Preview of Cambridge IGCSE™ and O Level Additional Mathematics Series

Dear Cambridge Teacher,

Welcome to the new, third edition of our Cambridge IGCSE™ and O Level Additional Mathematics series, which supports the revised Cambridge IGCSE and O Level Additional Mathematics syllabuses (0606/4037) for examination from 2025. We have developed this new edition through extensive research with teachers around the world to provide you and your learners with the support you need, where you need it. You can be confident that this series supports all aspects of the revised syllabuses. This Executive Preview contains sample content from the series, including:

• A guide explaining how to use the series

• A guide explaining how to use each resource

• The table of contents from each resource

‘Worked examples’ demonstrate a step-by-step process of working through a question or problem, and act as an entry to engaging exercise sets. Class discussion questions allow students to articulate their understanding of a skill or topic to a partner, a group or the whole class. We are pleased to include a series of investigative projects authored by NRICH (a collaboration between the Faculties of Mathematics and Education at the University of Cambridge). Four of these projects appear in the coursebook to further facilitate pair, small group and whole-class work. The digital teacher’s resource provides full support and guidance on these projects. A Practice Book for learners is offered to give extra opportunity to consolidate skills using additional questions. Our digital teacher’s resource supports with pedagogical approaches and ideas for how to teach the content in the syllabus – find out more in our resource guide pages.

We are happy to introduce Cambridge Online Mathematics, hosted on our Cambridge GO platform, to the Additional Mathematics resources. Cambridge Online Mathematics provides enhanced teacher and student support; it can be used to create virtual classrooms allowing you to blend print and digital resources into your teaching, in the classroom or as homework. Cambridge Online Mathematics contains all coursebook content in a digital format, additional quiz questions that can be automarked, worksheets, guided walkthroughs of new skills and reporting functionality for teachers. The platform is easy to use, tablet-friendly and flexible.

We hope you enjoy this new series of resources. Visit our website to view the full series or speak to your local sales representative. You can find their details here: cambridge.org/gb/education/find-your-sales-consultant

With best wishes from the Cambridge team,

Thomas Carter

Commissioning Editor for Cambridge IGCSE™ and O Level Additional Mathematics

Cambridge University Press

Cambridge Online Mathematics

Discover our enhanced digital mathematics support for Cambridge Lower Secondary, Cambridge IGCSE™ and Cambridge International AS & A Level Mathematics – endorsed by Cambridge Assessment International Education.

Available in 2023

New content to support the following syllabuses:

• Cambridge IGCSE Mathematics

• Cambridge IGCSE International Mathematics

• Cambridge IGCSE and O Level Additional Mathematics

Features can include:

• Guided walkthroughs of key mathematical concepts for students

• Teacher-set tests and tasks with auto–marking functionality

• A reporting dashboard to help you track student progress quickly and easily

• A test generator to help students practise and refine their skills – ideal for revision and consolidating knowledge

Free trials

A free trial will be available for Cambridge IGCSE Mathematics in 2023. In the meantime, please visit https://bit.ly/3TUGl4l for a free trial of our Cambridge Lower Secondary and Cambridge International AS & A Level Mathematics versions.

This highly illustrated coursebook covers the Cambridge IGCSE TM Additional Mathematics and O Level syllabuses (0606/4037). The course is aimed at students who are currently studying or have previously studied Cambridge IGCSE TM Mathematics (0580/0980) or Cambridge O Level Mathematics (4024).

Where the content in one chapter includes topics that should have already been covered in previous studies, a prerequisite knowledge section has been provided so that you can build on your prior knowledge.

‘Discussion’ sections have been included to provide you with the opportunity to discuss and learn new mathematical concepts with your classmates.

‘Challenge’ questions have been included at the end of most exercises to challenge and stretch you.

Towards the end of each chapter, there is a summary of the key concepts to help you consolidate what you have just learnt. This is followed by a ‘Past paper’ questions section, which contains questions taken from past papers for this syllabus.

A Practice Book is also available in the IGCSE TM Additional Mathematics series, which offers you further targeted practice. This book closely follows the chapters and topics of the coursebook, offering additional exercises to help you to consolidate concepts you have learnt and to assess your learning after each chapter.

How to use this book

Throughout this book, you will notice lots of different features that will help your learning. These are explained below.

THIS SECTION WILL SHOW YOU HOW TO:

These set the scene for each chapter, help with navigation through the Coursebook and indicate the important concepts in each topic.

PRE-REQUISITE KNOWLEDGE

This feature shows how your understanding or use of a topic covered in another area of the book will help you with the concepts in this chapter.

TIP

The information in this feature will help you complete the exercises, and give you support in areas that you might find difficult.

KEY WORDS

The key vocabulary appears in a box at the start of each chapter, and is highlighted in the text when it is first introduced. You will also find definitions of these words in the Glossary at the back of this book.

Exercises

Appearing throughout the text, exercises give you a chance to check that you have understood the topic you have just read about and practice the mathematical skills you have learned. You can find the answers to these questions in the digital version of the Coursebook.

CHALLENGE QUESTIONS

These exercises will stretch your skills in the topic you have just learned. You can find the answers to these questions in the digital version of the Coursebook.

ACTIVITY

Activities give you an opportunity to apply your understanding of a concept to a practical task. When activities have answers, you can find these in the digital version of the Coursebook.

WORKED EXAMPLE

These boxes show you the step-by-step process to work through an example question or problem, giving you the skills to work through questions yourself.

CLASS DISCUSSION

At certain points in the chapters you will be given opportunities to talk about your learning and understanding of the topic in a small group or with a partner.

REFLECTION

These activities ask you to think about the approach that you take to your work, and how you might improve this in the future.

Past paper questions

Questions at the end of each chapter provide a variety of past paper questions, some of which may require use of knowledge from previous chapters. Answers to these questions can be found in the digital version of the Coursebook.

SUMMARY

There is a summary of key points at the end of each chapter.

This icon shows you where you should complete an exercise without using your calculator.

How to use this series

This suite of resources supports learners and teachers following the Cambridge IGCSE™ and O Level Additional Mathematics syllabuses (0606/4037). Up-to-date metacognition techniques have been incorporated throughout the resources to meet the changes in the syllabus content and develop a complete understanding of mathematics for learners. All of the components in the series are designed to work together.

The coursebook contains sixteen chapters that together offer complete coverage of the syllabus. We have worked with NRICH to provide a variety of project activities, designed to engage learners and strengthen their problem-solving skills. Each chapter contains opportunities for formative assessment, differentiation and peer and self-assessment offering learners the support needed to make progress. Cambridge Online Mathematics is available through the digital/print bundle option or on its own without the print coursebook. Learners can review content digitally, explore worked examples and test their knowledge with quiz questions and answers. Teachers benefit from the ability to set tests and tasks with the added auto-marking functionality and a reporting dashboard to help track learner progress quickly and easily.

The digital teacher’s resource provides extensive guidance on how to teach the course, including suggestions for differentiation, formative assessment and language support, teaching ideas and PowerPoints. The Teaching Skills Focus shows teachers how to incorporate a variety of key pedagogical techniques into teaching, including differentiation, assessment for learning, and metacognition. Answers for all components are accessible to teachers for free on the Cambridge GO platform.

Mathematics

1 Characteristics & classification of living organisms

A Practice Book is available for learners that wish to have extra questions to work through. This resource which can be used in class or assigned as homework, provide a wide variety of extra maths activities and questions to help learners consolidate their learning and prepare for assessment. ‘Tips’ are also regularly featured to give learners extra advice and guidance on the different areas of maths they encounter. Access to the digital versions of the practice books is included, and answers can be found either here or in the back of the books.

A Worked Solutions Manual has been introduced to the series. This offers a fully worked solution, with annotated comments, to a selection of questions for teachers or learners to use as they work through the content.

Chapter 2 Simultaneous equations and quadratics

THIS SECTION WILL SHOW YOU HOW TO:

• solve simultaneous equations in two unknowns by elimination or substitution

• find the maximum and minimum values of a quadratic function

• sketch graphs of quadratic functions and find their range for a given domain

• sketch graphs of the function y = | f (x) | where f (x) is quadratic and solve associated equations

• determine the number of roots of a quadratic equation and the related conditions for a line to intersect, be a tangent or not intersect a given curve

• solve quadratic equations for real roots and find the solution set for quadratic inequalities.

PRE-REQUISITE KNOWLEDGE

Before you start…

Where it comes fromYou should be able to...Check your skills

Cambridge IGCSE/O Level Mathematics Solve simultaneous equations using the elimination method.

1 Use the elimination method to solve:

a 4x + 3y = 1

2x − 3y = 14

b 3x + 2y = 19

x + 2y = 13

Cambridge IGCSE/O Level Mathematics Solve simultaneous equations using the substitution method.

2 Use the substitution method to solve:

a y = 3x − 10 x + y = − 2

b x + 2y = 11 4y − x = − 2

Cambridge IGCSE/O Level Mathematics Solve quadratic equations using the factorisation method.

3 Solve by factorisation:

a x 2 + x − 6 = 0

b x 2 − 10x + 16 = 0

c 6x 2 + 11x − 10 = 0

Cambridge IGCSE/O Level Mathematics Solve quadratic equations by completing the square.

4 a Express 2x 2 + 7x + 3 in the form a(x + b)2 + c.

b Use your answer to part a to solve the equation 2x 2 + 7x + 3 = 0

Cambridge IGCSE/O Level Mathematics Solve quadratic equations using the quadratic formula

CLASS DISCUSSION

Solve each pair of simultaneous equations.

8x + 3y = 7 3x + y = 10 2x + 5 = 3y

3x + 5y = −9 2y = 15 − 6x 10 − 6y = −4x

5 Solve 2x 2 − 9x + 8 = 0 Give your answers correct to 2 decimal places.

KEY WORDS

parabola

minimum point

maximum point

Discuss your answers with your classmates. Discuss what the graphs would be like for each pair of equations.

CLASS DISCUSSION

Solve each of these quadratic equations.

x 2 − 8x + 15 = 0 x 2 + 4x + 4 = 0 x 2 + 2x + 4 = 0

Discuss your answers with your classmates. Discuss what the graphs would be like for each of the functions y = x 2 − 8x + 15, y = x 2 + 4x + 4 and y = x 2 + 2x + 4.

turning point

stationary point completing the square roots

discriminant tangent

2.1 Simultaneous equations (one linear and one non-linear)

In this section you will learn how to solve simultaneous equations where one equation is linear and the second equation is not linear.

The diagram shows the graphs of y = x + 1 and y = x 2 − 5.

The coordinates of the points of intersection of the two graphs are (−2, −1) and (3, 4).

We say that x = −2, y = −1 and x = 3, y = 4 are the solutions of the simultaneous equations

y = x + 1 and y = x 2 − 5.

The solutions can also be found algebraically:

y = x + 1 (1)

y = x 2 − 5 (2)

Substitute for y from (1) into (2):

x + 1 = x 2 − 5 rearrange

x 2 − x − 6 = 0 factorise

(x + 2)(x − 3) = 0

x = − 2 or x = 3

Substituting x = −2 into (1) gives y = −2 + 1 = −1.

Substituting x = 3 into (1) gives y = 3 + 1 = 4.

The solutions are: x = −2, y = −1 and x = 3, y = 4.

WORKED EXAMPLE 1

Solve the simultaneous equations.

2x + 2y = 7

x 2 + 4y 2 = 8

Answers

2x + 2y = 7 (1)

x 2 − 4y 2 = 8 (2)

From (1), x = 7 − 2y 2

Substitute for x in (2):

7 − 2y 2 )2 − 4y 2 = 8 expand brackets

49 − 28y + 4y 2 4 − 4y 2 = 8 multiply both sides by 4

49 − 28y + 4y 2 − 16y 2 = 32 rearrange

12y 2 + 28y − 17 = 0 factorise

(6y + 17)(2y − 1) = 0

y = − 2 5 6 or y = 1 2

Substituting y = − 2 5 6 into (1) gives x = 6 1 3

Substituting y = 1 2 into (1) gives x = 3 The

Exercise 2.1

Solve the following simultaneous equations.

19 Calculate the coordinates of the points where the line

20 The sum of two numbers x and y is 11. The product of the two numbers is 21.25.

a Write down two equations in x and y

b Solve your equations to find the possible values of x and y.

21 The sum of the areas of two squares is 818 cm 2.

The sum of the perimeters is 160 cm.

Find the lengths of the sides of the squares.

22 The line y = 2 − 2x cuts the curve 3x 2 − y 2 = 3 at the points A and B

Find the length of the line AB.

23 The line 2x + 5y = 1 meets the curve x 2 + 5xy − 4y 2 + 10 = 0 at the points A and B.

Find the coordinates of the midpoint of AB

24 The line y = x − 10 intersects the curve x 2 + y 2 + 4x + 6y − 40 = 0 at the points A and B. Find the length of the line AB.

25 The straight line y = 2x − 2 intersects the curve x 2 − y = 5 at the points A and B.

Given that A lies below the x-axis and the point P lies on AB such that AP : PB = 3 : 1, find the coordinates of P

26 The line x − 2y = 2 intersects the curve x + y 2 = 10 at two points A and B

Find the equation of the perpendicular bisector of the line AB.

2.2 Maximum and minimum values of a quadratic function

The general equation of a quadratic function is f(x) = ax 2 + bx + c, where a, b and c are constants and a ≠ 0.

The graph of the function y = ax 2 + bx + c is called a parabola. The orientation of the parabola depends on the value of a, the coefficient of x 2 .

If a . 0, the curve has a minimum point which occurs at the lowest point of the curve.

If a , 0, the curve has a maximum point which occurs at the highest point of the curve.

The maximum and minimum points are also called turning points or stationary points

Every parabola has a line of symmetry that passes through the maximum or minimum point.

WORKED EXAMPLE 2

f (x) = x 2 − 3x − 4 x ∈ ℝ

a Find the axis crossing points for the graph of y = f (x).

b Sketch the graph of y = f (x) and use the symmetry of the curve to find the coordinates of the minimum point.

c State the range of the function f (x).

Answers

a y = x 2 − 3x − 4

When x = 0, y = −4

When y = 0, x 2 − 3x − 4 = 0 (x + 1) (x − 4) = 0

x = − 1 or x = 4

Axes crossing points are: (0, −4), (−1, 0) and (4, 0).

b The line of symmetry cuts the x-axis midway between −1 and 4.

So, the line of symmetry is x = 1.5

When x = 1.5, y = 1.52 − 3(1.5) − 4

y = − 6.25

Minimum point = (1.5, − 6.25)

c The range is f(x) > −6.25

Completing the square

If you expand the expressions (x + d )2 and (x − d )2 you obtain the results: (x + d )2 = x 2 + 2dx + d 2 and (x − d )2 = x 2 − 2dx + d 2

Rearranging these give the following important results:

x 2 + 2dx = (x + d )2 − d 2

x 2 − 2dx = (x − d )2 − d 2

This is known as completing the square.

To complete the square for x 2 + 8x: 8 ÷ 2 = 4

x 2 + 8x = (x + 4)2 − 42

x 2 + 8x = (x + 4)2 − 16

To complete the square for x 2 + 10x − 3: 10 ÷ 2 = 5

x 2 + 10x − 3 = (x + 5)2 − 52 − 3

x 2 + 10x − 3 = (x + 5)2 − 28

To complete the square for 2x 2 − 8x − 14 you must first take a factor of 2 out of the expression:

2x 2 − 8x + 14 = 2[ x 2 − 4x + 7 ] 4 ÷ 2 = 2

x 2 − 4x + 7 = (x − 2)2 − 22 + 7

x 2 − 4x + 3 = (x − 2)2 + 3

So, 2x 2 − 8x + 6 = 2[ (x − 2)2 + 3 ] = 2(x − 2)2 + 6

You can also use an algebraic method for completing the square, as shown in Worked example 3

WORKED EXAMPLE 3

Express 2x 2 − 4x + 5 in the form p(x − q)2 + r, where p, q and r are constants to be found.

Answers

2x 2 − 4x + 5 = p(x − q)2 + r

Expanding the brackets and simplifying gives:

2x 2 − 4x + 5 = px 2 − 2pqx + pq 2 + r

Comparing coefficients of x2, coefficients of x and the constant gives:

2 = p (1) −4 = −2pq (2) 5 = pq 2 + r (3)

Substituting p = 2 in equation (2) gives q = 1.

Substituting p = 2 and q = 1 in equation (3) gives r = 3.

So 2x 2 − 4x + 5 = 2(x − 1)2 + 3.

Completing the square for a quadratic expression or function enables you to:

• write down the maximum or minimum value of the expression

• write down the coordinates of the maximum or minimum point of the function

• sketch the graph of the function

• write down the line of symmetry of the function

• state the range of the function.

In Worked example 3 you found that:

2x 2 − 4x + 5 = 2 (x − 1)2 + 3

This part of the expression is a square so it will always be ⩾ 0. The smallest value it can be is 0. This occurs when x = 1.

The minimum value of the expression is 2 × 0 + 3 = 3 and this minimum occurs when x = 1.

So, the function y = 2x 2 − 4x + 5 will have a minimum at the point (1, 3).

When x = 0, y = 5.

The graph of y = 2x 2 − 4x + 5 can now be sketched:

The line of symmetry is x = 1.

The range is y > 3.

The general rule is:

For a quadratic function f (x) = ax 2 + bx + c that is written in the form

f (x) = a(x − h)2 + k,

i if a > 0, the minimum point is (h, k)

ii if a < 0, the maximum point is (h, k).

WORKED EXAMPLE 4

f (x) = 2 + 8x − 2x 2 x ∈ ℝ

a Find the value of a, the value of b and the value of c for which f (x) = a − b (x + c)2

b Write down the coordinates of the maximum point on the curve y = f (x).

c Sketch the graph of y = f (x), showing the coordinates of the points where the graph intersects the x and y-axes.

d State the range of the function f (x).

Answers

a 2 + 8x − 2x 2 = a − b(x + c)2

2 + 8x − 2x 2 = a − b(x 2 + 2cx + c2)

2 + 8x − 2x 2 = a − bx2 − 2bcx − bc2

Comparing coefficients of x2, coefficients of x and the constant gives:

−2 = −b (1) 8 = −2bc (2) 2 = a − bc2 (3)

Substituting b = 2 in equation (2) gives c = −2.

Substituting b = 2 and c = −2 in equation (3) gives a = 10.

So, a = 10, b = 2 and c = −2.

b y = 10 − 2 (x − 2)2

The maximum value of the expression is 10 − 2 × 0 = 10 and this maximum occurs when x = 2. So, the function y = 2 + 8x − 2x 2 will have maximum at the point (2, 10).

c y = 2 + 8x − 2x 2

When x = 0, y = 2.

When y = 0, 10 − 2(x − 2)2 = 0 2(x − 2)2 = 10 (x − 2)2 = 5 x − 2 = ± √ 5

x = 2 ± √ 5

This part of the expression is a square so it will always be ⩾ 0. The smallest value it can be is 0. This occurs when x = 2. y

x = 2 − √ 5 or x = 2 + √ 5

(x = − 0.236 or x = 4.24 to 3 sf)

Axes crossing points are: (0, 2), (2 + √ 5 , 0) and (2 − √ 5 , 0)

d The range is f (x) < 10.

Exercise 2.2

1 Use the symmetry of each quadratic function to find the maximum or minimum points. Sketch each graph, showing all axis crossing points.

8 a Express 4x 2 + 2x + 5 in the form a (x + b)2 + c, where a, b and c are constants.

b Does the function y = 4x 2 + 2x + 5 meet the x-axis?

Explain your answer.

9 f (x) = 2x 2 − 8x + 1

a Express 2x 2 − 8x + 1 in the form a(x + b)2 + c, where a and b are integers.

b Find the coordinates of the stationary point on the graph of y = f (x).

10 f (x) = x 2 − x − 5 for x ∈ ℝ

a Find the minimum value of f (x) and the corresponding value of x

b Hence write down a suitable domain for f (x) in order that f −1 (x) exists.

11 f (x) = 5 − 7x − 2x 2 for x ∈ ℝ

a Write f(x) in the form p − 2(x − q)2, where p and q are constants to be found.

b Write down the range of the function f (x).

12 f (x) = 14 + 6x − 2x 2 for x ∈ ℝ

a Express 14 + 6x − 2x 2 in the form a + b (x + c)2, where a, b and c are constants.

b Write down the coordinates of the stationary point on the graph of y = f (x).

c Sketch the graph of y = f (x).

13 f (x) = 7 + 5x − x 2 for 0 < x < 7

a Express 7 + 5x − x 2 in the form a − (x + b)2, where a, and b are constants.

b Find the coordinates of the turning point of the function f (x), stating whether it is a maximum or minimum point.

c Find the range of f

d State, giving a reason, whether or not f has an inverse.

14 The function f is such that f (x) = 2x 2 − 8x + 3.

a Write f (x) in the form 2(x + a)2 + b, where a and b are constants to be found.

b Write down a suitable domain for f so that f −1 exists.

15 f (x) = 4x 2 + 6x − 8 where x > m

Find the smallest value of m for which f has an inverse.

16 f (x) = 1 + 4x − x 2 for x > 2

a Express 1 + 4x − x 2 in the form a − (x + b)2, where a and b are constants to be found.

b Find the coordinates of the turning point of the function f (x), stating whether it is a maximum or minimum point.

c Explain why f (x) has an inverse and find an expression for f −1(x) in terms of x

2.3 Graphs of y = |f (x)| where f (x) is quadratic

To sketch the graph of the modulus function y = | ax 2 + bx + c |, you must:

• first sketch the graph of y = ax 2 + bx + c

• reflect in the x-axis the part of the curve y = ax 2 + bx + c that is below the x-axis.

WORKED EXAMPLE 5

Sketch the graph of y = |x 2 − 2x − 3|.

Answers

First sketch the graph of y = x 2 − 2x − 3. When x = 0, y = −3. So, the y-intercept is −3.

When y = 0, x 2 − 2x − 3 = 0 (x + 1)(x − 3) = 0

x = − 1 or x = 3.

The x-intercepts are −1 and 3.

The x-coordinate of the minimum point = 1 + 3 2 = 1.

The y-coordinate of the minimum point = (1)2

The minimum point is (1, −4).

Now reflect in the x-axis the part of the curve y

that is below the x-axis.

A sketch of the function y = |

12 | is shown below.

Now consider using this graph to find the number of solutions of the equation

2 + 4x − 12 | = k, where k > 0.

The conditions for the number of solutions of the equation | x 2 + 4x − 12 | = k are:

of k k = 0 0 , k , 16 k = 16 k . 16

involving | f (x) |, where f (x) is quadratic, can be solved algebraically:

The graph of y = | x

+

x − 12 | is sketched here showing these three solutions.

Exercise 2.3

1 Sketch the graphs of each of the following functions.

a y = | x 2 − 4x + 3 | b y = |

d y = | x 2 − 2x − 8 |

2 f (x) = 1 − 4x − x 2

a Write f (x) in the form a − (x + b)2, where a and b are constants.

b Sketch the graph of y = f (x).

c Sketch the graph of y = | f (x) |.

3 f (x) = 2x 2 + x − 3

a Write f(x) in the form a (x + b)2 + c, where a, b and c are constants.

b Sketch the graph of y = | f (x) |.

4 a Find the coordinates of the stationary point on the curve y = | (x − 7) (x + 1) |.

b Sketch the graph of y = | (x − 7) (x + 1) |.

c Find the set of values of k for which | (x − 7) (x + 1) | = k has four solutions.

5 a Find the coordinates of the stationary point on the curve y = | (x + 5) (x + 1) |.

b Find the set of values of k for which | (x + 5) (x + 1) | = k has two solutions.

6 a Find the coordinates of the stationary point on the curve y = | (x − 8) (x − 3) |.

b Find the value of k for which | (x − 8) (x − 3) | = k has three solutions.

7 Solve these equations

8 CHALLENGE QUESTION

Solve these simultaneous equations.

2.4 Quadratic inequalities

You should already know how to solve linear inequalities. Two examples are shown below.

Solve 2x − 5 , 9 expand brackets 2x − 10 , 9 add 10 to both sides 2x , 19 divide both sides by 2

, 9.5

Solve 5 − 3x > 17 subtract 5 from both sides 3x > 12 divide both sides by −3 x < − 4

TIP

It is very important that you remember the rule that when you multiply or divide both sides of an inequality by a negative number then the inequality sign must be reversed. This is illustrated in the second of these examples, when both sides of the inequality were divided by −3.

CLASS DISCUSSION

Robert is asked to solve the inequality 7x + 12 x > 3.

He writes: 7x + 12 ⩾ 3x 4x ⩾ 12

So x ⩾ 3

Anna checks his answer using the number −4.

She writes: When x = 4, (7 × ( 4) + 12) ÷ ( 4) = ( 16) ÷ ( 4) = 4

Hence x = 4 is a value of x that satisfies the original inequality

So Robert’s answer must be incorrect!

Discuss Robert’s working out with your classmates and explain Robert’s error. Now solve the inequality 7x + 12 x > 3 correctly.

Quadratic inequalities can be solved by sketching a graph and considering when the graph is above or below the x-axis.

WORKED EXAMPLE 6

Solve x 2 − 3x − 4 . 0.

Answers

Sketch the graph of y = x 2 − 3x − 4.

When y = 0, x 2 − 3x − 4 = 0 (x + 1) (x − 4) = 0 x = − 1 or x = 4

So, the x-axis crossing points are −1 and 4.

For x 2 − 3x − 4 . 0 you need to find the range of values of x for which the curve is positive (above the x-axis).

The solution is x , −1 and x . 4.

WORKED EXAMPLE 7

Solve 2x 2 < 15 − x.

Answers

Rearranging: 2x 2 + x − 15 < 0.

Sketch the graph of y = 2x 2 + x − 15.

When y = 0, 2x 2 + x − 15 = 0

(2x − 5) (x + 3) = 0 x = 2.5 or x = − 3

So, the x-axis crossing points are −3 and 2.5

For 2x 2 + x − 15 < 0 you need to find the range of values of x for which the curve is either zero or negative (below the x-axis).

The solution is −3 < x < 2.5

Exercise

Find

Look back at this exercise.

a How confident do you feel in your understanding of this section?

b What can you do to increase your level of confidence?

2.5 Roots of quadratic equations

The solutions of an equation are called the roots of the equation. Consider solving the following three quadratic equations using the quadratic formula

The part of the quadratic formula underneath the square root sign is called the discriminant.

The sign (positive, zero or negative) of the discriminant tells you how many roots there are for a particular quadratic equation.

b2 − 4ac Nature of roots

. 02 real distinct roots = 02 real equal roots , 00 real roots

There is a connection between the roots of the quadratic equation ax 2 + bx + c = 0 and the corresponding curve y = ax 2 + bx + c

The curve cuts the x-axis at 2 distinct points.

The curve touches the x-axis at 1 point. ,

The curve is entirely above or entirely below the

Find the values of k for which x 2 − 3x + 6 = k(x − 2) has two equal roots.

Answers

x 2 − 3x + 6 = k(x − 2)

x 2 − 3x + 6 − kx + 2k = 0

x 2 − (3 + k)x + 6 + 2k = 0

For two equal roots, b2 − 4ac = 0.

(3 + k)2 − 4 × 1 × (6 + 2k) = 0

k2 + 6k + 9 − 24 − 8k = 0 k2 − 2k − 15 = 0

(k + 3) (k − 5) = 0

So k = −3 or k = 5.

WORKED EXAMPLE 9

Find

Exercise 2.5

1 State whether these equations have two distinct roots, two equal roots or no roots.

2.6 Intersection of a line and a curve

When considering the intersection of a straight line and a parabola, there are three possible situations.

2 points of intersection1 point of intersection

0 points of intersection

The line cuts the curve at two distinct points. The line touches the curve at one point. This means that the line is a tangent to the curve. The line does not intersect the curve.

You have already learned that to find the points of intersection of the line y = x − 6 with the parabola y = x 2 − 3x − 4 you solve the two equations simultaneously. This would give x 2 − 3x − 4 = x − 6 x 2 − 4x + 2 = 0.

The resulting quadratic equation can then be solved using the quadratic formula:

The number of points of intersection will depend on the value of b2 − 4ac.

The different situations are given in the table below.

b2 4ac Nature of roots Line and curve

. 02 real distinct roots2 distinct points of intersection

= 02 real equal roots1 point of intersection (line is a tangent)

, 00 real roots no points of intersection

The condition for a quadratic equation to have real roots is b2 − 4ac > 0.

WORKED EXAMPLE 10

Find the value of k for which y = 2x + k is a tangent to the curve y = x 2 −4x + 4.

Answers

x 2 − 4x + 4 = 2x + k

x 2 − 6x + (4−k) = 0

Since the line is a tangent to the curve, b2 − 4ac = 0.

(− 62) − 4 × 1 × (4 − k) = 0 36 − 16 + 4k = 0

k = − 5

WORKED EXAMPLE 11

Find the range of values of k for which y = x − 5 intersects the curve y = kx2 − 6 at two distinct points.

Answers

kx2 − 6 = x − 5

kx2 − x − 1 = 0

Since the line intersects the curve at two distinct points, b2 − 4ac . 0.

(− 1)2 − 4 × k × (− 1) . 0

1 + 4k . 0

k . − 1 4

WORKED EXAMPLE 12

Find the values of k for which y = kx − 3 does not intersect the curve y = x 2 − 2x + 1.

Answers

x 2 − 2x + 1 = kx − 3

x 2 − x(2 + k) + 4 = 0

Since the line and curve do not intersect, b2 − 4ac , 0.

(2 + k)2 − 4 × 1 × 4 , 0

k2 + 4k + 4 − 16 , 0

k2 + 4k − 12 , 0

(k + 6) (k − 2) , 0

Critical values are −6 and 2.

So −6 , k , 2.

Exercise 2.6

1 Find the values of k for which y = kx + 1 is a tangent to the curve y = 2x 2 + x + 3.

2 Find the value of k for which the x-axis is a tangent to the curve

y = x 2 + (3 − k)x − (4k + 3).

3 Find the values of the constant c for which the line y = x + c is a tangent to the curve y = 3x + 2 x

4 Find the set of values of k for which the line y = 3x + 1 cuts the curve

y = x 2 + kx + 2 in two distinct points.

5 The line y = 2x + k is a tangent to the curve x 2 + 2xy + 20 = 0.

a Find the possible values of k

b For each of these values of k, find the coordinates of the point of contact of the tangent with the curve.

6 Find the set of values of k for which the line y = k − x cuts the curve y = x 2 − 7x + 4 in two distinct points.

7 Find the values of k for which the line y = kx − 10 intersects the curve x 2 + y 2 = 10x

8 Find the set of values of m for which the line y = mx − 5 does not intersect the curve y = x 2 − 5x + 4.

9 The line y = mx + 6 is a tangent to the curve y = x 2 − 4x + 7. Find the possible values of m

SUMMARY

Completing the square

For a quadratic function f (x) = ax 2 + bx + c that is written in the form f (x) = a(x − h)2 + k, i if a . 0, the minimum point is (h, k) ii if a , 0, the maximum point is (h, k).

The curve cuts the x-axis at 2 distinct points.

curve touches the x-axis at 1 point.

The curve is entirely above or entirely below the x-axis.

Intersection of a quadratic curve and a straight line

Situation 1

Situation 2

Situation 3

2 points of intersection

The line cuts the curve at two distinct points.

Interpreting the discriminant

1 point of intersection

The line touches the curve at one point. This means that the line is a tangent to the curve.

0 points of intersection

The line does not intersect the curve.

Solving simultaneously the equation of the curve with the equation of the line will give a quadratic equation of the form ax 2 + bx + c = 0. The discriminant b2 − 4ac, gives information about the roots of the equation and also about the intersection of the curve with the line.

b2 4ac Nature of rootsLine and curve

. 02 real distinct roots2 distinct points of intersection

= 02 real equal roots1 point of intersection (line is a tangent)

, 0no real roots no points of intersection

The condition for a quadratic equation to have real roots is b2 − 4ac > 0.

Past paper questions

Worked

example

a Express 5x 2 − 14x − 3 in the form p(x + q)2 + r, where p, q and r are constants. [3]

b Sketch the graph of y = | 5x 2 − 14x − 3 | on the axes below. Show clearly any points where your graph meets the coordinate axes. [4]

c State the set of values of k for which | 5x 2 − 14x − 3 | = k has exactly four solutions. [2]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2018 Answers

b First sketch the graph of y = 5x 2 − 14x − 3.

When x = 0, y = − 3.

So the y-intercept is −3. When y = 0,

So, the x-intercepts are − 1 5 and 3.

Using the answer to part i, the minimum point on the curve is

c The values of k for which

| = k has exactly four solutions are 0

Simultaneous equations and quadratics

1 Find the set of values of k for which the line y = k (4x − 3) does not intersect the curve y = 4x 2 + 8x − 8. [5]

Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q4 Jun 2014

2 Find the set of values of x for which x(x + 2) , x [3]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q1 Jun 2014

3 a Express 2x 2 − x + 6 in the form p(x − q)2 + r, where p, q and r are constants to be found. [3]

b Hence state the least value of 2x 2 − x + 6 and the value of x at which this occurs. [2]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q5 Jun 2014

4 Find the range of values of k for which the equation kx2 + k = 8x − 2xk has 2 real distinct roots. [4]

Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q1 Nov 2015

5 a Find the set of values of x for which 4x 2 + 19x − 5 < 0. [3]

b i Express x 2 + 8x − 9 in the form (x + a)2 + b, where a and b are integers. [2]

ii Use your answer to part i to find the greatest value of 9 − 8x − x 2 and the value of x at which this occurs. [2]

iii Sketch the graph of y = 9 − 8x − x2, indicating the coordinates of any points of intersection with the coordinate axes. [2]

Adapted from Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2015

6 The curve 3x 2 + xy − y 2 + 4y − 3 = 0 and the line y = 2(1 − x) intersect at the points A and B.

i Find the coordinates of A and B. [5]

ii Find the equation of the perpendicular bisector of the line AB, giving your answer in the form ax + by = c, where a, b and c are integers. [4]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2017

7 a Write 9x 2 − 12x + 5 in the form p(x − q)2 + r, where p, q and r are constants. [3]

b Hence write down the coordinates of the minimum point of the curve y = 9x 2 − 12x + 5. [1]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q2 Jun 2020

8 The line y = 5x + 6 meets the curve xy = 8 at the points A and B.

a Find the coordinates of A and B. [3]

b Find the coordinates of the point where the perpendicular bisector of the line AB meets the line y = x. [5]

Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q6 Jun 2020

9 Solve the inequality (x − 1) (x − 5) . 12. [4]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q1 Nov 2017

10 Solve the equations y − x = 4

x 2 + y 2 − 8x − 4y − 16 = 0 [5]

Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q1 June 2018

11 Find the values of k for which the line y = kx + 3 is a tangent to the curve y = 2x 2 + 4x + k − 1. [5]

Cambridge IGCSE Additional Mathematics 0606 Paper 12 Q2 Mar 2020

12 Find the values of the constant k for which the equation kx 2 − 3(k + 1)x + 25 = 0 has equal roots. [4]

Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q2 Mar 2021

13 Do not use a calculator in this question. The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B

The mid-point of AB is the point C. Show that the point C lies on the line x + y = 11. [7]

Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q6 Nov 2019

14 a Show that 2x 2 + 5x − 3 can be written in the form a(x + b)2 + c, where a, b and c are constants. [3]

b Hence, write down the coordinates of the stationary point on the curve with equation y = 2x 2 + 5x − 3. [2]

c On the axes below, sketch the graph of y = | 2x 2 + 5x − 3 |, stating the coordinates of the intercepts with the axes.

d Write down the value of k for which the equation | 2x 2 + 5x − 3 | = k has exactly 3 distinct solutions. [1]

Cambridge IGCSE Additional Mathematics 0606 Paper 12 Q4 Mar 2021

15 i Write x 2 − 9x + 8 in the form (x − p)2 − q, where p and q are constants. [2]

ii Hence write down the coordinates of the minimum point on the curve y = x 2 − 9x + 8. [1]

iii On the axes below, sketch the graph of y = | x 2 − 9x + 8 |, showing the coordinates of the points where the curve meets the coordinate axes.

iv Write down the value of k for which | x 2 − 9x + 8 | = k has exactly 3 solutions. [1]

Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q4 Nov 2018

This practice book supports the Cambridge IGCSE TM Additional Mathematics and O Level syllabus (0606). It has been written by a highly experienced author, who is very familiar with the syllabus. The course is aimed at students who are currently studying or have previously studied Cambridge IGCSE TM Mathematics (0580/0980).

The practice book has been written to closely follow the chapters and topics of the coursebook, offering additional exercises to help you to consolidate what you have learnt.

At the start of each chapter, there is a list of learning intentions which tell you what you will learn in the chapter.

Worked examples are used throughout to demonstrate the methods for selected topics using typical workings and thought processes. These present methods to you in a practical and easy-to-follow way

The exercises offer plenty of opportunities for you to practice methods that have just been introduced.

Towards the end of each chapter, there is a summary of the key concepts to help you consolidate what you have learnt. This is followed by a questions section which brings together the methods and concepts from the whole chapter.

A Coursebook is available in the Additional Mathematics series, which includes class discussion activities, worked examples for every method, exercises and a ‘Past paper’ questions section. A digital Teacher’s Resource, to offer support and advice, is available on the Cambridge GO platform.

How to use this book

Throughout this book, you will notice lots of different features that will help your learning. These are explained below.

LEARNING INTENTIONS

These set the scene for each exercise and indicate the important concepts.

KEY WORDS

Definitions for useful vocabulary are given in bold throughout each chapter. You will also find definitions for these words in the Glossary at the back of this book.

Exercises

These help you to practise skills that are important for studying Cambridge IGCSE Mathematics.

There are two types of exercise:

• Exercises which let you practice the mathematical skills you have learned.

• Exercises which bring together all the mathematical concepts in a chapter, pushing your skills further.

WORKED EXAMPLE

Wherever you need to know how to use a formula to carry out a calculation, there are worked examples boxes to show you how to do this.

REMINDER

This feature highlights key concepts from the corresponding chapter in the coursebook.

TIP

This feature contains key equations or formulae that you will need to know.

SUMMARY

There is a summary of key points at the end of each chapter.

Chapter 1: Functions

LEARNING INTENTIONS

This section will show you how to

• understand and use the terms: function, domain, range (image set), one-one function, inverse function and composition of functions

• understand the relationship between

• solve graphically or algebraically equations of the type

• explain in words why a given function is a function or why it does not have an inverse

• find the inverse of a one-one function and form composite functions

• sketch graphs to show the relationship between a function and its inverse.

1.1 Mappings REMINDER

The table below shows one-one, many-one and one-many mappings.

For one input value there is just one output value.

Exercise 1.1

For two input values there is one output value.

For one input value there are two output values.

Determine whether each of these mappings is one-one, many-one or one-many.

1.2 Definition of a function

REMINDER

A function is a rule that maps each x value to just one y value for a defined set of input values.

This means that mappings that are either ⎧ ⎨ ⎩ one-one many-one are called functions.

The mapping x ↦ x + 1, where x ∈ ℝ, is a one-one function.

The function can be defined as f : x ↦ x + 1, x ∈ ℝ or f (x) = x + 1, x ∈ ℝ

The set of input values for a function is called the domain of the function. The set of output values for a function is called the range (or image set) of the function.

WORKED EXAMPLE 1

The function f is defined by f (x) = (x − 1)2 + 4, for 0 < x < 5. Find the range of f.

Answers

f (x) = (x − 1)2 + 4 is a positive quadratic function so the graph will be of the form

(x 1)2 + 4

This part of the expression is a square so it will always be > 0. The smallest value it can be is 0. This occurs when x = 1.

The minimum value of the expression is 0 + 4 = 4 and this minimum occurs when x = 1.

So the function f (x) = (x − 1)2 + 4 will have a minimum at the point (1, 4).

When x = 0, y = (0 − 1)2 + 4 = 5.

When x = 5, y = (5 − 1)2 + 4 = 20.

The range is 1 < f (x) <

Exercise 1.2

1 Which of the mappings in Exercise 1.1 are functions?

2 Find the range for each of these functions.

3 The function g is defined as g (x) = x 2 − 5 for x > 0. Find the range of g.

4 The function f is defined by f (x) = 4 − x 2 for x ∈ ℝ. Find the range of f.

5

function

is defined by

8 The function f is defined by f (

9 Find the largest possible domain for the following functions.

1.3 Composite functions REMINDER

• When one function is followed by another function, the resulting function is called a composite function.

• fg (x) means the function g acts on x first, then f acts on the result.

• f 2(x) means ff (x), so you apply the function f twice.

WORKED EXAMPLE 2 f : x ↦ 4x + 3, for x ∈ ℝ

(3).

Answers

fg (3) = f (2 × 32 − 5) g acts on 3 first and g (3) = 2 × 32 − 5 = 13. = f (13) = 4 × 13 + 3 = 55

WORKED EXAMPLE 3

g (x) = 2x 2 − 2, for x ∈ ℝ

h (x) = 4 − 3x, for x ∈ ℝ

Solve the equation hg (x) = −14.

Answers

hg (x) = h (2x 2 − 2) g acts on x first and g (x) = 2x 2 − 2. = 4 − 3(2x 2 − 2) h is the function ‘triple and take from 4’ = 4 − 6x 2 + 6 expand the brackets = 10 − 6x 2

hg (x) = − 14

14 = 10 − 6x 2 set up and solve the equation.

24 = 6x 2 4 = x 2 x = ± 2

Exercise 1.3

1 f (x) = 2 − x2, for x ∈ ℝ

g (x) = x 2 + 3, for x ∈ ℝ

Find the value of gf (4).

2 f (x) = (x − 2)2 − 2, for x ∈ ℝ

Find f 2(3).

3 The function f is defined by f (x) = 1 + √ x − 3 , for x > 3.

The function g is defined by g (x) = 3 x − 1, for x . 0.

Find gf (7).

4 The function f is defined by f (x) = (x − 2)2 + 3, for x . −2.

The function g is defined by g (x) = 3x + 4 x + 2 , for x . 2.

Find fg (6).

5 f : x ↦ 3x − 1, for x . 0

g : x ↦ √ x , for x . 0

Express each of the following in terms of f and g.

a x ↦ 3√ x − 1 b x ↦ √ 3x − 1

6 The function f is defined by f : x ↦ 2x − 1, for x ∈ ℝ.

The function g is defined by g : x ↦ 8 4 − x , for x ≠ 4.

Solve the equation gf (x) = 5.

7 f (x) = 2x 2 + 3, for x . 0

g (x) = 5 x , for x . 0

Solve the equation fg (x) = 4.

8 The function f is defined by f

Solve the equation fg (x) = 4.

9 The function g is defined by g (

for

1 for x > 0. Solve the equation gh (x) = −3, giving your answer(s) as exact value(s).

function h is defined by h (x) = 3x

10 The function f is defined by f : x ↦ x2, for x ∈ ℝ The function g is defined by g : x

each of the following as a composite function, using only f and g.

a Find the domain and range of g.

b Solve the equation g (x) = 0.

c Find the domain and range of fg.

1.4 Modulus functions REMINDER

• The modulus (or absolute value) of a number is the magnitude of the number without a sign attached.

• The modulus of x, written as | x |, is defined as

• The statement | x | = k, where k > 0, means that x = k or x = −k

Before writing your final answers, compare your solutions with the domains of the original functions.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK

WORKED EXAMPLE 4

giving your answers as exact values if appropriate.

Remember to check your answers to make sure that they satisfy the original equation.

1.5 Graphs of y = |f (x) | where f (x) is linear

Exercise 1.5

1 Sketch the graphs of each of the following functions, showing the coordinates of the points where the graph meets the axes.

a y = | x − 2 | b y = | 3x − 3 | c y = | 3 − x |

d y = | 1 3 x − 3 | e y = | 6 − 3x | f y = | 5 − 1 2 x |

2 a Complete the table of values for y = 3 − | x − 1 | . x −2−101234 y 1 3

b Draw the graph of y = 3 − | x − 1 |, for −2 < x < 4.

3 Draw the graphs of each of the following functions.

a y = | 2x | + 2 b y = | x | − 2 c y = 4 − | 3x|

d y = | x − 1 | + 3 e y = | 3x − 6 | − 2 f y = 4 − | 1 2 x |

4 Given that each of these functions is defined for the domain −3 < x < 4, find the range of

a f : x ↦ 6 − 3x b g : x ↦ | 6 − 3x |

5 Find the range of each function for −1 < x < 5.

a f : x ↦ 2 − 2x b g : x ↦ |

− 2x | c h : x ↦ 2 − | 2x |

6 a Sketch the graph of y = | 3x − 2 | for −4 < x < 4, showing the coordinates of the points where the graph meets the axes.

b On the same diagram, sketch the graph of y = x + 3.

c Solve the equation | 3x − 2 | = x + 3.

7 A function f is defined by f (x) = 2 − | 3x − 1 |, for −1 < x < 3.

a Sketch the graph of y = f (x).

b State the range of f.

c Solve the equation f (x) = −2.

8 a On a single diagram, sketch the graphs of x + 3y = 6 and y = | x + 2 |.

b Solve the inequality | x + 2 | , 1 3 (6 − x).

1.6 Inverse functions

REMINDER

• The inverse of the function f (x) is written as f −1(x).

• The domain of f −1(x) is the range of f (x).

• The range of f −1(x) is the domain of f (x).

• It is important to remember that not every function has an inverse.

• An inverse function f −1(x) can exist if, and only if, the function f (x) is a one-one mapping.

WORKED EXAMPLE 5

f (x) = (x + 3)2 −1, for x . −3

a Find an expression for f −1(x).

b Solve the equation f −1(x) = 3.

Answers

a f (x) = (x + 3)2 −1, for x . −3

Step 1: Write the function as y = … y = (x + 3)2 − 1

Step 2: Interchange the x and y variables. x = (y + 3)2 − 1

Step 3: Rearrange to make y the subject. x + 1 = (y + 3)2

√ x + 1 = y + 3 y = √ x + 1 − 3

f −1(x) = √ x + 1 − 3

b f −1(x) = 3. √ x + 1 − 3 = 3

√ x + 1 = 6 x + 1 = 36 x = 35

Exercise 1.6

1 f (x) = (x + 2)2 − 3, for x > −2.

Find an expression for f −1(x).

2 f (x) = 5 x − 2 , for x > 0.

Find an expression for f −1(x).

3 f (x) = (3x − 2)2 + 3, for x > 2 3

Find an expression for f −1(x).

4 f (x) = 4 − √ x − 2 , for x > 2. Find an expression for f −1(x).

5 f : x ↦ 3x − 4, for x . 0.

g : x ↦ 4 4 − x , for x ≠ 4. Express f −1(x) and g−1(x) in terms of x.

6 f (x) = (x − 2)2 + 3, for x . 2.

a Find an expression for f −1(x). b Solve the equation f −1(x) = f (4).

7 g (x) = 3x + 1 x − 3 , for x . 3

a Find an expression for g−1(x) and comment on your result.

b Solve the equation g−1(x) = 6.

8 f (x) = x 2 − 2, for x ∈ ℝ g (x) = x 2 − 4x, for x ∈ ℝ

a Find f −1(x).

b Solve fg (x) = f −1(x), leaving answers as exact values.

9 f : x ↦ 3x + 1 x − 1 , for x ≠ 1 g : x ↦ x − 2 3 , for x . −2

Solve the equation f (x) = g−1(x).

10 If f (x) = x 2 − 9 x 2 + 4 , x ∈ ℝ, find an expression for f −1(x).

11 If f (x) = 2√ x and g (x) = 5x, solve the equation f −1 g (x) = 0.01.

12 Find the value of the constant k such that f (x) = 2x − 4 x + k is a self-inverse function.

13 The function f is defined by f (x) = x3. Find an expression for g (x) in ter ms of x for each of the following:

a fg (x) = 3x + 2 b gf (x) = 3x + 2

14 Given that f (x) = 2x + 1 and g (x) = x + 1 2 , find the following.

a f −1 b g −1 c (fg)−1 d (gf )−1 e f −1 g −1 f g−1f −1

Write down any observations from your results.

15 Given that fg (x) = x + 2 3 and g (x) = 2x + 5, find f (x).

16 Functions f and g are defined for all real numbers.

g (x) = x 2 + 7 and gf (x) = 9x 2 + 6x + 8.

Find f (x).

TIP

A self-inverse function is one for which f (x) = f −1(x), for all values of x in the domain.

1.7 The graph of a function and its inverse

REMINDER

The graphs of f and f −1 are reflections of each other in the line y = x.

This is true for all one-one functions and their inverse functions.

This is because: ff −1(x) = x = f −1f (x)

Some functions are called self-inverse functions because f and its inverse f −1 are the same.

If f (x) = 1 x , for x ≠ 0, then f −1(x) = 1 x , for x ≠ 0

So f (x) = 1 x , for x ≠ 0, is an example of a self-inverse function.

When a function f is self-inverse, the graph of f will be symmetrical about the line y = x.

Exercise 1.7

1 On a copy of the grid, draw the graph of the inverse of the function y = 2 x

2 f (x) = x 2 + 5, x > 0.

On the same axes, sketch the graphs of y = f (x) and y = f −1(x), showing the coordinates of any points where the curves meet the coordinate axes.

3 g (x) = 1 2 x 2 − 4, for x > 0.

Sketch, on a single diagram, the graphs of y = g (x) and y = g−1(x), showing the coordinates of any points where the curves meet the coordinate axes.

4 The function f is defined by f (x) = 3x − 6 for all real values of x.

a Find the inverse function f −1(x).

b Sketch the graphs of f (x) and f −1(x) on the same axes.

c Write down the point of intersection of the graphs f (x) and f −1(x).

5 The function f is defined as: f (x) = x 2 − 2x, for x > 1.

a Explain why f −1(x) exists and find f −1(x).

b State the range of the function f −1(x).

c Sketch the graphs of f (x) and f −1(x) on the same axes.

d Write down where f −1(x) crosses the y-axis.

6 a By finding f −1(x), show that f (x) = 3x − 1 2x − 3 , x ∈ ℝ, x ≠ 3 2 , is a self-inverse function.

b Sketch the graphs of f (x) and f −1(x) on the same axes.

c Write down the coordinates of the intersection of the graphs with the coordinate axes.

SUMMARY

Functions

A function is a rule that maps each x-value to just one y-value for a defined set of input values. Mappings that are either one-one or many-one are called functions. The set of input values for a function is called the domain of the function. The set of output values for a function is called the range (or image set) of the function.

Modulus function

The modulus of x, written as | x |, is defined as

x if x . 0

| x | = 0 if x = 0

x if x < 0

Composite functions

fg (x) means the function g acts on x first, then f acts on the result.

f 2(x) means ff (x).

Inverse functions

The inverse of a function f (x) is the function that undoes what f (x) has done.

The inverse of the function f (x) is written as f −1(x).

The domain of f −1(x) is the range of f (x).

The range of f −1(x) is the domain of f (x).

An inverse function f −1(x) can exist if, and only if, the function f (x) is a one-one mapping.

The graphs of f and f −1 are reflections of each other in the line y = x.

Exercise 1.8

1 A one-one function f is defined by f (x) = (x − 2)2 − 3, for x > k.

a State the least value that x can take.

b For this least value of k, write down the range of f.

2 The function f (x) = x 2 − 4ax (where a is a positive constant) is defined for all real values of x.

Given that the range is > −8, find the exact value of a

3 f (x) = (2x − 1)2 + 3, for x . 0

g (x) = 5 2x , for x . 0

Solve the equation fg (x) = 7.

4 The function f is defined by f (x) = 1 − x2, for x ∈ ℝ

The function g is defined by g (x) = 2x − 1, for x ∈ ℝ

Find the values of x (in exact form) which solve the equation fg (x) = gf (x).

5 Solve these simultaneous equations.

y = 2x + 5

y = | 3 − x2|

6 a Sketch the graph of y = | 2x + 1 | for the domain −3 , x , 3, showing the coordinates of the points where the graph meets the axes.

b On the same diagram, sketch the graph of y = 3x

c Solve the equation 3x = | 2x + 1 |

7 a Sketch the graph of y = | x + 3 |.

b Solve the inequality | x + 3 | . 2x + 1.

8 f (x) = x 2 − 3, for x ∈ ℝ

g (x) = 3x + 2, for x ∈ ℝ

Solve the equation gf (x) = g−1(8).

9 f (x) = 2x + 3 and g (x) = 1 x + 1 , x ∈ ℝ, x ≠ 1.

a Find an expression for the inverse function f −1(x).

b Find an expression for the composite function gf (x).

c Solve the equation f −1(x) = gf (x) − 1.

10 The function f is defined as: f (x) = 2x + 1 x + 2 , x ≠ − 2

a Find f −1(x).

b Find the points of intersection of the graphs of f (x) and f −1(x).

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Assessment for learning

Developing language skills

Differentiation

Language awareness

Metacognition

Skills for life

Glossary

Coursebook answers

Practice Book answers

IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Introduction

Welcome to the third edition of our very popular Cambridge IGCSE™ Additional Mathematics series.

This new series has been designed around extensive research interviews and lesson observations with teachers and students around the world following the course. As a result of this research, some changes have been made to the new series, with the aim of solving and supporting your biggest classroom challenges and developing your students’ passion and excitement for Mathematics

As well as targeted support in the Coursebook, we have produced an updated Practice book, with exercises for each topic to provide more opportunities for students to consolidate their learning and develop their knowledge application skills.  We are introducing a Worked Solutions Manual to provide additional support for teachers and students to work through selected Coursebook questions.

As we develop new resources, we ensure that we are keeping up-to-date with best practice in pedagogies. For this new series we have added new features to the Coursebook, such as engaging projects to develop students’ collaborative skills and ‘pre-requisite knowledge’ guides to unlock students’ prior learning and help you to evaluate students’ learning starting points.

Finally, we have updated this Teacher’s Resource to make it as useful and relevant as possible to your dayto-day teaching needs. From teaching activity, assessment and homework ideas, to how to tackle common misconceptions in each topic, to a new feature developing your own teaching skills, we hope that this handy resource will inspire you, support you and save you much-needed time.

We hope that you enjoy using this series and that it helps you to continue to inspire and excite your students about this vital and ever-changing subject. Please don’t hesitate to get in touch if you have any questions for us, as your views are essential for us to keep producing resources that meet your classroom needs.

Thomas Carter Head of Mathematics, Cambridge University Press

CAMBRIDGE

TEACHER’S RESOURCE

About the authors

Julianne Hughes

Julianne Hughes has a first class honours degree in Pure Mathematics (Cardiff University 1991) and is qualified to teach mathematics in secondary and further education in the UK. She has been a teacher, tutor, mathematics consultant, author and resource creator since 1996. Julianne is now retired from teaching and tutoring and her main focus is resource creation, authoring materials and assessing.

How to use this Teacher’s Resource

This Teacher’s Resource contains both general guidance and teaching notes that help you to deliver the content in our Cambridge resources.

There are teaching notes for each unit of the Coursebook. Each set of teaching notes contains the following features to help you deliver the chapter.

At the start of each chapter there is a teaching plan (Figure 1). This summarises the topics covered in the chapter, including the number of learning hours recommended for each topic, an outline of the learning content, and the Cambridge resources from this series that can be used to deliver the topic.

Topic Order in chapter Learning content Resources

Each chapter also includes information on any background knowledge that students should have before studying this chapter, advice on helpful language support and a selection of useful links to digital resources

At the beginning of the teaching notes for the individual sections there is an outline of the learning objectives (Figure 2) for that section, as well as any common misconceptions that students may have about the topic and how you can overcome these.

Syllabus learning objectives / learning intentions Success criteria

For each section, there is a selection of starter ideas, main activities and plenary ideas. You can pick out individual ideas and mix and match them depending on the needs of your class. The activities include suggestions for how they can be differentiated or used for assessment.

Homework ideas give suggestions for tasks, along with advice for how to assess students’ work.

You will find answers to the Coursebook and Workbook questions and exercises at the end of each chapter in this Teacher’s Resource and answers to the Practical Workbook questions at the end of this resource.

This Teacher’s Resource also includes a set of PowerPoint presentations which include the worked examples from each chapter, plus some extra material, explanations and definitions. Every PowerPoint slide has additional explanatory notes and observations which are designed to help support your teaching.

Downloadable resources include differentiated Worksheets for each chapter, a sample lesson plan for each chapter which demonstrate how certain elements within each topic may be approached, and additional printed resource sheets.

How to use this Teacher’s Resource to supplement PD

We regularly hear from teachers that the Continuous Professional Development (CPD) they feel they get the most out of is face-to-face training. However, we also hear that not all teachers have the time or budget to get out of the classroom, so here’s some handy suggestions and information about how to use this teacher’s resource for your own professional development. After all, we are all lifelong learners!

Approaches to teaching and learning

Our teacher resources now contain guidance on the key pedagogies underpinning our course content and how we understand and define them. You can find detailed information for you to read in your own time about active learning, assessment for learning, metacognition, differentiation, language awareness and skills for life taken from our ‘Approaches to Learning and Teaching’ series.

Teaching activity ideas

This Teacher’s Resource contains a range of starter, main and plenary activity ideas for you to try in your classroom. Use them to support your creativity, breathe new life into a topic and build upon them with your own ideas.

How to use this Teacher’s Resource as CPD

We regularly hear from teachers that the Continuous Professional Development (CPD) they feel they get the most out of is face-to-face training. However, we also hear that not all teachers have the time or budget to get out of the classroom, so here’s some handy suggestions and information about how to use this teacher’s resource for your own professional development. After all, we are all lifelong learners!

Teaching skills focus

We have created a new ‘Teaching skills focus’ feature that appears once every chapter, covering a different teaching skill with suggestions of how you can implement it in the teaching of the topic. From differentiation, to assessment for learning, to metacognition, this feature aims to support you with trying out a new technique or approach in your classroom and reflecting upon your own practices.

Try it out once per teaching topic, or when you have time, and develop your skills in a supported and contextualised way.

Approaches to learning and teaching

Our teacher resources now contain guidance on the key pedagogies underpinning our course content and how we understand and define them. You can find detailed information for you to read in your own time about active learning, assessment for learning, metacognition, differentiation, language awareness and skills for life taken from our ‘Approaches to learning and teaching’ series.

Why not try reading each support document alongside the relevant Teaching skills focus for an extra bit of bedtime reading?

Teaching activity ideas

This teacher's resource provides plenty of engaging teaching ideas - from suggestions for starters, mains and plenaries to NRICH project guidance. You can choose what works best for your learners.

We want to include you in the Cambridge community of teachers. In this new resource, we have utilised up-todate pedagogy and our research in schools to cater to teachers and learners. Our authors are skilled teachers and we hope you enjoy their suggestions for activities to engage your learners.

Approaches to learning and teaching

The following are the pedagogical practices underpinning our course content and how we understand and define them.

Active learning

Active learning is a pedagogical practice that places student learning at its centre. It focuses on how students learn, not just on what they learn. We, as teachers, need to encourage students to ‘think hard’, rather than passively receive information. Active learning encourages students to take responsibility for their learning and supports them in becoming independent and confident students in school and beyond.

Assessment for Learning

Assessment for Learning (AfL) is a pedagogical practice that generates feedback which can be used to improve students’ performance. Students become more involved in the learning process and, from this, gain confidence in what they are expected to learn and to what standard. We, as teachers, gain insights into a student’s level of understanding of a particular concept or topic, which helps to inform how we support their progression.

Differentiation

Differentiation is usually presented as a pedagogical practice where teachers think of students as individuals and learning as a personalised process. Whilst precise definitions can vary, typically the core aim of differentiation is viewed as ensuring that all students, no matter their ability, interest or context, make progress towards their learning intentions. It is about using different approaches and appreciating the differences in students to help them make progress. Teachers therefore need to be responsive, and willing and able to adapt their teaching to meet the needs of their students.

Language awareness

For many students, English is an additional language. It might be their second or perhaps their third language. Depending on the school context, students might be learning all or just some of their subjects through English. For all students, regardless of whether they are learning through their first language or an additional language, language is a vehicle for learning. It is through language that students access the learning intentions of the lesson and communicate their ideas. It is our responsibility, as teachers, to ensure that language doesn’t present a barrier to learning.

Metacognition

Metacognition describes the processes involved when students plan, monitor, evaluate and make changes to their own learning behaviours. These processes help students to think about their own learning more explicitly and ensure that they are able to meet a learning goal that they have identified themselves or that we, as teachers, have set.

Skills for Life

How do we prepare students to succeed in a fast-changing world? To collaborate with people from around the globe? To create innovation as technology increasingly takes over routine work? To use advanced thinking skills in the face of more complex challenges? To show resilience in the face of constant change? At Cambridge, we are responding to educators who have asked for a way to understand how all these different approaches to life skills and competencies relate to their teaching. We have grouped these skills into six main Areas of Competency that can be incorporated into teaching, and have examined the different stages of the learning journey and how these competencies vary across each stage.

These six areas are:

• Creativity – finding new ways of doing things, and solutions to problems

• Collaboration – the ability to work well with others

• Communication – speaking and presenting confidently and participating effectively in meetings

• Critical thinking – evaluating what is heard or read, and linking ideas constructively

• Learning to learn – developing the skills to learn more effectively

• Social responsibilities – contributing to social groups, and being able to talk to and work with people from other cultures.

Cambridge learner and teacher attributes

This course helps develop the following Cambridge learner and teacher attributes.

Cambridge learners

Confident in working with information and ideas –their own and those of others.

Responsible for themselves, responsive to and respectful of others.

Reflective as learners, developing their ability to learn.

Innovative and equipped for new and future challenges.

Engaged intellectually and socially, ready to make a difference.

Cambridge teachers

Confident in teaching their subject and engaging each student in learning.

Responsible for themselves, responsive to and respectful of others.

Reflective as learners themselves, developing their practice.

Innovative and equipped for new and future challenges.

Engaged intellectually, professionally and socially, ready to make a difference.

Reproduced from Developing the Cambridge learner attributes with permission from Cambridge Assessment International Education.

2

Simultaneous equations and quadratics

Scheme of work

Topic Order in chapter Learning content

Solving quadratic equations for real roots

FirstSolve quadratic equations for real roots by factorising, formula, completing the square.

Resources

Coursebook:

Sections 2.2 and 2.5

PowerPoints:

2 recap b Factorising and quadratic formula

2.2b Completing the square recap

2.2c The parabola and quadratic function forms

Applications of completing the square

After solving quadratic equations

Find the maximum or minimum value of the quadratic function f : x ↦ ax2 + bx + c by completing the square.

Use the maximum or minimum value of f(x) = ax2 + bx + c to sketch the graph or determine the range for a given domain.

Coursebook: Section 2.2

PowerPoints:

2.2a Worked examples 2 & 4

Modulus functions

After solving quadratic equations; could be studied later in the course

Understand the relationship between y = f(x) and y = |f(x)|, where f(x) is quadratic.

Coursebook: Section 2.3

PowerPoints:

2.3a Modulus of a quadratic including Worked example 5

Solving quadratic inequalities

After solving quadratic equations

Find the solution set for quadratic inequalities. Coursebook: Section 2.4

PowerPoints:

2.4 Worked examples 6 & 7

Pdf files:

Chapter 2 Teacher notes, class discussion section 2.4

This resource is printable and/or editable

Chapter 2 Lesson Plan

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Solving simultaneous equations

After solving quadratic equations

Solve simple simultaneous equations in two unknowns, with one linear, by elimination or substitution.

Coursebook: Section 2.1

PowerPoints:

2 recap a Solving two linear simultaneous equations

2.1a Solving two simultaneous equations with one linear including Worked example 1

2.1b Solving two simultaneous equations both non linear

Nature of roots and intersections of lines and curves

After solving quadratics and inequalities and simultaneous equations

Learning plan

Syllabus

Know the conditions for ax2 + bx + c = 0 to have: (i) two real roots, (ii) two equal roots, (iii) no real roots

and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.

Coursebook:

Sections 2.5 and 2.6

PowerPoints:

2.3b Roots and intersections

2.5a Worked examples 8 to 12

2.5b Connecting the nature of roots with intersections of graphs

2.6 Simultaneous equations and quadratics further practice

Solve quadratic equations for real roots and find the solution set for quadratic inequalities. Make a simple sketch of the graph of a quadratic function using any roots and the y-intercept.

Students are able to complete the square for expressions of the form ax2 + bx + c where a is positive or negative and can interpret the results correctly.

Students can solve quadratic equations using an appropriate method for the problem being considered. They can use this information to make a sketch of the graph of the quadratic function. They understand how to use this skill to find the critical values needed to solve quadratic inequalities. They are also able to write the solution set for quadratic inequalities in the correct form. Find the maximum or minimum value of the quadratic function f : x ↦ ax2 + bx + c by completing the square.

Students can apply the methods of finding roots and completing the square to sketching graphs and to finding domains and ranges of quadratic functions.

Understand the relationship between y = f (x) and y = | f (x) |, where f (x) is quadratic.

Solve simple simultaneous equations in two unknowns, with at least one linear, by elimination or substitution.

Students can apply the methods of finding roots and completing the square to sketching graphs and to finding domains and ranges of quadratic functions.

Students can successfully apply the method of finding roots and sketch or draw accurately the graph of y = | ax2 + bx + c |. They can also use this to solve simple problems.

Students choose an appropriate method of solution and show the method of solution in full. They are able to understand that two lines can only intersect once and how the number of points of intersection changes when one of the equations is not linear.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Syllabus learning objectives / learning intentions Success criteria

Know the conditions for ax2 + bx + c = 0 to have:

(i) two real roots, (ii) two equal roots, (iii) no real roots and the related conditions for a given line to:

(i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.

BACKGROUND KNOWLEDGE

Students understand the relevance of the discriminant and are able to apply knowledge of the appropriate condition to solve simple algebraic problems. Students are able to combine all the necessary skills to solve simultaneous equations and connect the conditions for the nature of the roots of a quadratic equation to determine how a line intersects with a curve.

• The following table details what knowledge it is assumed that students already have from studying Cambridge IGCSE or O Level Mathematics. In Additional Mathematics, it is expected that students will be able to use these skills as part of a solution in a multi-step process, and the interpretation needed to do this should be of a greater challenge than that generally expected in the mathematics course.

What your students should be able to doExamples

Solve simultaneous equations using the elimination method.

Use the elimination method to solve these simultaneous equations.

a 4x + 3y = 1; 2x – 3y = 14

b 3x + 2y = 19; x + 2y = 13

Solve simultaneous equations using the substitution method.

Use the substitution method to solve these simultaneous equations.

a y = 3x – 10; x + y = –2

b x + 2y = 11; 4y – x = –2

Solve quadratic equations using the factorisation method.

Factorise and solve these equations.

a x2 + x – 6 = 0

b x2 – 10x + 16 = 0

c 6x2 + 11x – 10 = 0

Solve quadratic equations by completing the square.

Solve quadratic equations using the quadratic formula.

a Write 2x2 + 7x + 3 in the form a ( x + b ) 2 + c

b Use your answer to part a to solve the equation 2x2 + 7x + 3 = 0.

Solve 2x2 – 9x + 8 = 0.

Give your answers correct to 2 decimal places.

• The work in this chapter is essential to the whole course. The skill of solving quadratic equations or of factorising a quadratic expression is required in several other syllabus areas. It is highly recommended that this chapter is covered as soon as possible in the course. The skill of solving a pair of simultaneous equations also appears in other syllabus areas. For example, the work on the straight line, in equations, inequalities and graphs and in sequences and series. Some of the questions in this chapter require the use of skills that are considered in Chapter 6, Straight-line graphs. It may be sensible, therefore, to have looked at this chapter first or to work on them in sections, together.

BACKGROUND KNOWLEDGE

• This chapter starts with solving pairs of simultaneous linear equations by elimination and substitution. Solving quadratic equations is briefly recapped before solving pairs of equations in which only one of the equations is linear is studied. Quadratic expressions and functions are then considered more fully, including the shape of the graphs, maximum and minimum values, symmetry and modulus of the quadratic function. This is all essential to what comes after, that is, the solution of quadratic inequalities and using the discriminant to study the nature of the roots of quadratic equations and the points of intersections of graphs.

LANGUAGE SUPPORT

The definitions of key words and phrases are given in the glossary.

When considering the nature of the roots of quadratic equations it is important to model the correct language for the possible cases. These cases are:

• two roots that are real and distinct (sometimes written as real and different)

• two roots that are equal (sometimes written as real and equal or repeated)

• two roots that are real (this includes those that are real and distinct and real and equal)

• no real roots.

Model this language and these ideas for students as much as possible so that the interpretation needed to be successful is instinctive for them.

Links to Digital Resources

In worked examples 3 and 4, the completing of the square is done using the algebraic structure which has been given and then forming and solving equations. The language used in the worked examples is such that this method is fine, as students are simply required to write down the correct form or find the values of the constants given. However, if students need to show that a quadratic expression has a particular completed square form, then they should not form and solve equations in this way. Students should understand that using what you are trying to show as part of your solution is invalid. They should derive the correct completed square form using an approach similar to that used in the Coursebook prior to worked example 3 or as demonstrated in PowerPoint 2.2b. This is a key and important difference in the language used.

• WolframAlpha has a systems of equations solver and some step-by-step example solutions

• Purplemath has examples for solving systems of non-linear equations by considering graphs

• There are many useful videos on quadratics to be found at The Khan Academy.

• Maths is Fun has some real-world examples of quadratic equations

REAL-LIFE CONTEXT

Simultaneous equations can be used to represent and solve a variety of everyday problems in the real world. For example, deciding whether one mobile phone deal is better value than another, or finding the maximum profit available from making and selling goods. They are used in many applications in the study of various sciences and are an essential tool for any student of science or engineering, for example.

Quadratic equations are everywhere. They are used in business and finance, physics, architecture and the natural world, not just

Common Misconceptions and Issues

algebra classes. To engage your students in this topic, it may be helpful to start this section by looking at the sort of real-world situations that can be modelled by quadratic equations. The properties of the parabola give us satellite dishes and car headlamps, for example. The Sydney Opera House is distinctly parabolic in appearance. The motion of a pebble thrown up in the air and falling to the ground is also parabolic. Many features in design and modelling require the skills that are introduced in this syllabus. Knowing this may help students understand the importance of what they are studying.

Students are expected to have developed proficient algebraic methods of solving equations and inequalities. Graphs support the learning and help understanding, but algebra is the main key to a successful, efficient and, most importantly, accurate solution.

Misconception/issue How to identify How to avoid or overcome

Students are too dependent on their calculator to solve equations and do not demonstrate that they have mastered the techniques in the syllabus.

For example, students often find factors by first using their calculator to find roots and then working back.

Students often need to make a sketch of a function. Students need to be clear that drawing a sketch is not the same as drawing an accurate graph.

Students commonly reach for their calculator to solve quadratic equations. Calculators are an excellent checking tool, but no substitute for showing proper method.

Very often, factors such as 2x − 1 are written as x − 0.5, which is incorrect.

Students who are not sure about sketching often plot points and join them together. This can result in some very poor graph shapes.

Some icons have been used in the PowerPoint presentations to try to indicate when it is a useful time to check your working with a calculator, to allow you to emphasise this with your students.

Students should be clear that when a sketch is needed, the command word in the question will be sketch.

To draw a good sketch, it should be approximately correctly positioned. Any key points, such as intercepts or turning points should be marked if possible. All key features of the curve should be present.

Students should also be clear that when an accurate drawing is expected, the command word is likely to be draw.

This is likely should the graph then be used to solve an equation or inequality, for example.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Misconception/issue How to identify How to avoid or overcome

When considering whether a quadratic equation has real roots, students sometimes only consider either real and equal or real and different, but not both.

As with chapter 1, thinking that, for y = | f (x) |, the values of x cannot be negative.

Starter ideas

1 Alpha beta starter

Students will form a quadratic equation or use an incorrect inequality sign when forming an inequality.

This can be checked using PowerPoint 2.6.

This is very common when solving equations. Should a value of x be negative, students often think it should be rejected and will indicate this in their working.

Make sure you model the correct language when considering the nature of roots and ensure that students are experienced in using b2 − 4ac > 0 when real roots are specifically required.

This can be resolved by working on the graphs of absolute value functions so that students can clearly see that x can have negative values but that y cannot and then linking the graphs back to the equations they are solving.

Description and purpose: This is a good starter for any lesson involving the use of factorising quadratic expressions. It makes students think about products and sums of numbers, the need for which is relatively clear.

Resources:

• PowerPoint 2 starter: Alpha beta

• Pens and paper

Activity:

There are five questions.

Each question asks for a pair of numbers, alpha and beta, that have a given sum and product. A timer appears on the screen and runs for 1 minute (30 seconds green and 30 seconds blue). Students have this time to write down their answers. As soon as they have done this, they put up their hand.

If all hands are up before the timer runs out, click to reveal the answer. If not, the answer will appear once the time is up. Click to move to the next question.

This activity could possibly lead into: any activity that was dependent on factorising quadratics as a tool. This activity could be adapted: The numbers in each question can be changed if you wish to use the starter as a review as well as a starter – or if you wish to use it again with the same group for a different lesson.

2 What’s my equation?

Description and purpose: This activity can be used to recap the work on modulus functions when f(x) is linear, covered in Chapter 1, in readiness for extending to quadratic functions.

Resources:

• Geogebra or Desmos or other free graphing software.

Activity:

Present the class with the graph of a modulus function and ask them for the possible equations. To keep this as a short starter, limit the number of functions to one or two (or at the very most, three).

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Possible functions to draw are

To type these into Geogebra or Desmos, for example, enter y

This activity could possibly lead into: the study of modulus functions whose initial form was quadratic or revision of solving modulus equations.

This activity could be adapted: If technology is not available, draw the graph or graphs on a flip chart or a display board. Also, the exercise could be extended to include simple quadratic functions such as y = x2 , which could be a challenge.

Main teaching ideas

This topic could be taught with or without a calculator.

Much of the simultaneous equation solving students meet in this course requires them to be able to solve a quadratic equation to be able to complete the task. This is why these syllabus areas have been grouped into one chapter. You may choose to start with this, as the Coursebook does, and build on skills your students should already have. Alternatively, you could start with a recap of solving quadratic equations and build on that. Either approach is well supported by the Coursebook and teacher resources.

Students may have a good understanding of the methods used to solve quadratic equations. It is a good idea to make sure of this before progressing through the rest of the material in the chapter. Students need a good foundation on which to build their skills. Some of these ideas will last for more than one lesson. All the suggestions made have assessment for learning activities embedded within them.

1 Quadratic equations and the parabola

Learning intention:

• Solve quadratic equations for real roots.

• Make a simple sketch of the graph of a quadratic function using any roots and the y-intercept.

• Find the maximum or minimum value of the quadratic function f : x ↦ ax 2 + bx + c by completing the square.

• Use the maximum or minimum value of f (x) = ax 2 + bx + c to sketch the graph or determine the range for a given domain.

Resources:

• PowerPoint 2 recap b: Factorising and quadratic formula

• PowerPoint 2.2a: Worked examples 2 and 4

• PowerPoint 2.2b: Completing the square recap

• PowerPoint 2.2c: The parabola and quadratic function forms

• Coursebook Exercise 2.2

Description and purpose: Solving quadratic equations, which is an essential skill for the solution of the simultaneous equations, has been split into two sections. In PowerPoint 2 recap b the methods of factorising and using the quadratic formula are revised. Factorising is demonstrated using a reverse grid approach. This method reduces the amount of purely mental processing and allows the visual to help with thinking. Students will be able to ‘see’ the factors of the first and last terms in place in the grid. These skills should be sufficient for the work on solving simultaneous equations, which could then be studied if you wish. Before moving on to applications of solving quadratic equations, it is sensible for students to investigate the possible shapes of the graphs. This, as well as the different ways of presenting a quadratic function (vertex form, standard form, factorised form) are looked at in PowerPoint 2.2c.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

The method of completing the square is required as a skill in its own right, as well as being a useful method of solving equations. It also is useful for drawing graphs and for finding least and greatest values of functions, for example. A slightly different method to those described in the Coursebook is demonstrated in PowerPoint 2.2b which recaps the method of solving quadratic equations by completing the square using the first two terms of a 3-term quadratic expression only. The method used is called the square and compare method for completing the square. As with all the alternative methods given, it is offered as a useful alternative to support students who have not engaged with other approaches. Worked examples 2 and 4 have been put together in one resource in PowerPoint 2.2a. This is to allow you to dip into it, or not, as you wish. This leads neatly into Exercise 2.2 of the Coursebook.

Differentiation:

Support:

• Factorising using a reverse grid approach reduces the amount of purely mental processing and allows the visual to help with thinking.

• The square and compare method of completing the square also is visually supportive for students who have not engaged with other methods of doing this.

Challenge:

• Rearranging equations which include algebraic fractions and then solving.

• Deriving equations first and then solving − in a real-world context such as business. These could be given as investigation tasks for some students to use to self-study while other students master the more basic skills.

Assessment for Learning: There are many opportunities for discussion using the discussion points in the PowerPoints provided. There should also be opportunities for students to ask questions of each other and of the teacher, whilst working. Many of these skills will be knowledge that students already have, but try not to assume that they will all easily recall how to apply each technique. Allow students time to revise these skills and repair any skills that have not been recalled correctly.

2 Modulus functions

Learning intention:

• Understand the relationship between y = f(x) and y = |f(x)|, where f(x) is quadratic.

Resources:

• PowerPoint 2.3a: Modulus of a quadratic including Worked example 5

• Coursebook Exercise 2.3

Description and purpose: The section on modulus functions builds on the processes studied in Chapter 1. You may choose to look at it in Chapter 1, when the linear functions are considered, or even later in the course when other functions may then be included. The PowerPoint available for this section works through the examples in section 2.3. Students will need to be able to find the roots and y-intercept for a quadratic function and then apply their knowledge of the modulus to it when drawing graphs. Sometimes, students will need to be able to find the coordinates of the turning point to solve simple problems about points of intersection of the graph of the modulus function with another function. This may lead into a problemsolving exercise on solving equations of the type shown in question 7 of Exercise 2.3 in the Coursebook, where students could find the number of solutions using graphing software or by drawing accurately and then going on to solve.

Differentiation:

Support:

• It is vital that students understand the basic skill of solving a quadratic equation and its application to finding the roots here.

• It is also vital that students understand the shape of a parabola and how the modulus function acts on this.

• Try to use visual support, such as graphing software, whenever possible.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Challenge:

• Coursebook Exercise 2.3, Q7 and Q8 and similar questions should be a good challenge for students.

• The Purplemath website offers an example similar to those in Exercise 2.3 Q7 and also gives an interesting example of nested absolute value functions.

Assessment for Learning: Some opportunities for discussion arise through the use of the PowerPoint. Students may also be encouraged to peer mark and assess their answers to Exercise 2.3. Natural opportunities for Q&A sessions should arise when this exercise is being carried out. Try to ask questions that allow students to show you that they have understood the mathematics they are studying. You could review their knowledge using a piece of work that they had to mark and grade. Can they find errors? Can they discuss the impact of those errors?

3 Quadratic inequalities

Learning intention:

• Find the solution set for quadratic inequalities.

Resources:

• PowerPoint 2.4: Worked examples 6 and 7

• Chapter 2 Teacher notes: class discussion section 2.4

• Chapter 2 Lesson plan: Solving inequalities

• Coursebook Exercise 2.4

Description and purpose: A demonstration lesson plan has been given for a possible lesson covering quadratic inequalities. The lesson incorporates the recap and class discussion in section 2.4. The Chapter 2 teacher notes give some support for managing the discussion direction, if it is needed. Some ideas about students developing their own explanations are also given. There are links in the document to a video that may be useful to challenge students and a Wolfram inequality checker tool. The lesson leads into Exercise 2.4 of the Coursebook.

Differentiation:

Support: The focus should be on the algebraic process here but if students need visual support, access to graphing software may be supportive for some.

Challenge: The Khan Academy has a video on rearranging inequalities with algebraic fractions and then solving.

Assessment for Learning: Assessment for learning opportunities should arise naturally through observation, peer checking of answers, Q&A sessions and whole class discussion as well as the discussion points which arise in the PowerPoint of worked examples.

4 Simultaneous equations

Learning intention:

• Solve simple simultaneous equations in two unknowns, with at least one linear, by elimination or substitution.

Resources:

• PowerPoint 2 recap a: Solving two linear simultaneous equations

• PowerPoint 2.1a: Solving two simultaneous equations with one linear including worked example 1

• PowerPoint 2.1b: Solving two simultaneous equations both non-linear

• Coursebook Exercise 2.1

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Description and purpose: It is appropriate for the work in this chapter that students know how to find the points of intersection of a line and a curve, as these will be related to conditions for the nature of the roots of quadratic equations. The methodologies used to solve simultaneous equations are standard. These methods are tried and tested and students usually understand them well and use them with proficiency. However, we all forget things from time to time, and so the Coursebook offers plenty of revision of the key concepts. PowerPoint 2 recap a animates the recap of solving a pair of linear simultaneous equations at the start of the chapter in the Coursebook. PowerPoint 2.1a starts with the example in the Coursebook at the start of section 2.1 and then works through worked example 1. These techniques are sufficient to be able to solve simultaneous equations where at least one equation is linear and so lead into Exercise 2.1 of the Coursebook.

Students need to be familiar with solving simple simultaneous equations where neither equation is linear. PowerPoint 2.1b has some examples of these and includes an example where students are asked to solve a quartic equation that is quadratic in x 2. This can be used as an introduction for Chapter 5 section 5.5 or simply as a forerunner to the ideas which are explored more fully in Chapter 5.

Differentiation:

Support:

• Some students try to use the elimination method, when the method of substitution is far simpler when solving simultaneous equations. If this is the case, try to encourage your students to stick to one method of solution. Make sure that they know that, at this level, substitution is very much more useful as a method as it is more universal.

• Try to ensure that students use their calculator as a checking tool. If they have made an error, work with them through their solution to help them find it and correct it successfully.

Challenge: Some more challenging material is also provided in the exercises, to allow some students to develop their skills; for example, Exercise 2.1 Q19 to Q26.

Assessment for Learning: As well as the discussion opportunities which will naturally arise through the use of the resource materials and when your students are working through questions, assessment for learning can be carried out using a lesson review by students. At the end of the lesson, ask them what they have learned. Write their responses down for them to refer to in future work. This could then be used as part of a revision session later in the course.

5 Nature of roots

Learning intention:

• Know the conditions for ax 2 + bx + c = 0 to have: (i) two real roots, (ii) two equal roots, (iii) no real roots, and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.

Resources:

• PowerPoint 2.3b: Roots and intersections

• PowerPoint 2.5a: Worked examples 8 to 12

• PowerPoint 2.5b: Connecting the nature of roots with intersections of graphs

• Coursebook Exercise 2.5

• Coursebook Exercise 2.6

Description and purpose: You may choose to link the final sections on the nature of roots and the intersections of lines and curves. The resources and the Coursebook enable you to choose to separate or combine them as you prefer. PowerPoint 2.5b looks at six specific curves, their graphs and hence their roots and then identifies the discriminant in each case. Students are asked all through to explain what is happening on the basis of what they can see in the formula. The results are summarised. In the final three slides, more consideration of the connection between roots and intersections is given. Worked examples 8 to 12 have been combined in one resource, PowerPoint 2.5a. Again, you may wish to use part of it and come back to it at another point. It is not suggested that you use it all in one session. Exercise 2.5 of the Coursebook may be worked through after worked example 9, and Exercise 2.6 follows once the PowerPoint is complete, or both exercises may be looked at after the full set of worked examples in the PowerPoint has been considered.

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IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Differentiation:

Support: In the problem after worked example 5, some students may struggle to see why the number of intersections of the line and curve is the same as the number of solutions of the given equation. PowerPoint 2.3b looks at this very point. A statement is made regarding the roots of an equation and the points of intersection of two lines as being the same and the question ‘Why?’ is asked. This could be used as a class investigation. Many students will find it intuitively obvious, but some will not. Those to whom it is obvious may not find it easy to put their case in a watertight argument. There is more work on this point later with non-horizontal lines.

Challenge: Worked example 9 is solved using the discriminant. This is the most straightforward approach. There is an alternative calculus method which some students may prefer. You may wish to set this as an investigation for your students once you have covered simple differentiation.

Assessment for Learning: As part of assessment for learning for this topic, be careful to check that students understand the language used and are able to devise a method of solution based on that language. You can check this by marking written work or through general discussion, for example.

Review activities

1 Order review

Description and purpose: This review requires students to order the six steps needed to rearrange and solve a quadratic inequality. They then have to decide whether the solution that has been given is correct. The purpose of this review is to consolidate the logical steps practised in the lesson and also to remind students of the importance of sketching the graphs!

Resources:

• PowerPoint 2 review: Quadratics order order

Activity:

Students are given this scenario:

Disaster! Neena’s pet bookworm has shredded her homework! She cannot tell which part of her answer comes first…. help Neena by sorting her work into the correct order! Now tell her if she was correct!

Allow students to discuss and write down what they think the correct order is. They can use the A, B, C…. marked alongside.

Answer: The correct order on the PowerPoint is CEFBDA.

When everyone is ready, click to move the statements to the correct place. Click again to move to a second slide where the sketch graph of the quadratic is revealed for students to judge the solution. This activity could possibly lead from: a lesson focused on solving quadratic inequalities. This activity could be adapted: The six statements that need to be ordered could be written on large pieces of paper in exactly the same way as in the PowerPoint and pinned onto a board. Labelling them A, B, C etc. will give students an easy way to describe which order is the correct order. As another alternative, the statements could be printed or written in regular sized font/print and the class split into pairs or groups and given a set each which they can then move around to form the correct order.

2 Check my graph review

Description and purpose: This task is designed to make students think about the shape of the graphs they draw and how accuracy and attention to detail can improve a solution. It can be used as an assessment for learning exercise. It should be a useful tool in assessing whether they have fully understood what they have learned.

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

Resources:

• Check my graph

• Check my graph teacher notes

Activity:

The Check my graph file can either be printed as a handout or displayed on a flip chart or interactive whiteboard.

It has two graphs for your students to check for accuracy and award marks. They should identify and be able to explain how to correct any errors. They then have to find two good comments about the work and set one target for improvement. Suggested comments and marks are made in the Check my graph teacher notes file. Your students may think of others!

This activity could possibly lead from: a lesson on sketching the graphs of the modulus of a quadratic function.

This activity could be adapted: The graphs could be updated or added to, and this could also be used with other types of modulus function.

Homework ideas

1 Coursebook: Quadratic equations and the parabola, Exercise 2.2

Completion of this exercise should give students a good amount of practice of finding roots, sketching graphs and completing the square.

2 PowerPoint 2.6

Completion of the questions in this PowerPoint should measure the ability of students when solving problems involving the nature of the roots of a quadratic equation. It will also check whether they can solve quadratic inequalities successfully and whether they can make the connections between points of intersection and roots. This PowerPoint has two versions. The first version has no model answers included, but does have some hints in the teacher notes for each slide. These can be removed if you do not want to give any hints at all. The second version includes animated answers and is very supportive of those who need greater modelling of what is needed. Again, hints, and also details of what each animation will reveal, are included in the Teacher notes for each slide. As well as a useful homework tool, this PowerPoint can be used as a revision exercise, for self-study, further practice in class or as part of a bank of resources students can access at any point throughout the course when needed. This practice material is also available as a PDF file in case technology is not available to your students.

Curriculum framework correlation grid

These learning objectives are reproduced from the Cambridge IGCSE and O Level syllabuses Additional Mathematics (0606/4037) for examination from 2025. This Cambridge International copyright material is reproduced under licence and remains the intellectual property of Cambridge Assessment International Education.

The following table shows how the learning objectives map to the Cambridge IGCSE Additional Mathematics Coursebook, Workbook, Worked Solutions Manual and Teacher’s Resource.

1. Functions

1.1 understand the terms: function, domain, range (image set), one – one function, many – one function, inverse function, and composition of functions

1.2 find the domain and range of functions

1.3 recognise and use function notations

1.4 understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic, cubic or trigonometric

1.5 explain in words why a given function does not have an inverse

1.6 find the inverse of a one – one function

1.7 form and use composite functions

1.8 use sketch graphs to show the relationship between a function and its inverse

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

2. Quadratic functions

2.1 find the maximum or minimum value of the quadratic function f : x ↦ ax2 + bx + c by completing the square or by differentiation

2.2 use the maximum or minimum value of f(x) to sketch the graph of y =f(x) or determine the range for a given domain

2.3 know the conditions for f(x) = 0 to have:

(i) two real roots

(ii) two equal roots

(iii) no real roots and the related conditions for a given line to:

(i) intersect a given curve

(ii) be a tangent to a given curve

(iii) not intersect a given curve

2.4 solve quadratic equations for real roots

2.5 find the solution set for quadratic inequalities either graphically or algebraically

3. Factors of polynomials

3.1 know and use the remainder and factor theorems

3.2 find factors of polynomials

3.3 solve cubic equations

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

4. Equations, inequalities and graphs

4.1 solve equations of the type

• |ax + b| = c (c > 0)

• |ax + b| = cx + d

• |ax + b| = |cx + d|

• |ax 2 + bx + c| = d

Using algebraic or graphical methods

4.2 solve graphically or algebraically inequalities of the type

• k|ax + b| . c (c > 0)

• k|ax + b| < c (c > 0)

• k|ax + b| < |cx + d| where k . 0

• |ax + b| < cx + d

• |ax 2 + bx + c| . d

• |ax 2 + bx + c| < d

4.3 use substitution to form and solve a quadratic equation in order to solve a related equation

4.4 sketch the graphs of cubic polynomials and their moduli, when given as a product of three linear factors

4.5 solve graphically cubic inequalities of the form

• f(x) > d

• f(x) . d

• f(x) < d

• f(x) , d

where f(x) is a product of three linear factors and d is a constant

✓

5. Simultaneous equations

5.1 solve simultaneous equations in two unknowns by elimination or substitution

6. Logarithmic and exponential functions

6.1 know and use simple properties and graphs of the logarithmic and exponential functions, including lnx and ex

6.2 know and use the laws of logarithms, including change of base of logarithms

6.3 solve equations of the form ax = b

7. Straight line graphs

7.1 use the equation of a straight line

7.2 know and use the condition for two lines to be parallel or perpendicular

7.3 solve problems involving mid-point and length of a line, including finding and using the equation of a perpendicular bisector

7.4 transform given relationships to and from straight line form, including determining unknown constants by calculating the gradient or intercept of the transformed graph

8. Coordinate geometry of the circle+

8.1 know and use the equation of a circle with radius r and centre (h, k)

8.2 solve problems involving the intersection of a circle and a straight line

8.3 solve problems involving tangents to a circle

8.4 solve problems involving the intersection of two circles

9. Circular measure

9.1 solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure

10. Trigonometry

10.1 know and use the six trigonometric functions of angles of any magnitude

10.2 understand and use the amplitude and period of a trigonometric function, including the relationship between graphs of related trigonometric functions

10.3 draw and use the graphs of

y = a sin bx + c

y = a cos bx + c

y = a tan bx + c

where a is a positive integer, b is a simple fraction or integer, and c is an integer

10.4 use the relationships:

• sin2 A + cos2 A = 1

• sec2 A = 1 + tan2 A

• cosec2 A = 1 + cot2 A

10.5 solve, for a given domain, trigonometric equations involving the six trigonometric functions

10.6 prove trigonometric relationships involving the six trigonometric functions

11. Permutations and combinations

11.1 recognise the difference between permutations and combinations and know when each should be used’

11.2 know and use the notation n! and the expressions for permutations and combinations of n items taken r at a time

11.3 answer problems on arrangement and selection using permutations or combinations

12. Series

12.1 use the binomial theorem for expansion of (a + b)n for positive integer n

12.2 use the general term (n r)an − rbr, 0 < r < n

12.3 recognise arithmetic and geometric progressions and understand the difference between them

12.4 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions

12.5 use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression

13. Vectors in two dimensions

13.1 Understand and use vector notation

13.2 know and use position vectors and unit vectors

13.3 find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars

13.4 compose and resolve velocities

14. Calculus

14.1 understand the idea of a derived function

14.2 use the notations

f′(x), f″(x), dy dx , d2y dx2 = [ d dx ( dy dx )] dx, δx → 0, dy

x

14.3 know and use the derivatives of the standard functions xn (for any rational n), sin x, cos x, tan x, ex, ln x.

14.4 differentiate products and quotients of functions

14.5 Use differentiation to find gradients, tangents and normals

14.6 Use differentiation to find stationary points

14.7 apply differentiation to connected rates of change, small increments and approximations

14.8 apply differentiation to practical problems involving maxima and minima

14.9 use the first and second derivative tests to discriminate between maxima and minima

14.10 understand integration as the reverse process of differentiation

14.11 integrate sums of terms in powers of x including 1 x

CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE

14.12 integrate functions of the form:

• (ax + b)n for any rational n

• sin (ax + b)

• cos (ax + b)

• sec2 (ax + b)

• eax + b .

14.13 evaluate definite integrals and apply integration to the evaluation of plane areas

14.14 apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration

14.15 make use of the relationships in 14.14 to draw and use the following graphs:

• displacement–time

• distance–time

• velocity–time

• speed–time

• acceleration–time.