1.
The temperature in degrees Celsius during a 24 hour period is shown on the graph and is given
by the function f(t) = a cos (bt) + c, where a, b and c are constants, t is the time in hours
and (bt) is measured in degrees.
(a)
Write down the value of a.
(1)
(b)
Find the value of b.
(2)
(c)
Write down the value of c.
(1)
(d)
Write down the interval of time during which the temperature is increasing from
–4 °C to –2 °C.
(2)
(Total 6 marks)
2.
The graph of the function f(x) = a sin (bx) + c is shown below for –360° ≤ x ≤ 1080°.
(a)
Write down the period of f(x).
(1)
(b)
Write down the value of
(i)
a;
(ii) b;
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(iii)
c.
(3)
P is one of the points where the graph y = f(x) intersects the x-axis. The x-coordinate of P lies
between –180° and 180°.
(c) (i)
Mark and label the point P on the graph above.
(ii) Estimate the x-coordinate of P.
(2)
(Total 6 marks)
3.
The diagram below shows the graph of the functions:
f(x) = p cos (qx), where p, q 
and g(x) = sin(x) – 1.
(a)
Write down the period of f(x).
(1)
(b)
Write down the value of p.
(1)
(c)
Calculate the value of q.
(2)
(d)
Use your graphic display calculator to find any solutions to the equation f(x) = g(x) in the
interval 180° ≤ x ≤ 360°.
(2)
(Total 6 marks)
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4.
Consider the function y = 3 cos(2x) + 1.
(a) Sketch the graph of this function for 0 ≤ x ≤ 180°.
(3)
(b)
Write down the period of the function.
(1)
(c)
Using your graphic display calculator find the smallest possible value of x,
0 ≤ x ≤ 180°, for which 3 cos(2x) + 1 = 2.
(2)
(Total 6 marks)
5.
The graph of the function f(x) = 2 cos (4x) – 1, where 0° ≤ x ≤ 90°, is shown on the diagram
below.
(a)
On the diagram draw the graph of the function g(x) = sin(2x) – 2,
for 0° ≤ x ≤ 90°.
(2)
(b)
Write down the number of solutions to the equation f(x) = g(x), for 0° ≤ x ≤ 90°.
(1)
(c)
Write down one value of x for which f(x) > g(x), for 0° ≤ x ≤ 90°.
(1)
f(x) < g(x) in the interval a < x < b.
(d) Use your graphic display calculator to find the value of
(i)
a;
(ii) b.
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(2)
(Total 6 marks)
6.
The graph of the function f(x) = a sin (bx) + c is shown below for 0° ≤ x ≤ 360°
(a)
Write down the period of f(x).
(1)
(b)
Write down the value of
(i)
a;
(ii) b;
(iii) c.
(3)
(c)
Find the number of solutions to the equation f(x) = 3 in the interval 90° ≤ x ≤ 270°.
(2)
(Total 6 marks)
7.
The following diagram shows the graph of y = a cos bx for 0° ≤ x ≤ 240°.
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(a)
Write down
(i)
the period;
(ii) the amplitude.
(2)
(b)
Find the value of
(i)
a;
(ii) b.
(4)
(Total 6 marks)
8.
The height, h(t), in centimetres, of a bicycle pedal above the ground at time, t, seconds is a
cosine function of the form h(t) = A cos(bt) + C, where (bt) is measured in degrees.
The graph of this function for 0 ≤ t ≤ 4.3 is shown below.
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(a)
Write down the maximum height of the pedal above the ground.
(1)
(b)
Write down the minimum height of the pedal above the ground.
(1)
(c)
Find the amplitude of the function.
(2)
(d)
Hence or otherwise, find the value of A and of C.
(2)
(e)
(i)
(ii)
Write down the period of the function h(t).
Hence find the value of b.
(3)
(f)
Use your graphic display calculator to determine the first value of t for which the height
of the pedal above the ground is 30 cm.
(3)
(g)
Calculate the number of times the pedal rotates in one minute.
(2)
(Total 14 marks)
9.
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.
(c)
Find the period of d. Give your answer in hours.
(1)
(2)
On Tuesday, the minimum value of d occurs at 14:00.
(d) Find when the next maximum value of d occurs.
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(2)
(Total 6 marks)
10.
The graph below represents the temperature (T Celsius) in Washington measured at midday
during a period of thirteen consecutive days starting at Day 0.
These points also lie on the graph of the function
T(x) = a + b cos (cx), 0  x  12,
where a, b and c  .
T
23
22
21
20
19
18
17
16
0
(a)
0
2
Find the value of
(i)
a;
(ii) b.
4
6
8
10
12 x
(2)
(b)
Show that c = 30.
(c)
Using the graph, or otherwise, write down the part of the domain for which the midday
temperature is less than 18.5.
(1)
(2)
(Total 5 marks)
11.
The diagram below shows the graphs of two sine functions, f(x) and g(x) , for –180° ≤ x ≤ 180°.
(a)
Write down
(i)
the equation of f(x);
(ii) the equation of g(x).
(4)
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(b)
Use your graphic display calculator to solve the equation f(x) = g(x) in the interval
–90° ≤ x ≤ 90°.
(2)
(Total 6 marks)
12.
The graph of y = a sin 2x + c is shown below, −180  x  360, x is measured in degrees.
3
2
1
–180 –120 –60
(a)
(b)
(c)
60
120 180 240 300 360
–1
State:
(i)
the period of the function;
(ii) the amplitude of the function.
Determine the values of a and c.
Calculate the value of the first negative x-intercept.
(Total 6 marks)
13.
The figure below shows the set U of all functions of x.
A is the set of all sine and cosine functions of x (measured in degrees).
B is the set of all functions of x with period 120.
C is the subset of A, containing functions whose amplitude is 3.
U
A
C
B
Write the following functions in the correct place in the Venn diagram.
(You may use the numbers (i) to (vi) if you wish, to achieve a better fit.)
(i)
sin (x);
(ii) x2;
(iii) 3 cos (x);
(iv) 2 sin (3x);
(v) 3 cos (3x);
(vi) −3 sin (2x).
(Total 6 marks)
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14.
(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
–1
–2
(b) Write down the period of the function.
(1)
(c)
Write down the amplitude of the function.
(1)
(Total 6 marks)
15.
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
–1
(a)
(b)
(c)
(d)
Determine how many times the temperature is exactly 0C during this 24 hour period.
Write down the time at which the temperature reaches its maximum value.
Write down the interval of time in which the temperature changes from −1C to 2C.
Calculate the value of b.
(Total 6 marks)
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16.
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)
17.
(a)
Sketch the graph of the function f : x  1 + 2 sin x, where x 
, –360° ≤ x ≤ 360°.
(4)
(b)
Write down the range of this function for the given domain.
(2)
(c)
Write down the amplitude of this function.
(1)
(d)
On the same diagram sketch the graph of the function g : x  sin 2x, where x 
–360° ≤ x ≤ 360°.
,
(4)
(e)
Write down the period of this function.
(f)
Use the sketch graphs drawn to find the number of solutions to the equation
f (x) = g (x) in the given domain.
(1)
(1)
(g)
Hence solve the equation 1 + 2 sin x = sin 2x for 0° ≤ x ≤ 360°.
(4)
(Total 17 marks)
18.
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.
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Graph A
Graph B
a
2
y
0
90
180
x
270
360
–a
90
180
x
270
360
Graph C
a
0
90
180
270
360
–a
(a)
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.
(Total 8 marks)
19.
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y
4
3
2
1
0
90°
180°
270°
360°
x
–1
–2
–3
–4
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.
(Total 8 marks)
20.
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
–180–150–120 –90 –60 –30 0
–0.5
(a)
Find the range of f (x).
(b)
Find the amplitude of f (x).
IB Questionbank Mathematical Studies 3rd edition
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60
90 120 150 180 210 240 x
13
(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?
(Total 8 marks)
21.
The diagram shows the graph of y = sin ax + b.
y
2
1
0
30°
60°
x
90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
(a)
Using the graph, write down the following values
(i)
the period;
(ii) the amplitude;
(iii) b.
(b)
Calculate the value of a.
(Total 8 marks)
22.
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.
(Total 8 marks)
The graph below shows the tide heights, h metres, at time t hours after midnight, for Tahini
island.
Tide heights for Tahini
h
5
Height in metres
23.
4
3
2
1
0
(a)
1
2
3
4
5
Number of hours after midnight
6
7
8 t
Use the graph to find
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(i)
(ii)
the height of the tide at 03:15;
the times when the height of the tide is 3.5 metres.
(3)
(b)
The best time to catch fish is when the tide is below 3 metres. Find this best time, giving
your answer as an inequality in t.
(3)
Due to the location of Tahini island, there is very little variation in the pattern of tidal heights.
The maximum tide height is 4.5 metres and the minimum tide height is 1.5 metres. The height h
can be modelled by the function
h(t) = a cos(bt°) + 3.
(c) Use the graph above to find the values of the variables a and b.
(4)
(d)
Hence calculate the height of the tide at 13:00.
(e)
At what time would the tide be at its lowest point in the second 8 hour period?
(3)
(2)
(Total 15 marks)
24.
The diagrams below show the graphs of two functions, y = f(x), and y = g(x).
(a)
(b)
y
y
1.5
y = f(x)
y = g(x)
1
2
0.5
1
0
–360º
1
x
2
–180º –0.5
–1
180º
360º
x
–1.5
State the domain and range of
(a) the function f;
(b)
the function g.
(Total 8 marks)
25.
The diagram below represents a stopwatch. This is a circle, centre O, inside a square of side
6 cm, also with centre O. The stopwatch has a minutes hand and a seconds hand. The seconds
hand, with end point T, is shown in the diagram, and has a radius of 2 cm.
A
D
6 cm
B
T
O
p
q
C
r
(a)
When T is at the point A, the shortest distance from T to the base of the square is p.
Calculate the value of p.
(2)
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(b)
In 10 seconds, T moves from point A to point B. When T is at the point B, the shortest
distance from T to the base of the square is q. Calculate
(i)
the size of angle AOB;
(ii) the distance OD;
(iii) the value of q.
(5)
(c)
In another 10 seconds, T moves from point B to point C. When T is at the point C, the
shortest distance from T to the base of the square is r. Calculate the value of r.
(4)
Let d be the shortest distance from T to the base of the square, when the seconds hand has
moved through an angle . The following table gives values of d and .
Angle
0°
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330

Distance
d
p
4.7
q
3
r
1.3
1
1.3
r
3
q
4.7
360°
p
The graph representing this information is as follows.
d
p
q
r
90º 120º 150º 180º 210º 240º 270º 300º 330º 360º
angle
The equation of this graph can be written in the form d = c + k cos().
(d) Find the values of c and k.
0º
30º
60º
(4)
(Total 15 marks)
26.
The graph below shows part of the function y = 2 sin x + 3.
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y
6
5
4
3
2
1
0
90º
0º
270º
180º
360º
450º
(a)
Write the domain of the part of the function shown on the graph.
(b)
Write the range of the part of the function shown on the graph.
x
(Total 4 marks)
27.
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
–1
–2
–3
Use the graph to find the values of
(a) c;
(b) a.
(Total 4 marks)
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28.
(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
(Total 4 marks)
29.
(a)
A function f is represented by the following mapping diagram.
–1
0
1
2
3
–5
–2
1
4
7
Write down the function f in the form
f : x  y, x  {the domain of f}.
(b)
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.
(Total 4 marks)
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