Maths Quest Maths B Year 12 for Queensland
WorkSHEET 5.2
1
Chapter 5 Periodic functions WorkSHEET 5.2
Periodic functions
Sketch the graph of each of the following for
one complete cycle stating the amplitude,
period and range.
(a) y  3 cos 2 x
(b)
(c)
y
1
Name: _________________________
(a)
y  3 cos 2 x
amplitude = 3, period =
range = [3, 3]
2

2
1
1
sin 
2
4
y   cos 3x
(b)
y
1
1
sin 
2
4
amplitude = 0.5, period = 2 
range = [0.5, 0.5]
(c)
y   cos 3x
amplitude = 1, period =
range = [1, 1]
2
3
1
 8
4
Maths Quest Maths B Year 12 for Queensland
2
Chapter 5 Periodic functions WorkSHEET 5.2
State the horizontal translation and the vertical
translation from the basic graphs of
y  sin x and y  cos x for each of the
following and then sketch each graph from
0 to 2.


y  3 sin  x    5
(a)
4

(b)


y  4 cos  x    1
2

(c)


y  4  3 sin 2 x  
3

(a)


y  3 sin  x    5
4


to the right.
4
vertical translation is 5 units upwards.
horizontal translation is
(b)


y  4 cos  x    1
2


to the left.
2
vertical translation is 1 unit downwards.
horizontal translation is
(c)


y  4  3 sin 2 x  
3


to the right.
3
vertical translation is 4 units upwards of
an inverted sine graph.
horizontal translation is
2
Maths Quest Maths B Year 12 for Queensland
3
Chapter 5 Periodic functions WorkSHEET 5.2
State the maximum and minimum values for
each of the following:


y  7 sin  x    6
(a)
3

1
7
(b) y  cos x    
2
2


(c)
y  2  5 cos  x  
6

(a)
(b)
(c)
4
Find the new equations when each of the
following equations undergoes the translations
described.

y  5 cos x is translated
(a)
units to the
3
right and 4 units down.
(b)
5
1

x is translated
units to the
2
4
left and 3 units up.
y  3 sin
Use addition of ordinates to sketch the graph of
y  3 sin x  cos 3x for x  [0, 2 ] .
(a)
3


y  7 sin  x    6
3

maximum value is 7 + 6 =13
minimum value is 7 + 6 = 1
1
7
cos x    
2
2
maximum value is 0.5  3.5 = 3
minimum value is 0.5  3.5 = 4
y


y  2  5 cos  x  
6

maximum value is 5 + 2 = 7
minimum value is 5 + 2 = 3
y  5 cos x


becomes y  5 cos  x    4
3

(b)
y  3 sin
1
x
2
becomes y  3 sin
1

x 3
2
4
Maths Quest Maths B Year 12 for Queensland
Chapter 5 Periodic functions WorkSHEET 5.2
6
Period =
2
3
2 2

3
n
n3
Range is [4, 4], amplitude = 4 = a
The graph above shows a function of the form
y  a sin n x . Determine the values of a and n.
Period = 3
2
3 
n
2
n
3
Inverted cosine curve with Range [0.5, 0.5],
amplitude = 0.5
a = 0.5
7
The graph above shows a graph of the form
y  a cos n x . Determine the values of a and n.
8
4
3
4 2

3
n
3
b
2
downward shift of 4 which means d = 4

c

shift to the right of
which means  
b
3
3

and so c 
2
range = [2, 6], amplitude is 2 = a
Period 
The graph above shows a function of the form
y  a sin (bx  c)  d . Determine the values of
a, b, c, and d.
4
Maths Quest Maths B Year 12 for Queensland
9
Chapter 5 Periodic functions WorkSHEET 5.2
Determine the equation in the form
y  a cos (bx  c)  d , if the amplitude is 6,
there is an upward shift of 2 and the period is 8.
y  a cos (bx  c)  d
a6
d 4
period 
b
2
8
b

4
no horizontal shift which means c  0

y  6 cos x  2
4
10
1Determine the equation in the form
0y  a sin n ( x  )  c if the amplitude is 5,
there is an downward shift of 4, a shift to the

left of , and the period is 8 .
2
y  a sin (bx  c)  d
a=5
d = 4
2
period  8 
n
1
b
4
c 


c 
b 2
8

1
y  5 sin  x    4
8
4
5