Contents
4.1 Features of periodic functions 4.2 Predictions using periodic functions 4.3 Models of periodic functions Chapter summary Chapter review
Syllabus subject matter
Periodic functions and applications
■ Definition of a periodic function, the period and amplitude ■ Applications of periodic functions Quantitative concepts and skills
■ Calculation and estimation with and without instruments ■ Plotting points using Cartesian coordinates Syllabus
Guide Chapter 4
Waves and functions
Waves and functions
4.1
Features of periodic functions
Many natural phenomena are cyclic in that they go through a repetitive, predictable cycle. Examples include the phases of the moon, the height of the tide, hormone levels in the
bloodstream, the heart’s electrical activity, the times at which the sun rises and sets, and monthly temperature variations as shown below.
It is possible to model many natural cyclic phenomena using mathematical functions. Consider the function shown below.
It repeats the same pattern after a distance of 4 on the x-scale. That is,
f(x+ 4) =f(x)
Periodic functions are all around us. The ways in which waves move across water, light moves through space, sound moves through air and earthquakes travel around the Earth are all periodic functions. Radio and TV transmissions use periodic waves to send sound and pictures to our living rooms. It is very important to understand periodic functions in order to analyse what happens when you use a switch on an appliance; otherwise the ‘transients’ could wreck the electronics. In this chapter you will start to examine periodic functions.
15 20 25 30 35
Month
T
emperature (°C)
Mean maximum temperature for Brisbane
5 10
0
Feb Mar Apr May Jun Jul Aug Sep Oct No
v
Dec
Jan Feb Mar Apr May Jun Jul Aug Sep Oct No
v
Dec
Jan
4
3
2
1
0
−1
−2
1 2 3 4 5 6 7
y
x Period
Amplitude
Amplitude
A function that repeats values in this way is called periodic. The minimum change of x for which it repeats is called the period.
For this function the value of y varies between −2 and 4, with an average of 1 at the centre. It is 3 units from the centre to the top, and 3 units again to the bottom. We call this distance (3) the amplitude.
The smallest value of T such that g(x + T) = g(x) for all values of x is called the period of the periodic function g(x).
The section of the function over one period is called a cycle.
The amplitude of a periodic function is the distance from the average value to the peaks and troughs. It is half the distance between the maximum and minimum values.
The frequency f of a periodic function is the inverse of the period: f =
Periodic functions are often functions of time.
1
T
---!
Find the period and amplitude of the height of a pendulum from the graph below.
Solution
The graph reaches a minimum at about 0.6 s. It returns to the minimum at about 2.1 s and 3.6 s. There are 2 cycles from 0.6 s to 3.6 s.
2T = 3 s
T = 3 s 2
= 1.5 s
The repeated minimum value is 10 mm and the repeated maximum value is 50 mm.
Amplitude =
= 20 mm
Write the answer. The period is 1.5 s and amplitude 20 mm.
Pendulum height
50
40
30
20
1 2 3
Time (s) 10
0
Height (mm)
50–10 2
---Example
1
Waves and functions
In practical examples of periodic functions, it may be necessary to calculate the period and amplitude by averaging the values over several cycles.
Find the period and amplitude of the tide height at Brisbane from the graph below, which shows the height from Wednesday 1 January to Friday 3 January.
Solution
To find the amplitude, read the maximum peak and trough values in each cycle from the graph, and put them in a table. Then calculate the peak-to-trough differences. There are 3 cycles from 12:20 am Wednesday to 2:36 am Saturday.
3T = 3 days 2 h 16 min
T ≈ 1 day 45.3 min
≈ 1.03 days
Then find the average peak-to-trough difference. Average ≈
≈ 1.89 m
Halve to get the amplitude. Amplitude ≈
≈ 0.95 m
Write the answer. The period is about 1.03 days and the amplitude is about 0.95 m.
8 4 pm MN
4 am MD 8 4am 8
2.5
2.0
1.5
1.0
0.5
0
MN MD4 pm 8 MN4 am 8 MD4 pm 8 MN 4 am
T
ide height (m)
3.0
Tide heights at Brisbane
Wed. 1 Jan. Thu. 2 Jan. Fri. 3 Jan.
12:20 am 2:36 am
Value of peak/trough (m)
Distance from previous peak/trough (m)
0.40
2.15 1.75
0.40 1.75
2.25 1.85
0.35 1.90
2.35 2.00
0.25 2.10
Total 11.35
11.35 6
---1.89 2
---Example
2
Exercise 4.1
Features of periodic functions
Modelling and problem solving
1 The level of a certain hormone in a patient’s bloodstream fluctuates between an almost undetectable concentration at around 8:00 am and 90 ng/mL just before 8:00 pm as shown.
a Does the concentration of hormone in the bloodstream appear to be a periodic function of time?
b Estimate the time at which the concentration of hormone peaks.
c When does the concentration first appear to be at its lowest?
d What is the period of the function?
e What is the amplitude?
2 The graph below shows the monthly mean (average) maximum and minimum temperatures for Brisbane over 2 years.
a What is Brisbane’s hottest month?
b What is Brisbane’s coldest month?
c Calculate the period for mean maximum temperature.
d What is the period for mean minimum temperature?
e Calculate the amplitude for mean maximum temperature.
f What is the amplitude for mean minimum temperature?
g Do mean maximum or minimum temperatures appear more variable?
Additional Exercise
4.1
Hours (h) 8 am
100
80
60
40
20
0
Concentration of hormone (ng/mL)
4 am Noon 4 pm 8 pm MN
MN 4 am 8 am Noon
Concentration of hormone in bloodstream
Feb Mar Apr May Jun Jul Aug 15
20 25 30 35
Month
T
emperature (°C)
5 10
0
Sep Oct Nov Dec
Jan Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Maximum
Minimum
Mean maximum and minimum temperatures in Brisbane
Waves and functions
3 The graph below shows the voltage fluctuations in the domestic electricity supply measured at a 240 V outlet (ms = milliseconds).
a Calculate the period and amplitude.
b 1 hertz (Hz) is 1 cycle per second. Calculate the frequency in hertz.
4 The tide heights at the Brisbane bar are recorded over 4 days as shown below.
a What is the highest tide height over this period? b What is the lowest tide height?
c Calculate the amplitude for the tide height. d What is the period for tide height?
5 The graph of the displacement of air molecules due to the sound wave produced by the tone C3 of an organ pipe is shown below.
100
0
−100 −200
−300
V
oltage (V)
200 300
10 20 30 40 50 60 70
Time (ms)
Domestic electricity supply
MN
Day 1 MN
2.5
2.0
1.5
1.0
0.5
0
MN MN
Height (m)
Day 2 Day 3 Day 4
MN
Tide heights at Brisbane bar
−1.0 −0.5 0.5 1.0
Time (ms) 2
−2 −4 −6 4 6 8 Displacement (mm)
Displacement of air molecules by C3 sound wave
a Does the displacement graph due to the sound wave appear to behave like a periodic function?
b Calculate the amplitude for the displacement function.
c What is the period of the function?
6 The graphs below represent the results of a 90-year study of native quoll and fox populations in a remote location in eastern Australia. The fox preys on the quoll.
a Which line refers to the quoll?
b Do the population levels of the fox and quoll appear to be periodic?
c Calculate the amplitude for the fox and the quoll populations.
d What is the period for each population?
7 The average monthly surface water temperature of a lake in southeastern Australia is measured over 3 years. The results are shown below.
a Draw a graph of the average surface water temperature (vertical axis) and the number of months following the commencement of the study. (You can use a spreadsheet or graphics calculator or draw it by hand.)
b Does the surface water temperature appear to be periodic?
c Calculate the amplitude of the surface water temperature.
d What is the period of the surface water temperature?
Month Temp (ºC) Month Temp (ºC) Month Temp (ºC)
1 5.9 13 5.3 25 1.0
2 6.1 14 6.9 26 1.4
3 7.0 15 6.3 27 8.9
4 15.0 16 13.5 28 14.0
5 19.9 17 19.2 29 19.2
6 23.7 18 23.7 30 24.7
7 27.5 19 26.3 31 27.2
8 28.1 20 28.0 32 25.9
9 26.0 21 23.6 33 22.8
10 19.1 22 17.0 34 17.2
11 11.7 23 12.9 35 11.8
12 12.9 24 8.3 36 6.0
Time (years) 20
160
120
80
40
0
Number of animals
10 30 40 50 60 70 80 90
140
100
60
20
Quoll and fox populations
Waves and functions
4.2
Predictions using periodic functions
We can use the graph of a periodic function to predict future values. We can also use the period to work out when particular values will recur.
The data below shows measurements of the distance the water came up the beach at Coolangatta on one particular day. Measurements were taken at 0.1-minute intervals for the first few waves. Then they were taken at the highest and lowest points for successive waves. The period of the waves remained approximately the same throughout.
1 Work in groups to draw a graph of the distance the waves came up the beach. 2 Work out the average period and amplitude of the waves.
3 Is there any evidence from this information that waves come in ‘sets’? 4 How long should a surfer wait to get the best wave?
Time (min) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Distance (m) 3 3.8 4.5 5.0 5.3 5.2 4.8 4.2 3.4 2.6 1.9 1.3 0.9 0.8 1.1 1.6
Time (min) 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1
Distance (m) 2.2 3.0 3.7 4.4 4.8 5.0 4.9 4.5 3.9 3.3 2.6 2.0 1.5 1.3 1.3 1.6
Time (min) 3.2 3.3 3.4 3.5 3.8 4.5 5.3 6.1 6.9 7.8 8.6 9.4 10.3 11.1 12.0 12.8
Distance (m) 2.0 2.6 3.2 3.8 4.4 1.7 4.3 1.5 4.7 1.1 5.2 0.8 5.3 0.9 5.0 1.3
Investigation
Waves
The graph below shows the vertical displacement of a mass suspended on a spring from its rest position over time.
a Find the period and amplitude of the displacement.
b Predict the displacement at 5 s.
c Find all the times up to 8 s when the displacement will be 10 cm below its rest position (shown as negative on the graph).
10
0
−10 −20
Displacement (cm)
20
Time (s)
0.8 1.6 2.4
0.4 1.2 2.0 Rest position
Negative displacement
Positive displacement
Displacement of a mass on a spring
Example
3
Exercise 4.2
Predictions using periodic functions
Modelling and problem solving
1 The graph below shows the horizontal displacement of a pendulum from its rest position over time.
a Find the period and amplitude of the movement.
b Predict the displacement at 10 s.
c Find all the times up to 20 s when the displacement will be 5 cm to the right (shown as positive on the graph).
Solution
a Find the time between successive troughs.
Period = 2.4 s − 0.8 s
= 1.6 s
Find the distance from peak to trough. Peak to trough distance = 40 cm
Halve to get the amplitude. Amplitude =
= 20 cm
b Write an expression for displacement. Displacement= D(t) where t is in seconds.
The period is 1.6 s. D(t) = D(t ± 1.6)
Write values equal to D(5). D(5)= D(5 − 1.6) = D(3.4)
D(5)= D(5 − 2 × 1.6) = D(1.8)
D(1.8) can be read from the graph. D(5)= D(1.8) ≈ 14 cm
Write the answer. The displacement at 5 s will be about 14 cm.
c Find times on the graph for −10 cm. D(0.5) ≈−10 D(1.1) ≈ −10
All other times can be found by adding multiples of the period to the times.
D(0.5 + 1.6) ≈−10 D(1.1 + 1.6) ≈−10
D(0.5 + 3.2) ≈−10 D(1.1 + 3.2) ≈−10
D(0.5 + 4.8) ≈−10 D(1.1 + 4.8) ≈−10
D(0.5 + 6.4) ≈−10 D(1.1 + 6.4) ≈−10
Add the multiples and write the times. The displacement will be −10 cm at 0.5, 1.1, 2.1, 2.7, 3.7, 4.3, 5.3, 5.9, 6.9 and 7.5 s.
40 2
---Additional Exercise
4.2
8
4
−4
−8
Displacement (cm)
Pendulum displacement
Time (s)
1 2 3 4
0
Displacement
Movement of pendulum
Waves and functions
2 A reflector on a bicycle wheel is fixed at a distance of 25 cm from the centre of the wheel. The wheel has a of diameter 66 cm. For the starting position of the wheel, the reflector is at its lowest height.
Height
How it moves
a What is the least height of the reflector from the ground as the wheel turns?
b What is the greatest height of the reflector as the wheel turns?
c How far does the wheel roll along the ground when it makes one complete turn?
d Draw a graph to show the height of the reflector as the wheel rolls along.
e From your graph, estimate how far the wheel rolls before the reflector is first at a height of 33 cm.
f Estimate how far the wheel rolls before the reflector is first at a height of 40 cm.
3 The graph below shows the voltage of electricity supplied over 45 ms.
a Find the period and amplitude of the voltage.
b What is the voltage at 11 ms?
c Use the information from the graph to predict the voltage at 0.108 s.
d At what times (up to 25 ms) will the voltage be 140 V?
4 A cable-car ride can take tourists to an observation deck 240 m above the base level in 15 minutes if it travels without stopping. The cable cars travel in an approximately straight line from the base level to the observation deck and return, stopping for 2 minutes at either end of the trip. There are four cable cars on the cable.
100
0
−100 −200
V
oltage (V
)
200
Time (ms)
10 20 30
5 15 25
300 400
−300 −400
35 40
Electricity supply
a How many times does a cable car stop between the same points on a round trip?
b How long does a round trip take?
c Draw a graph showing the height of a cable car for two round trips.
d How many round trips are made by a single cable car in 12 daylight hours?
e If each car can take 10 passengers, how many people can do the return trip in one 12-hour day, assuming that cable cars going to the top are full on every trip?
f How long is it from the time a cable car is at 200 m until it is again at this height?
5 The electrical impulses of the heart are shown in the diagram below.
a What is the period? b What is the frequency?
c How is the frequency (in min−1) related to pulse rate?
d How long does it take for the heart to beat 1000 times?
6 The graph below shows how the drag over the wing of an insect in flight varies over time.
a What is the period? b What is the amplitude of the drag?
c At what times up to 3 ms will the drag be greatest?
d The period of the drag represents one beat of the insect’s wings. How many times will the insect’s wings beat in 10 s?
Contraction of the ventricles
Recovery of the ventricles Contraction
of the atria
Contraction of heart muscle Relaxation of
heart muscle
Diastole Systole
Milli
v
olts (mV)
Time (s)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0
Electrical impulses of the heart
1
0
−1
−2
Drag (mN)
2
Time (ms)
0.2 0.4 0.6 0.8 1.0 1.2
Drag over insect’s wing
Waves and functions
4.3
Models of periodic functions
A graphics calculator can be used to model some periodic functions. The sine and cosine functions can be used for this purpose. These are accessed using the and keys. When using a graphics calculator for these functions, the calculator should be set in RADIAN mode. You will learn more about radian measure later in this course.
The graph in Example 4 is the same as the one in question 1 in Exercise 4.2.
sin cos
a Show the graph of y = 8 sin on a graphics calculator over x values from 0 to 5.
b What is the period of the function?
c What is the amplitude?
Solution
a If you are not sure how to enter a function, look back to pages 106–7 in the last chapter. The value of π is available on all the calculators.
On the Casio fx-9860G AU, press .
On the TI-84, press .
On the Sharp EL-9990, press .
Enter the function in Y1 as Y1=8 . Set the view window so that the x values are from Xmin = 0 to Xmax = 5, with y values from Ymin =−10 to Ymax = 10. The scales on the graphs will show these ranges.
GRAPH the function. You should obtain the graph shown.
The TRACE function can be used on all the calculators to move along the curve.
b The graph repeats after an x-length of 4. The period is 4.
c The graph goes from −8 to 8. Peak to trough = 16
Halve to get the amplitude. Amplitude = 8 2 x( +0.5)π
4
---SHIFT EXP
2nd
2nd F VARS
2 X( +0.5)π⁄4
( )
sin
Example
4
In general, the graph of
y = a sin
has amplitude a and period c.
The value of b determines where the graph starts. 2 x( +b)π
c
---!
Exercise 4.3
Models of periodic functions
1 Draw the following functions on a graphics calculator for x = 0 to 10 and y = −10 to 10.
a y = 4 sin 2πx b y = 7 sin 2πx c y = 5 cos 2πx
d y = 5 cos x e y = 9 sin x f y = 9 cos x
g y = 6 cos x h y = 8 sin x i y = 3 cos x
2 Draw the following functions on a graphics calculator for x = 0 to 10 and y = −10 to 10.
a y = 4 sin b y = 7 cos c y = 9 sin
d y = 2 sin e y = 6 cos f y = 5 sin Find a model for the graph shown at
right, and check your model using a graphics calculator or graphing computer program.
Solution
Draw the graph on a calculator and try various values of b. Eventually you should obtain b = 0.5 as the best match. The final display is shown.
A program such as Graphmatica can also be used to display the graph, as shown below. The graph has a period of 3 s. c = 3 in the model.
The amplitude is 2 cm. a = 2 in the model.
The value of b is unclear. Try b = 0 in the model.
Try y = 2 sin .
Change the model to the correct variables. h = 2 sin
1
0
−1 −2 h (cm)
2
t (s)
1 2 3
0.5 1.5 2.5 3.5 4
2 x( +0)π 3
---2 t( +0.5)π 3
---Example
5
Additional Exercise
4.3
3π 2
--- 3π
2
--- π
2
---π
2
---2πx
5
--- 3πx
8
--- πx
7
---4πx
5
--- 7πx
8
--- πx
Waves and functions
3 Draw the following pairs of functions on the same set of axes using a graphics calculator for
x = 0 to 10 and y = −10 to 10.
a y = 8 sin and y = 8 sin b y = 6 cos and y = 6 cos
c y = 5 sin and y = 5 sin d y = 9 cos and y = 9 cos
e y = 3 sin and y = 3 sin f y = 7 cos and y = 7 cos
4 Draw the following functions on a graphics calculator for x = 0 to 10 and y = −10 to 10.
a y = 4 sin b y = 6 sin c y = 8 sin
d y = 9 sin e y = 7 sin f y = 6 sin
5 Draw the following pairs of functions on the same set of axes using a graphics calculator for x = 0 to 10 and y = −10 to 10.
a y = 4 sin and y = 4 sin
b y = 8 sin and y = 8 sin
c y = 7 sin and y = 7 sin
d y = 2 sin and y = 8 sin
Modelling and problem solving
6 Find a model for each of the periodic functions shown below. Check your answers using a graphics calculator, spreadsheet or other computer program.
πx
3
--- 2πx 5
--- 2πx
5
--- 4πx 5
---2πx
7
--- 2πx 5
--- 2πx 3
--- 2πx 5
---2πx
9
--- 5πx 9
--- 2πx 3
--- 7πx 3
---2π(x+1) 3
--- 2π(x–1) 5
--- 2π(x+2) 7
---2π(x–2) 5
--- 2π(x+4) 7
--- 2π(x–3) 11
---2π(x+1) 3
--- 2π(x–1) 3
---2π(x+1) 3
--- 2π(x+1) 9
---π(x+1) 2
--- 5π(x+1) 2
---2π(x+1) 3
--- 2π(x+1) 3 ---a b c d e f −4 −2 4 2 0 2 3 −8 −4 8 4 0 1 2 −2 2 0 1 2 −4 4 0 3 7 −8 −4 8 4 0 1 4
1 2 4 5 6 8 9
3 7
1 2 4 5 6 8 9
−2 −1 2 1 0 3 7
1 2 4 5 6 8 9
7 A satellite is sent into orbit from Cape Canaveral and then it goes into an orbit that takes it north and south of the Equator. The path it traces over the surface of the Earth as shown below is a periodic function of time.
The satellite reaches its maximum distance of 4000 km north of the Equator after 10 minutes, and 45 minutes later it is 4000 km south of the Equator.
a Use this information to find a model of the distance that the orbit of the satellite is north and south of the Equator.
b Use your model to predict how far the satellite is from the Equator after 160 minutes.
c Based on your model, how far north is Cape Canaveral from the Equator?
−2 −1 2
1
0
6 14
2 4 8 10 12 16 18
−4 −2 4
2
0
i j
g h
−8 −4 8
4
0
2 4
1 3 5 1 2 3 4 5 6 7 8 9
−2 2
0
1 2 3 4
Chapter summary
■ A periodic function repeats values.
■ The smallest value for which it repeats is called the period, T:
f (x + T) = f (x) for all values of x
■ The frequency, f, is the inverse of the period:
f =
■ A cycle is the section of the function over one period.
■ The amplitude of a periodic function is the distance from the average value to the maximum or minimum. It is half the distance between the maximum and minimum values.
■ The repetitive nature of periodic functions can be used to predict future values.
■ The sine and cosine functions can be used to model many periodic functions.
The graph of y = a sin has amplitude a and period c.
The value of b determines the point in the cycle at which the graph begins. 1
T
---2 x( +b)π c
Chapter review
Knowledge and procedures
1 The sound level produced by a note from a trombone over time is shown in the graph below, which is taken from an oscilloscope connected to a microphone near the trombone.
a Does the sound level appear to be a periodic function of time?
b At about what time does the sound level first seem to peak?
c When does the sound level first appear to be at its lowest?
d What is the period?
e What is the amplitude?
2 The graph below shows the horizontal displacement of a pendulum from its rest position over time.
a Calculate the period and amplitude of the movement of the pendulum.
b Predict the displacement at 8 s.
c Find all the times up to 4 s when the displacement will be 6 cm.
3 Draw the following functions on a graphics calculator, showing at least 1 cycle.
a y = 4 sin 2πx b y = 5 cos 2πx
Ex 4.1
Time (ms) 2
4.5
3.5
2.5
1.5
0
Oscilloscope reading (V)
1 3 4 5 6 7 9
4.0
3.0
2.0
1.0
8 0.5
Sound level from trombone note
Ex 4.2
4
0
−4
−10
Displacement (cm)
8 14
−14
Pendulum displacement
2 6 10 12
−2
−6 −8
−12
0.5 1.0 1.5 2.0
Time (s)
Chapter review
4 Draw the following functions on a graphics calculator, showing at least 1 cycle.
a y = 7 sin b y = 6 cos
Modelling and problem solving
5 The monthly temperature in a city in the northern hemisphere can be modelled by the equation
T = 6.7 sin + 11.1
where T is the temperature in degrees Celsius and t is the number of months since the end of April.
a What does the model assume about the length of months?
b Which is the coldest month?
c What is the temperature range?
d What would you expect the average temperature to be in spring and autumn?
e During which months is the average temperature less than 7.75ºC?
f At what times of the year would you expect the temperature to be closest to 15ºC?
6 At a particular time of the year, the depth of water in a harbour can be modelled by the equation
d = 2.8 cos + 11.6
where t is the hours after midnight. A ship requires that there is a depth of at least 13 m of water in order to enter the harbour. During what time can it enter the harbour?
7 The distance of a wave from a point on the beach can be modelled by the sine curve
d = a sin bt + c
where d is the distance in metres, t is the time in minutes, and a, b and c are constants. At 2 pm, waves wash from 12 m below a point on the beach to 2 m below the point every half a minute.
a What are the maximum and minimum distances of the wave from the point on the beach?
b How many waves wash up on the beach in an hour?
c Find the values of a, b and c.
8 Find a model (equation) for each of the periodic functions shown below, and check your answers using a graphics calculator, spreadsheet or other computer program.
Ex 4.3
2π(x–3) 5
--- 2π(x+4) 3
---Ex 4.2
πt
6
---Ex 4.2
6πt
37
---Ex 4.3
Ex 4.3
a b
4
2
1 2 3
6 7
5
3
1 y
x 0
3
1
1 2 3
5 6
4
2 y
x
−1
