1.
(a)
(b)
For y = 0.5 cos 0.5 x, find
(i)
the amplitude;
(ii)
the period.
Let y = –3 sin x + 2, where 90° ≤ x ≤ 270°.
By drawing the graph of y or otherwise, complete the table below for the given
values of y.
x
y
–1
2
Working:
Answers:
(a) (i) ……………………………………..
(ii) ……………………………………..
(Total 4 marks)
1
2.
(a)
A function f is represented by the following mapping diagram.
?
0
1
2
3
?
?
1
4
7
Write down the function f in the form
f : x  y,
(b)
x  {the domain of f}.
The function g is defined as follows
g : x  sin 15x°,
{x 
and 0 < x ≤ 4}.
Complete the following mapping diagram to represent the function g.
Working:
Answer:
(a) …………………………………………..
(Total 4 marks)
2
3.
The diagram below shows the graph of y = – a sin x° + c,
0 ≤ x ≤ 360.
y
5
4
3
2
1
x
0
90
180
270
360
?
?
?
Use the graph to find the values of
(a)
c;
(b)
a.
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks)
3
4.
Consider the function f (x) = 2 sin x – 1 where 0 ≤ x ≤ 720°.
(a)
Write down the period of the function.
(b)
Find the minimum value of the function.
(c)
Solve f (x) = l.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
4
5.
The curve shown in the figure below is part of the graph of the function, f (x) = 2 + sin (2x),
where x is measured in degrees.
f(x)
4
3.5
3
2.5
2
1.5
1
0.5
? 80? 50? 20 ? 0 ? 0 ? 0 0
? .5
30
60
90 120 150 180 210 240 x
(a)
Find the range of f (x).
(b)
Find the amplitude of f (x).
(c)
Find the period of f (x).
(d)
If the function is changed to f (x) = 2 + sin (4x) what is the effect on the period, compared
to the period of the original function?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ..................................................................
(Total 8 marks)
5
6.
The diagram shows the graph of y = sin ax + b.
y
2
1
0
(a)
(b)
30
60
90
120
150
180
210
240
270
300
330
360
x
Using the graph, write down the following values
(i)
the period;
(ii)
the amplitude;
(iii)
b.
Calculate the value of a.
Working:
Answers:
(a) (i) ...........................................................
(ii) ...........................................................
(iii) ...........................................................
(b) ..................................................................
(Total 8 marks)
6
7.
Below is a graph of the function y = a + b sin (cx) where a, b and c are positive integers and x is
measured in degrees.
7
y
6
5
4
3
2
0
50
100
150
200
250
300
x
350
Find the values of a, b and c.
(Total 6 marks)
7
8.
The graphs of three trigonometric functions are drawn below. The x variable is measured in
degrees, with 0 ≤ x ≤ 360°. The amplitude 'a' is a positive constant with 0 < a ≤ 1.
Graph A
Graph B
a
2
y
0
90
90
180
x
270
180
x
270
360
360
Graph C
a
0
90
(a)
180
270
360
Write the letter of the graph next to the function representing that graph in the box below.
FUNCTION
GRAPH
y = a cos (x)
y = a sin (2x)
y = 2 + a sin (x)
(b)
State the period of the function shown in graph B.
(c)
State the range of the function 2 + a sin (x) in terms of the constant a.
Working:
Answers:
(b) ...................................................
(c) ...................................................
(Total 8 marks)
8
9.
y
4
3
2
1
0
90
180
360
270
x
?
?
?
?
The graph represents the function y = 4 sin (3x).
(a)
(i)
Write down the period of the function.
(ii)
Write down the amplitude of the function.
(b)
Draw the line y = 2 on the diagram.
(c)
Using the graph, or otherwise, solve the equation 4 sin (3x) = 2 for 0  x  90.
Working:
Answers:
(a) (i)..............................................
(ii).............................................
(c) ....................................................
(Total 8 marks)
9
10. The temperature (C) during a 24 hour period in a certain city can be modelled by the function
T (t) = – 3 sin (bt) +2, where b is a constant, t is the time in hours and bt is measured in degrees.
The graph of this function is illustrated below.
T (C)
5
4
3
2
1
t (hours)
0
?
(a)
Determine how many times the temperature is exactly 0C during this 24 hour period.
(b)
Write down the time at which the temperature reaches its maximum value.
(c)
Write down the interval of time in which the temperature changes from −1C to 2C.
(d)
Calculate the value of b.
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) .....................................................
(Total 6 marks)
10
11.
The graph of y = a sin 2x + c is shown below, −180  x  360, x is measured in degrees.
3
2
1
? 80 ? 20 ? 0
60
120 180 240 300 360
?
(a)
State:
(i)
the period of the function;
(ii)
the amplitude of the function.
(b)
Determine the values of a and c.
(c)
Calculate the value of the first negative x-intercept.
Working:
Answers:
(a)
(i)...........................................
(ii)..........................................
(b) ...................................................
(c) ...................................................
(Total 6 marks)
11
12.
(a)
Sketch the graph of the function y =1+
sin (2 x)
for 0  x  360 on the
2
axes below.
(4)
y
2
1
90
180
270
x
360
?
?
(b)
Write down the period of the function.
(1)
(c)
Write down the amplitude of the function.
(1)
Working:
Answers:
(b) ...................................................
(c) ...................................................
(Total 6 marks)
12
13.
The depth, in metres, of water in a harbour is given by the function d = 4 sin (0.5t) + 7, where t
is in minutes, 0  t  1440.
(a)
Write down the amplitude of d.
(1)
(b)
Find the maximum value of d.
(1)
(c)
Find the period of d. Give your answer in hours.
(2)
On Tuesday, the minimum value of d occurs at 14:00.
(d)
Find when the next maximum value of d occurs.
(2)
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) .....................................................
(Total 6 marks)
13