intercepts.

b

Find the period of the function.

c

State the equation of the function.

14

Consider the graph of f \left( x \right) = \tan \left( \alpha x\right), and the graph of g \left( x \right) = \tan \left( \beta x\right) below:

-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
x
-2
-1
1
2
y

Which is greater: \alpha or \beta? Explain your answer.

15

A table of values for the the first period of the graph y=\tan x for x \geq 0 is given in the first table on the right:

a

Complete the second table given with equivalent values for x in the the first period of the graph y = \tan \left(\dfrac{x}{6}\right) for \\x \geq 0.

b

Hence, state the period of y = \tan \left(\dfrac{x}{6}\right).

x0\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3\pi}{4}\pi
\tan x01\text{undefined}-10
x
\tan\left(\dfrac{x}{6}\right)01\text{undefined}-10
d

State the first three asymptotes of the function for x \geq 0.

e

State the first asymptote of the function for x \leq 0.

16

For each of the following functions of the form y=A \tan bx:

i

Find the value of the function at x=\dfrac{\pi}{4b}.

ii

Find the period of the function in radians.

iv

State the first asymptote of the function for x \geq 0

v

State the first asymptote of the function for x \leq 0

vi

Sketch a graph of the function for -2\pi \leq x \leq 2\pi.

a
y = 3\tan 2 x
b
y = 0.5\tan 3 x
c
y = -5\tan \left(\dfrac{x}{2}\right)
d
y = 2\tan \left(\dfrac{x}{4} \right)
Phase shifts
17

Consider the function y = \sin \left(x + \dfrac{\pi}{2}\right).

a

Find the maximum value of the function.

b

Find the minimum value of the function.

18

Consider the function f \left( x \right) = \cos x and g \left( x \right) = \cos \left(x - \dfrac{\pi}{2}\right).

a

Complete the table of values for both functions:

b

Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).

c

Sketch a graph of g \left( x \right) for 0 \leq x \leq 2\pi.

x0\dfrac{\pi}{2}\pi\dfrac{3\pi}{2}2\pi
f(x)
g(x)
19

For each of following functions:

i

State the amplitude of the function.

ii

Describe the transformation required to obtain the graph of y =f\left(x\right) from the graph of y = \sin x.

iii

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

a
f\left(x\right) = \sin \left(x - \dfrac{\pi}{2}\right)
b
f\left(x\right) = \sin \left(x + \dfrac{\pi}{4}\right)
20

For each of following functions:

i

State the amplitude of the function.

ii

Describe the transformation required to obtain the graph of y =f\left(x\right) from the graph of y = \cos x.

iii

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

a
f\left(x\right) = \cos \left(x + \pi \right)
b
f\left(x\right) = \cos \left(x - \pi\right)
c
f\left(x\right) = \cos \left(x + \dfrac{3\pi}{2}\right)
21

Find the values of c in the region - 2 \pi \leq c \leq 2 \pi that make the graph of y = \sin \left(x - c\right) the same as the graph of y = \cos x.

22

Sketch the graph of the following for -2\pi \leq x \leq 2\pi.

a
y = \sin \left(x + \dfrac{\pi}{2}\right)
b
y = \cos \left(x + \dfrac{\pi}{2}\right)
23

True of false: The graph of f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) is a phase shift of y=\tan x by \dfrac{\pi}{6} units to the right.

24

For each of the functions below, state the horizontal translation required to obtan the graph of g \left( x \right) from the graph of f \left( x \right) = \tan x:

a
g \left( x \right) = \tan \left(x + \dfrac{\pi}{2}\right)
b
g \left( x \right) = \tan \left(x - \dfrac{\pi}{5}\right)
c
g \left( x \right) = \tan \left(x - 1.5\right)
d
g \left( x \right) = \tan \left(x + \dfrac{3\pi}{4}\right)
25

Consider the function f\left(x\right)=\tan x and functions of the form g\left(x\right)=\tan \left(x-a\right).

a
Sketch the function y=f\left(x\right) for -2 \pi \leq x \leq 2\pi.
b
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f\left(x\right).
c

Hence, sketch each of the following functions for -2 \pi \leq x \leq 2 \pi:

i
g \left( x \right) = \tan \left(x - \dfrac{\pi}{4}\right)
ii
g \left( x \right) = \tan \left(x - \dfrac{\pi}{2}\right)
iii
g \left( x \right)= \tan \left(x + \dfrac{\pi}{6}\right)
iv
g \left( x \right)= \tan \left(x - \dfrac{\pi}{3}\right)
26

The function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) is to be graphed on the interval \left[\dfrac{2 \pi}{3}, \dfrac{8 \pi}{3}\right].

a

Find the asymptotes of the function that occur on this interval.

b

Find the x-intercepts of the function that occur on this interval.

c

Sketch a graph of the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) over the interval \left[\dfrac{2 \pi}{3}, \dfrac{8 \pi}{3}\right].

27

For each of the following functions:

i

Find the y-intercept.

ii

Find the period of the function.

iii

Find the distance between the asymptotes of the function.

iv

State the first asymptote of the function for x > 0.

v

State the first asymptote of the function for x \leq 0.

vi

Sketch a graph the function for -\pi \leq x \leq \pi.

a
y = \tan \left(x - \dfrac{\pi}{2}\right)
b
y = \tan \left(x - \dfrac{\pi}{6}\right)
c
y = \tan \left(x + \dfrac{\pi}{3}\right)
28

Consider the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{7}\right).

a

Compared to the equation y = \tan x, state the phase shift of f \left( x \right).

b

State the equations of the first four asymptotes of f \left( x \right) to the right of the origin.

29

Consider functions of the form f\left(x\right)=\tan \left( x-h\right).

a

Sketch a graph of each of the following functions for -\pi \leq x \leq \pi:

i

p \left( x \right) = \tan \left(x - \dfrac{2 \pi}{3}\right)

ii

r \left( x \right) = \tan \left(x + \dfrac{4\pi}{3}\right)

iii

s \left( x \right) = \tan \left(x + \dfrac{\pi}{3}\right)

iv

r \left( x \right) = \tan \left(x - \dfrac{53 \pi}{3}\right)

b

Compare the graphs in part (a) and explain any similarities noted.

Combined translations and dilations
30

Consider the function y = \cos 3 x + 2.

a

Find the period of the function, giving your answer in radians.

b

State the amplitude of the function.

c

Find the maximum value of the function.

d

Find the minimum value of the function.

e

Sketch a graph of the function for 0 \leq x \leq 2\pi.

31

For each of the following functions:

i

State the domain of the function.

ii

State the range of the function.

iii

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

a

y = \sin 2 x - 2

b

y = - 5 \sin 2 x

c

y = \sin \left(\dfrac{x}{3}\right) + 5

d

y = \cos \left(\dfrac{x}{2}\right) - 3

32

Sketch a graph of each of the following the functions for 0 \leq x \leq 2\pi:

a
y = 4 \cos \left(x - \dfrac{\pi}{2}\right)
b
y = \tan \left(\dfrac{\pi}{2} - x\right)
c
y = \tan \left( 2 \left(x - \dfrac{2 \pi}{3}\right)\right)
33

Consider the graphs of y = \cos x and y = 3 \cos \left(x - \dfrac{\pi}{4}\right).

-\frac{9}{4}π
-2π
-\frac{7}{4}π
-\frac{3}{2}π
-\frac{5}{4}π
-1π
-\frac{3}{4}π
-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
\frac{9}{4}π
x
-3
-2
-1
1
2
3
y

Describe the series of transformations required to obtain the graph of y = 3 \cos \left(x - \frac{\pi}{4} \right) from the graph of y=\cos x.

34

Consider the function f\left(x\right) = - 5 \sin \left(x + \dfrac{\pi}{3}\right).

a

Describe the series of transformations required to obtain the graph of f\left(x\right) from the graph of y=\sin x.

b

Sketch a graph of y = f \left(x\right) for -2\pi \leq x \leq 2\pi.

35

The graph of y = \cos x undergoes the series of transformations below:

  • The graph is reflected across the x-axis.

  • The graph is horizontally translated to the left by \dfrac{\pi}{6} radians.

  • The graph is vertically translated upwards by 5 units.

Find the equation of the transformed graph in the form y = a \cos \left(x + c\right) + d, where c is the lowest positive value in radians.

36

For each of the following graphs, find the equation of the function given that it is of the form y = a \sin b x or y = a \cos b x, where b is positive:

a
-\frac{3}{2}π
-1π
-\frac{1}{2}π
\frac{1}{2}π
\frac{3}{2}π
x
-3
-2
-1
1
2
3
y
b
-1π
-\frac{3}{4}π
-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
x
-2
-1
1
2
y
c
-\frac{5}{3}π
-\frac{4}{3}π
-1π
-\frac{2}{3}π
-\frac{1}{3}π
\frac{1}{3}π
\frac{2}{3}π
\frac{4}{3}π
\frac{5}{3}π
x
-3
-2
-1
1
2
3
y
d
-\frac{3}{2}π
-1π
-\frac{1}{2}π
\frac{1}{2}π
\frac{3}{2}π
x
-4
-3
-2
-1
1
2
3
4
y
37

The functions f \left( x \right) and g \left( x \right) = f \left( x - k \right) - j have been graphed on the same set of axes below:

-1π
-\frac{5}{6}π
-\frac{2}{3}π
-\frac{1}{2}π
-\frac{1}{3}π
-\frac{1}{6}π
\frac{1}{6}π
\frac{1}{3}π
\frac{1}{2}π
\frac{2}{3}π
\frac{5}{6}π
x
-4
-3
-2
-1
1
y
a

Find the value of j.

b

Find the smallest positive value of k.

c

Describe the series of transformations required to obtain the graph of g\left(x\right) from the graph of f\left(x\right).

38

Describe the series of transformations required to obtain the graph of y = \sin \left(x - \dfrac{\pi}{4}\right) + 2 from the graph of y = \sin \left(x\right).

39

State whether the following functions represent a change in the period from the function y = \sin x:

a

y = \sin \left( 5 x\right)

b

y = \sin \left( x - 5 \right)

c

y = 5 \sin x

d

y = \sin \left( \dfrac{x}{5} \right)

e

y = \sin x + 5

40

For each of the following functions:

i

State the amplitude of the function.

ii

Describe the transformation required to obtain the graph of y = f\left(x\right) from the graph of y = \sin \left(x\right).

iii
Sketch a graph of the function for 0 \leq x \leq 2\pi.
a
f\left(x\right) = 4 \sin \left(x - \pi\right)
b
f\left(x\right) = - 4 \sin \left(x + \dfrac{\pi}{2}\right)
41

To obtain the graph of y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right) from y = \tan x, two transformations were applied:

  • Horizontal translation of \dfrac{\pi}{4} units to the right.

  • Horizontal dilation by a scale factor of \dfrac{1}{2}.

a

State the transformation that was applied first.

b

Sketch the graph of each of the following functions for -\pi \leq x \leq \pi:

i

y = \tan 2 x

ii

y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right)

42

Consider the function f\left(x\right) = \cos \left( 2 x + \dfrac{\pi}{3}\right).

a

Describe the series of transformations required to obtain the graph of f\left(x\right) from the graph of y=\cos x given that a horizontal dilation was performed first.

b

Sketch the graph of y = f \left( x\right) for 0 \leq x \leq 2\pi.

43

Describe the series of transformations required to obtain the graph of y = \tan \left( 2 x - \dfrac{\pi}{4}\right) from y = \tan x given that a horizontal dilation was performed first.

44

Consider the function y = \cos \left( 3 x + \pi\right).

a

Find the period of the function.

b

Describe the series of transformations required to obtain the graph of y=f\left(x\right) from the graph of y=\cos x given that a horizontal dilation was performed first.

c

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

45

For each of the following functions:

i

Find the period.

ii

State the amplitude.

iii

Find the maximum value.

iv

Find the minimum value.

v

Graph the function for 0 \leq x \leq 2\pi.

a
y = \cos 3 x + 2
b
y = 2 \cos \left(x - \dfrac{\pi}{2}\right) + 3
c
y = 3 \sin \left(x - \dfrac{\pi}{3}\right) + 2
46

Consider the function f \left( x \right) = \tan \left( 3 x - \dfrac{\pi}{4}\right).

a

State the domain of f \left( x \right).

b

State the range of f \left( x \right).

47

Consider the function y = - 4 \tan \dfrac{1}{5} \left(x + \dfrac{\pi}{4}\right).

a

Find the period of the function.

b

Describe a series of transformations to obtain the graph of y=f\left(x\right) from the graph of y=\tan x.

c

Find the range of the function.

48

Consinder the function f \left( x \right) = 3 \tan \left( 3 x\right) + 2.

a
Sketch a graph of y=f\left(x\right) for -\pi \leq x \leq \pi.
b

State the domain of f \left( x \right).

c

State the range of f \left( x \right).

d

The graph of f \left( x \right) has its domain restricted to \left( - \dfrac{\pi}{6} , 0\right], state the range of the restricted graph.

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Outcomes

MA12-1

uses detailed algebraic and graphical techniques to critically construct, model and evaluate arguments in a range of familiar and unfamiliar contexts

MA12-5

applies the concepts and techniques of periodic functions in the solution of problems involving trigonometric graphs

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