Topics
7. Trigonometric Functions
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AustraliaQLD
Year 11

7.06 Transformations of sine and cosine

Lesson
Worksheet
Practice
Worksheet
Vertical translations
1

Consider the graphs of the functions y = \sin x and y = \cos x and determine the following:

a

Amplitude

b

Period

c

Range

d

Midline

2

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

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

-2Ο€
-\frac{3}{2}Ο€
-1Ο€
-\frac{1}{2}Ο€
\frac{1}{2}Ο€
1Ο€
\frac{3}{2}Ο€
2Ο€
x
-3
-2
-1
1
y
3

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

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

-2Ο€
-\frac{3}{2}Ο€
-1Ο€
-\frac{1}{2}Ο€
\frac{1}{2}Ο€
1Ο€
\frac{3}{2}Ο€
2Ο€
x
-1
1
2
3
y
4

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

5

The function y = \sin x is translated 2 units down.

a

State the equation of the new function after the translation.

b

Find the minimum value of the new function.

6

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.

7

Consider the graph of y = \cos x + 3.

a

State the y-intercept.

b

State the coordinates of the minimum points on the domain -2\pi \leq x \leq 2\pi.

c

State the equation of the midline.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
1
2
3
4
y
8

State the equation of the midline of the graphs of the following functions:

a
y = \sin x - 5
b
y = 2\cos x + 6
c
y = 4\sin x - 11
d
y = \dfrac{1}{2}\cos x + 8
9

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 in degrees.
c

State the maximum value of the function.

d

State the minimum value of the function.

10

Joanna wishes to sketch the graph of y = \sin x + 3.

a

State the equation of the midline for the graph.

b

State the coordinates of the y-intercept.

c

State the coordinates of the maximum point on the domain 0 \leq x \leq 2\pi.

d

State the coordinates of the minimum point on the domain 0 \leq x \leq 2\pi.

e

Hence sketch the graph of the function on the domain -2\pi \leq x \leq 2\pi.

11

The point C with coordinates \left( \pi, 2 \right) lies on the graph of f \left( x \right). State the coordinates of C after the following transformations:

a

Vertical translation 4 units up.

b

Vertical translation 3 units down.

c
f \left( x \right) + 5
d
f \left( x \right)-7
Horizontal translations
12

Consider the graphs of y = \cos x and \\y = \cos \left(x + \dfrac{\pi}{2}\right) graphed on the same axes:

a

State the amplitude of the functions.

b

State the y-intercept of y = \cos x .

c

How can the graph of y = \cos x be transformed into the graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right)?

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
13

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

a

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

b

Find the smallest positive value of k.

-1Ο€
-\frac{3}{4}Ο€
-\frac{1}{2}Ο€
-\frac{1}{4}Ο€
\frac{1}{4}Ο€
\frac{1}{2}Ο€
\frac{3}{4}Ο€
1Ο€
x
-1
1
y
14

Consider the following graph of a function in the form y = \sin \left(x - c\right):

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

State the coordinates of the y-intercept of the base function y=\sin x.

b

For y=\sin (x-c), state the coordinates of the x-intercept closest to the orgin.

c

Hence determine the equation of the graphed function, where c is the least positive value.

15

Consider the following graph of a function in the form y = \cos \left(x - c\right):

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

State the coordinates of the y-intercept of the base function y=\cos x.

b

For y=\cos (x-c), state the coordinates of the maximum point closest to the orgin.

c

Hence determine the equation of the graphed function, where c is the least positive value.

16

The graph of y = \cos x is translated \dfrac{\pi}{3} units to the left. Determine the following attributes of the new function:

a

Equation

b

Amplitude

c

Period

d

y-intercept

17

Find the values of c in the domain - 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.

18

The point X with coordinates \left( \pi, 1 \right) lies on the graph of f \left( x \right). State the coordinates of X after the following transformations:

a

Horizontal translation \dfrac{\pi}{2} units left.

b

Horizontal translation \pi units right.

c
f(x+\pi)
d
f(x-\dfrac{\pi}{3})
Reflection about the x-axis
19

Consider the graph of the function y = \sin x for 0 \leq x < 2 \pi.

a

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

b

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

\frac{1}{2}Ο€
1Ο€
\frac{3}{2}Ο€
2Ο€
x
-1
1
y
20

Consider the graph of the function y = \cos x for 0 \leq x < 2 \pi.

a

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

b

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

\frac{1}{2}Ο€
1Ο€
\frac{3}{2}Ο€
2Ο€
x
-1
1
y
21

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

a

State the amplitude of the function.

b

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

c

Find the y-intercept.

d

Find the maximum value of the function.

22

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

a

State the maximum value of the function.

b

State the minimum value of the function.

c

State the amplitude of the function.

d

State the y-intercept.

e

Describe the two transformations required to obtain the graph of y = -\cos x + 2, from the graph of y = \cos x.

23

The point A with coordinates \left( \dfrac{\pi}{3}, 4 \right) lies on the graph of f \left( x \right). State the coordinates of A after the following transformations:

a

Reflection about the x-axis

b
-f\left( x \right)
Vertical dilations
24

Compare the graphs of y = 4 \cos x and y = \cos x in terms of:

  • period
  • amplitude
  • maximum
  • minimum
25

State the amplitude of each of the following functions:

a

y = \cos 3 x

b

y = 2\sin \left( x - 3 \right)

c

y = \cos x + 3

d

y = -3 \sin x

26

State the equation of the resulting graph if y = \sin x is vertically dilated by a factor of 9 from the x-axis.

27

State the amplitude of the function f \left( t \right) = - \dfrac{1}{9} \sin t.

28

Describe the transformation required for the following:

a

y = \sin x to become y = \dfrac{1}{7} \sin x.

b

y = \sin x to become y = 5 \sin x.

c

y = \cos x to become y = \dfrac{1}{4} \cos x.

d

y = \cos x to become y = 9 \cos x.

29

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

a

State the amplitude of the function.

b

State the value of A.

-1Ο€
1Ο€
x
-8
-6
-4
-2
2
4
6
8
y
30

Find two possible equations of the graphed function:

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-3
-2
-1
1
2
3
y
31

The function y = k \sin x has a maximum value of 5. Find the value of k when:

a
k > 0
b
k < 0
32

The point B with coordinates \left( \dfrac{\pi}{2}, 3 \right) lies on the graph of f \left( x \right). State the coordinates of B after the following transformations:

a

Vertical dilation factor 4

b

Vertical compression factor \dfrac{1}{3}

c
2f \left( x \right)
d
-5f \left( x \right)
Horizontal dilations
33

Describe the transformation required for the function y = \sin x to become the following:

a

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

b

y = \sin \left(\dfrac{1}{2} x \right)

c

y = \sin 8x

d

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

34

For each of the following graphs of the form y = \sin b x or y = \cos b x, where b is positive:

i

State the period.

ii

Determine the equation of the function.

a
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
b
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
c
1\pi
2\pi
3\pi
4\pi
5\pi
6\pi
7\pi
8\pi
9\pi
x
-1
1
y
d
1\pi
2\pi
3\pi
4\pi
5\pi
6\pi
7\pi
8\pi
9\pi
x
-1
1
y
35

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

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

b

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

c

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

d

Hence find the value of k.

36

The point Y with coordinates \left( \dfrac{\pi}{2}, 4 \right) lies on the graph of f \left( x \right). State the coordinates of Y after the following transformations:

a

Horizontal dilation factor 2

b

Horizontal compression factor \dfrac{1}{2}

c
f \left( 3x \right)
d
f \left( \dfrac{1}{4}x \right)
Combined transformations
37

The graph of y = \sin x undergoes the following series of transformations:

  • Reflected about the x-axis.

  • Horizontally translated to the left by \dfrac{\pi}{4} radians.

  • Vertically translated downwards by 2 units.

Determine the equation of the transformed graph.

38

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

  • Reflected about the x-axis.

  • Horizontally translated to the left by \dfrac{\pi}{6} radians.

  • Vertically translated upwards by 5 units.

Determine the equation of the transformed graph.

39

Determine the equation of the new function after performing the following transformations:

a

The curve y = \sin x is reflected about the x-axis and translated 4 units down.

b

The curve y = \cos x is translated \dfrac{\pi}{3} units to the right, and then reflected about the x-axis.

40

For each of the following functions:

i

Describe the transformations that could be applied to y = \cos x to form the function.

ii

The point \left(0, 1 \right) lies on the graph of y = \cos x. Determine the coordinates of the point, after the transformations have been applied.

a
y = 2 \cos \left( x + \pi\right)
b
y = - \cos x + 3
c
y = - 5 \cos \left( 4 x\right)
d
y = \dfrac{1}{2} \cos \left( x - \dfrac{\pi}{2}\right)+1
41

Consider the functions f\left(x \right) = \cos x and g\left(x \right) as displayed on the following graph:

a

Describe the transformations required to turn the graph of f\left(x \right) into g\left(x \right).

b

Hence write the equation for g\left(x \right).

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-\frac{2}{3}
-\frac{1}{3}
\frac{1}{3}
\frac{2}{3}
1
\frac{4}{3}
\frac{5}{3}
2
\frac{7}{3}
\frac{8}{3}
3
\frac{10}{3}
\frac{11}{3}
y
42

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

a

Determine the period of the function in radians.

b

Find the x-intercepts of the function within the domain 0 \leq x \leq 4 \pi.

c

Determine the coordinates of the maximum value of the function within the domain \\0 \leq x \leq 4 \pi.

d

Describe the transformations required for y = \sin x to become y = 3 \sin \left(\dfrac{x}{2}\right).

43

Consider the graphs of y = \sin x and y = 5 \sin \left(x + \dfrac{\pi}{4}\right).

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-5
-4
-3
-2
-1
1
2
3
4
5
y

Describe the series of transformations required to obtain the graph of y = 5 \sin \left(x + \dfrac{\pi}{4}\right) from the graph of y=\sin x.

44

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

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

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

45

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).

46

Consider the functions f \left( x \right) and g \left( x \right) = jf \left( kx \right), graphed on the same set of axes:

-8\pi
-6\pi
-4\pi
-2\pi
2\pi
4\pi
6\pi
8\pi
x
-1
-\frac{2}{3}
-\frac{1}{3}
\frac{1}{3}
\frac{2}{3}
1
y
a

Determine the equation for f \left( x \right).

b

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

c

Determine the value of j.

d

Determine the value of k.

e

Hence state the equation for g \left( x \right).

47

Consider the graph of y = \sin x.

The first maximum point for x \geq 0 is indicated at \left(\dfrac{\pi}{2}, 1\right).

By considering the transformation that has taken place, find the coordinates of the first maximum point of each of the given functions for x \geq 0:

a

y = 5 \sin x

b

y = - 5 \sin x

c

y = \sin x + 2

d

y = \sin 3 x

e

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

f

y = 5 \sin x + 2

\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
48

Consider the functions f \left( x \right) and g \left( x \right) = f \left( x - k \right) - j, graphed on the same set of axes:

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

Determine the equation for f \left( x \right).

b

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

c

Determine the value of j.

d

Determine the smallest positive value of k.

e

Hence state the equation for g \left( x \right).

49

Consider the graph of y = \cos x.

Its first maximum point for x \geq 0 is indicated at \left(0, 1\right).

By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the given functions for x \geq 0:

a

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

b

y = 5 \cos \left(x - \dfrac{\pi}{3}\right)

c

y = 2 - 5 \cos x

d

y = \cos \left(\dfrac{x}{4}\right)

e

y = 5 \cos 4 x - 2

f

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

\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
50

Consider the graphs of y = \sin x and y = g \left( x \right) graphed on the same set of axes:

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

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

b

Hence state the equation of g \left( x \right) .

51

Consider the graphs of y = \cos x and y = g \left( x \right) graphed on the same set of axes:

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

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

b

Hence state the equation of g \left( x \right) .

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Outcomes

1.2.1.4

examine transformations of the graphs of 𝑓(π‘₯), including dilations and reflections, and the graphs of 𝑦=π‘Žf(π‘₯) and 𝑦=𝑓(𝑏x), translations, and the graphs of 𝑦=𝑓(π‘₯+𝑐) and 𝑦=𝑓(π‘₯)+𝑑; π‘Ž,𝑏,𝑐,𝑑 ∈ R

2.3.2.4

investigate the effect of the parameters 𝐴,𝐡,𝐢 and 𝐷 on the graphs of 𝑦=𝐴sin(𝐡(π‘₯+𝐢))+𝐷, 𝑦=𝐴cos(𝐡(π‘₯+𝐢))+𝐷 with and without technology

2.3.2.5

sketch the graphs of 𝑦=𝐴sin(𝐡(π‘₯+𝐢))+𝐷, 𝑦=𝐴cos(𝐡(π‘₯+𝐢))+𝐷 with and without technology

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