Additional Mathematics Module

CHAPTER 1 – FUNCTIONS

LEARNING AREA : FUNCTIONS
Learning Objectives : Understand the concept of relations
Learning Outcomes : Student will be able to

1.1 Represent relation using a) arrow diagram , b) ordered pairs, c)graphs
1.2 Identify domain, co domain, object, image and range of a relation.
1.3 Classify a relation shown on the mapped diagram as : one to one many to one ,

one to many or many to many

1.1a Representing a relation between two sets by using an arrow diagram

Example 1

Suppose we have set A = { 0,1,2,3 } and set B = { 0,1,2,3,4,5,6} .Let us examine the relations “is one more than “
from set A to Set B . One way to show the relations is to draw an arrow diagram as shown below . The arrows
relate the elements in A to the elements in B

AB In the space below give other example to show a relation

is one more than  6
5

34

23

12

0. . 1
0

FIG 1.1

A relation from set A to set B is an association of elements of A with elements of B .

Exercise 1 2. What relation from set S to set T is illustrated in
diagram below .Write your answer in the box below
1 Two set of numbers, P and Q are shown below
Complete the arrow diagram to show the relations “ is ST
less than “ from set P to set Q
36
P is less then Q 24
12
10  8 -1 . .  -2
83
62
1. . 1

3. Construct an arrow diagram to show the relation 4. Construct an arrow diagram to show the relation
“ is a factor of “ from set A = {1,4,15,35,40} to set B “ is a factor of “ from set A ={1,2,3,4,5,6 } to

={2,5,7,13 } set B = {2,3,5}

1

1.1(b) Representing a relation between two sets by using an ordered pairs

The relation can also be shown concisely by ordered pairs (x , y ) . The elements in the pair are in order since the

first element comes from the first set A and the second element comes from the second set B .We have
A = {0,1,2,3} and B = {0,1,2,3,4,5,6, If x  A ( this means that x is a member of A) and y  B, the set of all ordered
pairs (x,y) is { (0,1),(1,2),( 2,3),(3,4)} . This set of ordered pairs defines the relations “ is one less than “ from set A

to set B

Example 2

1. A = { 1,2,3,4,5,6} and B = {2,3,5 }. 2. Write down the relation as a set of ordered pairs
Show the relation “ is a factor of” from A to B “ is multiple of “from set A ={2,5,7,13 } to set B =

as a set of ordered pairs {1,4,15,35,40}
Solution Solution

The ordered pairs ={ (2,2),(3,3),(4,2),(5,5),(6,2),(6,3) } The ordered pairs = { (2,4) (2,40), (5,15),(5,35),(5,40),(7,35) }

Exercise 1.1 (b)

1. A relation between two sets is defined by the set of ordered 2. A relation R is defined by { (1,3),(2,6),( 3, 9),(4,12) }
pairs , { (-1,2),(1,4) ,(3,6) (5,8) , (7,10 ) } Express the relation in function notation..
Express the relation in function notation..

3. Given that X ={ 0,1,2,3,4,5,6 } and Y = { 0,1,2,3,4 } 4. Given that A = {2,4,6,8} and B = { 1,2,3,4,5 }

If x  X and y  Y , list the set of ordered pairs in the relation If x  A and y  B , list the set of ordered pairs in the relation x is

x “is one less than “ y half of “ y

1.1(c) Representing a relation between two sets using a graphs

We can plot an ordered pairs in a Cartesian graph of the relations

Example 3

1. We have A = {0,1,2,3} and B = {0,1,2,3,4,5,6, If x  A and 2. Given that A = B = { 1,2,3,4,5 }
y  B, the set of all ordered pairs ( x,y) is If x  A and y  B , and the relation x “ less than “ y

(0,1),(1,2),(2,3),(3,4)} Illustrate the relation by means of a Cartesian graph.

we can plot a Cartesian graph to show the relation.

Set B 
Set B

 Set A

Set A 2

Exercise 1.1(c)

1. Given that A = B = { 1,2,3,4,5,6,} Given that A = {2,4,6 } and B = { 1,2,3,4,5 }
If x  A and y  B , and the relation x “ is a factor of “ y
If x  A and y  B , and the relation x “ is double “ y
Illustrate the relation by means of a Cartesian graph
Illustrate the relation by means of a Cartesian graph

Set B
Set B

Set A Set A

Homework : Textbook Exercise 1.1.1 page 3

1.2 Identifying domain, codomain, object, image and range of a relation.

Domain is the set of x-coordinates of the set of points on a graph or the set of x-coordinates of a given set of
ordered pairs. The value that is the input in a function or relation
Range is the y-coordinates of the set of points on a graph, or the y-coordinates of a given set of ordered pairs.
The range is the output in a function or a relation. A graph of a relation (a set of ordered pairs) is given below.
(Fig 1.2)

Note that the five points on the graph have the ordered pairs

{(-2,1),(3,2),(4,0),(2,-3),(-2,-3)}.

The domain of this relation is the set {-2,2,3,4}. Notice that although -
2 is an x-coordinate twice, we need only list it in the domain once.

The range of this relation is the set {-3,0,1,2}. Notice that although -3
is a y-coordinate twice, we need only list it in the range once.

For the ordered pair (  2 , 1 ) the object is  2 and the image is 1
so the objects are -2,2,3,4 and the images are -3,0,1,2

FIG 1.2(a)

1 2 Refer to figure 1.2 (b)
The codomain is the set { 2,4,6,8,10 }
3 4 The domain is the set { 1,3,5,7,9,}
5 6 The range is the set { 2,4,6,8,}
7 The images are 2,4,6,8
9 8 The object s are 1,3,7,9

10 3
FIG 1.2(b)

Exercise 1.2 Ali 84
Bakar 90
Complete the table below based on the diagram given Samy

1 2.

-2 0
0 4
2 8
4 10

16

Domain Codomain Object Image Range Domain Codomain Object Image Range

Homework : Textbook Exercise 1.1..2 page 4

1.1.3 Classifying a relation

Example 3

Observe the table below and list the characteristic of each relation .

Relations Arrow diagrams Ordered pairs Graph

23 {(2,3),(4,5), (6,7)} y
45
One to one 67 7

5
3

x

24 6

y

One to many 6 
5 
14 {(1,4),(3,5), (3,6)}
35 4

6 13

Many to one {(7,6),(9,6), (11,14)} 14 
Many to many
76 10
9 10
11 14 6 

ap 7 9 11
bq
cr {(a,p),(b,p),(b,q),(c,s),(c,t)} t
s
s r
t q
p 

abc

4

Exercise 1.2
Complete the table below and determine the type of relations

Bil Arrow diagram Ordered Pairs Graf Relation
1 {(-2,1),(-2,3) (0,3)(1,4),(3,6)} One to one
{(a,2),(b,3) (c,3),(d,5)} 9
-3 4 4
15 1
26
-3 1 2

2 1 6
3 4
-2 4 3
0 6 1
1
3 -2 1

3

3

a 2 5
b 3
c 5 3 
d
2

a bc d

4 a {(4,2),(25,5)} 
b 
1 c 
2 d

5 2
5 12
4
25 5
36
2

4 25 36

Homework : Textbook Exercise 1.1..3 page 5 and Skill Practice 1.2 page 6

2.0 Learning Objective : Understand the concept of functions
Learning outcomes : Student will be able to
2.1 Recognise functions as a special relation
2.2 Express functions using functions notation
2.3 Determine domain,object,image and range of a function
2.4 Determine the image of a function given the object and vice versa.

5

2.1 Recognising functions as a special relation
Note: A special type of relations between two set A and B can exist when each and every members of A is related
to one and only one member of B, although some of the member of B may not related to any member of A
This type of relation is known as a function. Itis important to relise that a function is a relation but a relation may or
may not be a function . In other words, a function is a special type of relation

Example 4

Determine which of these relations is a functions

a) b) c) d)
A A
AB AB B B

-2 1 76 a 2 76
03 9 10 b 3 9 10
14 11 14 c 5 11 14
36 d

A function because each Not a function A function although 9 and
No image for d 11 in A are both linked to
Not a function – 2 has two member in A is related to 10

images member in B d)

Exersice 2.1
1) Determine which of this relations is a functions

a) b) c)

14 76 ap a
35 9 10 b2
11 14 bq c3
6 cr d5

s

Let’s think

Which type of relations ; one to one , one to many, many to one and many to many is a a function.
Discuss it.
Homework : Textbook Exercise 1.2.1 page 7

2.2 Expressing function using function notation

We have learnt that relations is also a function or mapping as each and every member of set A is linked to one and
only one member of set B . Symbolically , we write f : x  x  2 which means that ” the function f maps x onto ” x
+ 2 ” . In functional notation, we write f (x)  x  2 . f (x) is read as ’ function of x ’.

The set of elements in A for the mapping is known as the domain while the set of images in B is known as the
range . In functional notation, if f (x)  x  2 and x =3 , then f (3)  3  2  5 ( 3 is an object while 5 is the image)

Homework : Textbook Exercise 1.2..2 page 8

2.3 Determining domain,object,image and range of a function

Example 5 Object Images
x
” times 2 ” 2 f(x)
3
1. 2 4 -1 f(2) = 2(2) = 4
a
4 8 f(3) = 2(3) = 6
a+1
x 2x x2 f(-1) = 2(-1)= - 2
x3
f(a)= 2(a) = 2a
x2 - 1
f(a+1)= 2(a+1) = 2a + 2
6 f(x2) = 2x2
f(x3) = 2x3
f(x2 – 1) = 2(x2 -1) = 2x2-2

f : x 2x
f(x) = 2x

Example 6 Functions 2 f : x  4x
Bil Functions notations { (1,4), (2,8), (3,12)}
f : x  x2 f(p) = 3
1 0 3 f(q) = 1
1 f(x) = x2 f(r) = 5
-2 4 7
-1 f(-2) = 4 5
0 3
1 f(2) = 4
2 1
f(0) = 0

Exerise 2.2 pq r
Complete the folowing table

Functions x value Calculations image
a) f : x  2x + 1 -2 f(-2) = 2(-2)+1= -3
0 f(0) = 2(0)+1 = 1
1

b) g; x  2x  3 2
7 -3
5

c) h : x  4x - x2 -4
4

-2

2.4 Determining the image of a function given the object and vice versa

A Find the image for each of the following functions. b) Given that f(x) = (x2 - 2)2 (x + 1 )-10 , find the
a) Given that f(x) = 2x2 - 4x + 5 , find the image image for each of the following.

For each of the following .

(i)f(-1)=2(-1)2 – 4(-1) + 5 ii) f(2) i) f(0) ii)f(2)
= 2+4+5
= 11

iii) f( 2 ) iv) f(-4) v) f(-1) vi f (-3)
3

B . Find the object for each of the following functions

c) Given that f(x) = 2x + 6 , find the object when d) Given that f(x) = 2x 8 , find the object when
the image is,
5

the image is,

(i) 2 ii) - 5 i) 5 ii) - 3

7

7 iv)  3 8 3
5 2
iii) v)

3 9

C. b) Given that f (x) = 4 - 2x, find the value m

a) Given h : x  3x-12 , Find If f(m) = 10

i) the object when the image is 6
ii) the object which mapped to it’s self
Solution :
(i) (ii)

Homework : Textbook Exercise 1.2.4 page 10

D. (i) The diagrams below shows part of the mapping f (x) : ax  b

Find the values of a and b
(i) the image of 3 under f
(ii) the object whose image is -4
(iii)

the values of a and b Calculations the object whose image is -4

the image of 3 under f

a) x  a x + b

4

7

1

8

b)

8

D (ii) b)
a)

The diagram above shows part of the mapping The diagram above shows part of the mapping

f (x) :  ax2  bx  c . Find f : x  72 . Find ,

(a) f(x) . ax  b
(i) the values of a and b
(b) the image of - 4 under f (ii) the image of 10 under f
(iii) the object of 4

Example 6

a) A function f is defined by f : x  12 , x   b b) A function f is defined by f: x  a  b, x  k
ax  b a
x
Given that f(4) = -3 and f (10) = 6, Calculate Given that f(2) and f(5) = -1

a) the value of a and b a) state the value of k
b) the value of x for which f(x) = -x b) find the value a and b

c) find f(4)

Homework : Textbook skill Practice 1.2 page 10

9

Absolute Valued Function
Note :

f(x) = x is called absolute valued function. The absolute value of a number is the distance the
number is from zero on the number line. We write the absolute value of -2 as | -2 |.

The absolute value of a number is found by determining how many units the number is from zero. Since
distance is always thought of as positive, the absolute value of any number is positive. For example, -2
is 2 units from zero, so | -2 | = 2. In the picture below, you can see that | 4 | = 4 and | -4 | = 4, because both
4 and -4 are 4 units from zero on the number line.

In equalities of a bsolute Valued Function
(i) x < k  - k < x < k
(ii) ax  b < k  - k < ax + b < k

(iii) ax  b > k  ax + b < -k or ax + b > k

Example 10

a) Given that f(x) = 3x  2 , find the value of b) Given that f : x  3x  5 . Find the domain of

i) f(2) ii) f ( 1 ) the following functions
2
i) f(x)  4 ii) f(x) > 3

Exercise 2.4

a) Given that f (x) =  x3 – 4  , find b) Given that f :x = 2x - 8  . Find the domain of the

i) f( 3) ii) f(-2 ) following functions

i) f(x)  2 ii) f(x) > 4

c) Given the function f :x = 2x - 5 ,find the values d) Given the function f :x = 2x - 5 ,find the range of
of x such that f(x) = 7 values of x such that f(x) < 7

10

Example 11: -2  x  3
Sketch the following function . Hence, state the range that match with the domain given .

a) f : x  x  3 , - 4  x  2 b) f: x  x - 3, -1  x  3 c) f: x  x  2 + 1

2.5 Exercise
Sketch the following function . Hence, state the range that match with the domain given .

a) f : x  x  2 for -1  x  3 b) g : x  2x  1 - 2 for -1  x  2 c) h : x  4  x2 for -1  x  3

Homework 2. . Sketch the graph of function   2x-5 

1. Sketch the graph of function f:x   2x-3  for 0  x  6 . Find the value of x if f(x)  4.
for 0  x  4. Hence state the range that

match with the domain given

3. Given that f(x) =  7 – 2x  for 0  x  8 . find the 4. A function f is defined by f : x  3x-5 
a) find f(4) , f(10) and f( -5)
range that match to the domain given. b) if f (a) = 26, find the possible value of a

11

Learning Objective : Understand the concept of composite functions.
Learning Outcomes : Student will be able to

3.1 Determine composition of two functions.
3.2 Determine the image of composite functions given the object and vice versa.
3.3 Determine one of the functions when the composite function and the other

function are given.

3.0 Understanding the concept of composite functions.

Note : 1. Composite functions is a function composed of two or more algebraic functions.

2. Functions can be combined to give a composite function. If a function f is followed by a function g ,

we obtain the composite function g f . In general gf  fg .
f followed by g followed by h is donated by hgf . The order is relevant and important.

3.1 Determine composition of two functions.
Example 3. 1 Determining the composite functions.

a) Two functions f and g are defined by b) Two functions f and g are defined by f : x  5x - 2

f : x  x - 2 and g : x  x2 – 1. Obtained and g :x  x2 – x Find,

expressions for i ) g2 ii) f 2 g 2

(i) fg , (ii) gf

Exercise 3.1 b) Two functions f and g are defined by

a) Two functions f and g are defined by f : x  x2 + 6 and g : x  3x + 4.

f : x  x2 + 1 and g: x  4 . Find Find the values of x if gf (x) = fg (x)
x

i) gf and g2 ii) fg and f 2

Homework : Textbook Exercise 1.3.1 page 1

12

3.2 Determining the image of composite functions given the object and vice versa.

Example 3.2

Two functions f and g are defined by i) Two functions f and g are defined by

f : x  x2 + 1 and g: x  4 . Find f : x  2x + 6 and g : x  a x + b
x Given that f g (1) = 10 and g f (2) = 8 . Calculate the
values of a and b
(i) gf (2) and g2 (4) ii) fg (-4) and f 2 ( 3)

Exercise 3.2 d) Two functions f and g are defined by f : x  a x - b

c) Two functions f and g are defined by and g :x  2x - 5 . Given that f g (1) = 10 and
g f (2) = 8 . Calculate the values of a and b
f : x  x - 2 and g : x  x2 – 1. Obtained

expressions for fg (2) , gf (-2) , f 2 (2) , g 2 (8)

Note : It is common to express ff (x) (x) as f2 (x) , fff (x) as f3(x) etc.

Homework : Textbook Exercise 1.3.2 page 14

Example 3.2(ii) The function g is defined by g:x  1 x , x  1. Express in their simplest forms each of the
1 x

following functions b) g 3(x) c) g4(x) c) g16 (x)
a) g2(x)

13

Exercise 3.2(ii) The function g is defined by f: x  5 , x  0 Express in their simplest forms each of the
x

following functions

a) f 2(x) b) f3(x) c) f4(x) d) f31 (x)

3.3 Determining one of the functions when the composite function and the other function are given .

Example 3.3 b) If f (x) = x +1 find the function g such that
g f (x) = x2 + 2x + 12
a) If g(x) = 1 (x 1) find the function f such that
2

g f (x) = 2x 1 ,

3

Exercise 3.3

a) If f (x) = x + 4 find the function g such that b) if f(x) =x2 + 1 and g f (x) = x4 + 2x2+ 9. Find the
function g . [ Ans g(x)=x2 + 8 ]
f g(x) = 2(x 1) , x  2 [g(x) 2(3  x) , x  2 ]
x2 x2

x d) Given that fg(x) = 5 – x and f(x) = 8x +3. Find the
function g
c) Given that hg(x) = 2x + 1 and h(x) = . Find the

4

function g

14

Learning objective 4.0 Understand the concept of inverse functions.

Learning outcomes : Student will be able to:
4.1 Find the object by inverse mapping given its image and function.

4.2 Determine inverse functions using algebra.

4.3 Determine and state the condition for existence of an inverse function.

Homework : Textbook Exercise 1.3.3 page 15 and Skill Practice 1.3 page 16.

4.0 Understanding the concept of inverse functions.

Suppose f is a function “multiply by 3” .If. this function is applied to x, then the images of x is 3x What function should we
apply to 3x in order to get back to x ?.This function which maps the image back to its initial value is known as the inverse

function of f, and it is denoted by f 1 . Symbolically we write f (x)  3x and f 1(x)  3
x

If f is a many – to – one function, then its inverse is not a function but a one- to- many relation. Only one – to – one functions will
give one – to – one inverse functions.

4.1 Finding the object by inverse mapping given its image and function.

f = x–1 f -1

3 2 2 3

4 3 3 4

5 4 4 5

6 5 5 6

f(x) = x – 1 f -1( 2) = 3
f(3) = 2 f -1( 3) = 4
f(4) = 3 f -1( 4) = 5
f(5) = 4 f -1( 5) = 6

Example 4.1
Find the value of a and b in the following diagram by the inverse mapping

a) f x : x  4  3x b) f x : x  2x  5

3  - 5 a  - 3
a 13
b 8 4 13
1  1 b 17
1  7

15

Exercise 4.1 Find the value of a and b in the following diagram by the inverse mapping

a) f x : x  2  x b) f x : x  2x 1

3  - a  - 3
4 9
1 b 17
a 1  3

1
b

8
1

1

Homework : Textbook Exercise 1.4.1 page 17 b) Given f : x  3x  1 , x  2. Find f -1 (x) ,
x2
4.2 Determining inverse functions using algebra
Example 4. 2 (i) f 1(4) , f 1(2) and f 1(1)
a) Find the inverse function for the function 3

f : x  3x + 4 then find f 1(2)

Exercise 4.2 b) Given that f :x  hx  k , x  2 and its inverse function f -1
a) Given h : x  3  x , find h – 1(x) , h1(3) x2

2 : x  2x  9 , x  4 . Find the value of h and k.
x4
and h1( 1 )
4

Homework : Textbook Exercise 1.4.2 page 18

16

Example 4.2(i) : b) A function f is defined by f : x  x + 2. Find
a) A function f is defined by f : x  x + 3. Find the function g such that gf: x  x + 9

(i) ff 1

(ii) the function g such that gf : x  x2 + 6x + 2.

(i) f f -1 ii) function g

A function f is defined by f: x  2x-1. . Find the function g such that
i) f g : x  7 -6x ii) g f: x  5
2x

4.3 Determining and stating the condition for existence of an inverse function.
.

(a) (b)

Diagram (a) shows function g(x) = x2 , which is not a one to one function. Reversing the arrows, you do not have a
one to one function.. In general, for a function f to have an inverse function, f must be a one to one function.
Note : If f is a many – to – one function, then its inverse is not a function but a one – to – many relation.
Only one – to – one function will give one- to-one functions

Remember : If a function f is one to one, then the inverse function f 1 does exist !!!
17

Example 4.3

Determine whether the inverse of the following function is a function or not .

a) f(x) = (x+1)2 b) g(x)  1  4 , x0

x

Homework : Textbook Exercise 1.4.3 page 20 and Skill Practice 1.4

FUNCTIONS – SPM QUESTIONS

2004 – P1 2. Given the functions h : x 4x + m and
1. Diagram 1 shows the relation between set P and
h- 1 : x 5
set Q.
2kx + , where m and k are
dw
8
ex
constants , find the value of m and of k .

[3 marks]

fy

Set P z 3. Given the function h(x) = 6 , x ≠ 0 and
Set Q x
.
DIAGRAM 1 [2 marks] composite function hg(x) = 3x,
State Find (a) g(x) ,

(a) the range of the relation, (b) the value of x when gh(x) = 5 .
(b) the type of the relation [4 marks]

18

2003 – P1 8. The following information refers to the functions h
4.
and g.
P={1,2,3}
Q = { 2 , 4 , 6 , 8 , 10 } h : x  2x  3

Based on the above information, the relation g : x  4x 1
between P and Q is defined by the
set of ordered pairs { (1,2) , (1,4) , (2,6) , (2,8) } . Find gh- 1(x).

State (a) the image of 1 , [3 marks]
(b) the object of 2 ,
[2 marks]

5. Given that g : x 5x + 1 and 2006 – P1
h : x x2 – 2x + 3 , find 9. In Diagram 1, set B shows the images of certain
(a) g – 1 (3) , elements of set A.
(b) hg(x) .
5
[4marks] 4 25
-4
- 5 16

2005 – P1 Set A Set B
6. In Diagram 1, the function h maps x to y and the
function g maps y to z. Diagram 1

xh yg z (a) State the type of relation between set A and
8 set B.

(b) Using the function notation, write a relation
between set A and set B.

5

2 10. Diagram 2 shows the function
Diagram 1
h : x  m  x , x  0, where m is a constant.
Determine (a) h1 (5), x mx
(b) gh(2).
x
[2 marks] x

8 DIAGRAM 2

7. The function w is defined as Find the value of m. 1
2
w(x)  5 , x  2.
2x [2 marks]

(a) w1 (x),

(b) w1 (4).

[3 marks]

19

2006 – P2 15. Given the functions f : x ax + b , a > 0 and
f2:x
11. Given that f : x  3x  2 and g : x  x  1, 9x – 8 . Find
5
(a) the value of a and of b
find
(b) ( f – 1)2 (x) [3 marks]
(a) f 1 (x),
[3 marks]

[1 mark]

(b) f 1g(x),

[2 marks]

(c) h(x) such that hg(x)  2x  6.

[3 marks]

2000 – P1

16. Given that g – 1 (x) = 5  kx and f(x) = 3 x2 – 5
3

2002 – P1 . Find
12. (a) Given that f : x
(a) g(x) [2 marks]

3x + 1 , find f – 1 (5) , (b) the value of k such that g(x2 ) = 2 f (- x)
[2 marks] [3 marks]

(b) Given that f(x) = 5 – 3x and g(x) = 2ax + b

where a and b are constants .
If fg(x) = 8 – 3x , find the value of a and of b

[3 marks]

13. (a) Find the range of value of x if x < 5 . 2007 – P1
17. Diagram 1 shows the linear function h.
[2 marks]
x h(x)
(b) Sketch the graph of the function
01
f : x 2x  3 for the domain 0  x  4. 12

Hence, state the corresponding range for the m4

domain. [3 marks] 56

Diagram 1

(a) State the value of m.
(b) Using the function notation, express h in terms

of x. [2 marks]

14. Given that f(x) = 4x – 2 and g(x) = 5x + 3 . Find
(i) fg – 1 (x)
x
(ii) the value of x such that fg – 1  2  = 2 .
5

[5 marks]

2001 – P1

20

18. Given the function f : x  x  3 , find the 20. Diagram 1 shows the graph of the function

values of x such that f(x) = 5 . [2 marks] f (x)  2x 1 , for the domain
21. Given the functions g : x  5x  2 and
19. The following information is about the function h
and the composite function h2 . h : x  x2  4x  3 , find
(a) g 1(6)
h : x  ax  b, where a and b are constants, (b) hg(x)
a > 0 h2 : x  36x  35
[4 marks]
Find the value of a and of b . 08P1. 2

[3 marks] 22. Given the functions f (x)  x 1 and
g(x)  kx  2 , find
2008 P1
(a) f(5),
y (b) the value of k such that gf(5) = 14.

[3 marks]
08P1. 3

1 5 x
t

(a) the value of t,
(b) the range of f(x) corresponding to the given domain.

21

CHAPTER 2 – QUADRATIC EQUATIONS

In this subtopic you will learn to :
1. Understand the concept of quadratic equation and its roots

1.1 Recognise a quadratic equation and express it in general form.
1.2 Determine whether a given value is the root of a quadratic equation by

a) substitution;
b) inspection.
1.3 Determine roots of quadratic equations by trial and improvement method

.

1. Understand the concept of quadratic equation and its roots.
IMPORTANT NOTES :

(i) The general form of a quadratic equation is ax2 + bx + c = 0; a, b, c are constants and a ≠ 0.
(ii) Characteristics of a quadratic equation:

(a) Involves only ONE variable,
(b) Has an equal sign “ = ” and can be expressed in the form ax2 + bx + c = 0,
(c) The highest power of the variable is 2
Recognising Quadratic Equations
EXAMPLES1

No Quafratic Equations (Q.E.) NON Q.E. WHY?

1. x2 + 2x -3 = 0 2x – 3 = 0 No terms in x2 ( a = 0)

2. x2 = ½ x2  2 = 0 2
3. 4x = 3x2 x
Term
x3 – 2 x2 = 0
x

Term x3

4. 3x (x – 1) = 2 x2 – 3x -1 + 2 = 0 Term x -1

5. p – 4x + 5x2 = 0, p constant x2 – 2xy + y2 = 0 Two variables

Exercise 1
1. State whether each of the following equations is a quadratic equation or not.

Equations yes or no Give your reason
1. 2x + 4 = 0 no The highest power of x is one .It’s a linear equation

2. x2 + 4 = 0

3. 3x2 + 2x + 3 = 0

4. 5x = 2 - 3x
5. y(2 – 3y) = 7
6. p(3p – 2) = 4 + 2p

7. 1 – 2x =0
x2
8. (x – 7)2 = 6

9. 5xy + 6 = 0

22

1 Recognise a quadratic equation and express it in general form.
1.1 Rewrite the following quadratic equations in general form and find the value of a, b and c.

Example 1 Example 2 Example 3
x2 – 2x = 3 (3x + 1)(x-3) = 4
x2  1  10 x
x2 – 2x – 3 = 0 3

a = 1, b = – 2, c = – 3

a) 2x2 = 3x – 4 a) x(2x – 1) = x + 5 a) 2 (x2  3x)  x
3

b) n(2n -1 ) = 3n b) (3x + 2)2 = 8 b) 2x2 1  1 x
4

c) x2 1  5x  3p c) 3 x2 – 5 = 4x(1 –x) c) 2x2  x  5
d) 3x2  4x  1 2 p 3
d) x2  2mx  3x  5
d) 2x2  5x  p(1 2x)

Homework : Text Book page 26 Exercise 2. 1.1

1.2 Determine whether a given value is the root of a quadratic equation by

a) substitution;

b) inspection.
Note : The root of a quadratic equation is the value o the unknown in the equation which satisfies the equation .
If a value is given, it can be determined whether it is a root by substitution or inspection.

1.2.1 Determine whether the x value given are the roots of the following quadratic equations .

Quadratic equations x value Conclusions
12 – 2(1) – 3 = –4 Not satisfy the quadratic
x = 1, equations so x =1 is not
the root of QE

1. x2 – 2x – 3 = 0 x = –1, (–1)2 – 2(–1) – 3 = 0 Yes

x = 3, 32 – 2(3) – 3 = 0

x = –3, (–3)2 – 2(–3) – 3 = 12

23

Quadratic equations x value Conclusions

x = 1,

2. 3x2 – 5x – 12 = 0 x = 3,
x = –3,

x=–4 ,
3

x = 1,

3. (2 x + 1 ) ( x- 4 ) = 0 x = 4,
x = –3,

x=–1 ,
2

Homework : Text Book page 28 Exercise 2. 1.2

1.3 Determine roots of quadratic equations by trial and improvement method.
Trial and improvement method is a primitive method of repeated substitution of integers into a function or

polynomials to find solutions. (Synonymous to trial and error method)

Example 1.3
Find the roots of the quadratic equation x2 - 5x + 6 = 0 by using trial and improvement method

Solution: x x2 - 5x + 6 Trial x x2 - 5x + 6
Trial

First -3 Fourth 2

Second 3 Fifth -2

Third 1 Sixth 6

Conclusion : The roots of quadratic equation x2 - 5x + 6 =0 are ………………………………………………………
Homework : Text Book page 29 Exercise 2. 1.3 and Skill Practice 2.1 page 29

What we should learnt in this subtopic are :

2. Understand the concept of quadratic equations

. 2.1 Determine the roots of a quadratic equation by
a) factorisation;
b) completing the square
c) using the formula.

2.2 Form a quadratic equation from given roots

24

2.1 Determine the roots of a quadratic equation by

a) factorisation; b) completing the square c) using the formula d) using calculator

2. SOLVING QUADRATIC EQUATIONS

2.1 Factorisation

Example 1 Example 2

.x2 + 6x + 5 = 0 x+3 3x 4(x +3) = x(2x – 1)
( x + 3)(x + 2) = 0 x+2 2x 4x + 12 = 2x2 - x
.x + 3 = 0 or x + 2 = 0 2x2 – 5x – 12 = 0
.x = -3 x = - 2 x2 + 6 5x (2x + 3)(x – 4) = 0
2x + 3 = 0 or x – 4 = 0
Therefore, The roots of the equation are
.x = –3 and –2 3
.x = x = 4

2
Therefore, The roots of the equation are

.x =  3 and 4
2

Exercises 3
Solve the following quadratic equation by factorisation

1. x2 + 3x – 4 = 0 2. x2 –2x = 15

x  1,4 x  3,5
4. 3x2 – 7x + 2 = 0
3. 4x2 + 4x – 3 = 0

x  1, 3 x  1, 2
22 3

5. 8x2 + 10x – 3 = 0 6) . 6x2 + 5x – 4 = 0 x  1, 4
23
x  1, 3
42 25

2.2. Completing the square

Example 1 Example 2

x2 – 6x + 7 = 0 Rearrange in the 2x2 – 5x – 1 = 0
x2 – 6x = –7 2x2 – 5x = 1
form
.x2 + px = q x2 – 5 x = 1
22
x2 – 6x +   6 2 = – 7 +   6 2 Change the

coefficient
of x2 to 1

2 2 x2 – 5 x +   5 2 = 1 +   5 2
2 4 2 4
x2 – 6x + (–3)2 = –7 + (–3)2 Add

(x – 3)2 = 2  coefficien t..of .x 2  x  5 2 = 1 + 25
2   4  2 16
x–3 =  2

x= 3  2 To both sides

x = 3 + 2 or 3 – 2 = 33
16

x = 4.414 or 1.586 x – 5 =  33 =  33
4 16 4

5 33 or 5 – 33
x= +
44 44

x = 2.686 or – 0.186

Exercise 4
Solve the following quadratic equations by completing the square:

1. (x + 1)(x – 5) = 4 2. 3x2 + 6x – 2 = 0

x  5.6056, 1.6056 x  0.2910, 2.2980
4. 2x2 – 3x – 4 = 0
3. 5x2 – 7x + 1 = 0

x  1.2385, 0.1615 x  2.3508,  0.8508

26

5. 1 x2  1 x  3 6. (5x – 4)2 = 24
42

x = 2.606 or x = - 4.606 x 1.780,  0.1798

2.3 Quadratic formula

EXAMPLE : Solve the following quadratic equation by using formula  b  b2  4ac

x=

2a

and give your answer correct to 4.s.f

1. x (x – 3 ) = 5 2. 3x2 = 5(x+2)

x = 4.193 , x = –1.193 x = 2.840 or x = - 1.174
. 27

Exercise 5
Solve the following quadratic equations by using the quadratic formula

1. 3x2 + 6x – 2 = 0 2. – x2 – 3x + 5 = 0

x  0.2910, 2.2980 x  4.1923, 1.1926
4. (x – 1)(4x – 9) + 7 = 10x
1 x 7
3. + 3 =

x x5

x  0.4037,  3.0963 x  4.9403, 0.8097

Note : Solve all quadratic equations above using calculator
Homework : Text Book page 32 Exercise 2. 2.1

2.2 Form a quadratic equation from given roots

1. If a and b are the roots of a quadratic equation then 28
x = a or x = b

x – a = 0 or x – b = 0
(x – a)(x – b) = 0, hence x 2 – (a + b)x + ab = 0

Therefore , the quadratic equation with roots P and q is
x2 – ( a+b) x + ab = 0

2. The Step of forming a quadratic equation from given roots are

i. Find the sum of the roots
ii. Find the product of the roots
iii. Form a quadratic equation by writing in a following form

x2 – ( sum of the roots ) x + product of the roots = 0

Example 2.2(i) Form the quadratic equation whose roots are shown below

a). 3 and -5 2 c) 3 and 2

b). 4 and

3

Exercise 2.2 b). 1 and  2 1 c) 4r and 5r
23
a) .  1
7

Example 2.2(ii) State the sum and product of the roots of the following quadratic equations.

a) . x2 - 9x - 4 = 0 b). 3x2 + 5x + 4 = 0 c) x(x – 1) = 2(1 – x)

Exercise 2.2(ii) State the sum and product of the roots of the following quadratic equations.

a) . x2  4x  5  0 b). 2x2 - 6x + 3 = 0 c) ). 2x2 +( t +2) x + t2 = 0

Example 2.2(ii)

If  and  are the roots of the equation 2x2 + 5x – 6 = 0, Form the equations whose roots are

a)  1 ,  1 b) 2 , 2


29

Exercise 2.2(ii)

If  and  are the roots of the equation 3x2 - 2x + 4 = 0, Form the equations whose roots are

a) 2 1 , 2 1 b) 1 , 1


Example 2.2(iii) Solve the following problems. 2. If one root of the equations 2x2 + x – c = 0 is two
times the other, find the value of c
1. If One root of the equations 27x2 + kx – 8 = 0 is
square the other .Find the value of k

.

3. Find q if the equation 3x2 - 4x + q = 0 has equal 4. If the roots of the equation x2 + px + 7 = 0 are
roots
denoted by  and  , and  2   2  22 .find the

possible values of p (camb)

30

5. Given that  and  are the roots of the equation 6. . Given that  and  are the roots of the equation

x2 – 2x + 3 = 0, Find a quadratic equation whose roots 2x2 – 3x + 4 = 0,Write down the value of

are  2   2 and 2 (Camb    and 2 . Find an equation whose roots

are   1 and   1 (camb)


Homework : Text Book page 34 Exercise 2. 2.2 and Skill Practice 2.2

3. Understand and use the conditions for quadratic equations to have
a) two different roots;
b) two equal roots;
c) no roots / no real roots

3.1 Determine types of roots of quadratic equations from the value of b2 - 4ac.

3.2 Solve problems involving

b2 - 4ac in quadratic equations to:

a) find an unknown value;
b) derive a relation.

3.1
Determine types of roots of quadratic equations from the value of b2  4ac.
For the quadratic equation ax2 + b x + c = o, the discriminant of the equation is b2  4ac
Types of roots of quadratic equations from the value of b2  4ac
(i) b2  4ac > 0 ….Two different roots ( the roots are distinct)
(ii) b2  4ac = 0 …Two same roots
(iii) b2  4ac < 0 …. No real roots

Example 3.1 Determine the type of the roots of the following quadratic equations

ax2 + bx + c = 0 abc b2 -4ac Type of roots
Two different roots
1. x2 + 5x + 6 = 0 156 1

2. x2 + 6x + 9 = 0

3. 4x2 - 4x + 1 = 0

4. 2x2 - 4x - 5 = 0

5. 2x2 - 5x + 4 = 0

Homework : Text Book page 36 Exercise 2. 3.1.

31

3.2 Solve problems involving b2 - 4ac in quadratic equations to:

a) find an unknown value;
b) derive a relation.

The value b2  4ac can be used to find the unknown value of coefficients or to derive a relation which involves

unknown in the quadratic equations

Example 3.2

a) Find k if x2 + 8x + k = 0 has equal roots b) Find p if 3x2 + 2x + 3p = 0,has two different roots

Exercise 3.2 Find The range of value of p if
a) Find The range of value of p if x2 + 2x +9 = p(2x – p) has two different roots

3x2 – 1 = 6x – 2p has two distinct roots

[p<2] [p > - 4]
c) .The quadratic equation 3x2 + 2x + h = 0 has
d) Show that the roots of the equations
equal roots . Find the value of h
6x – 6 -2px2 = x2 1

are complex if p >
4

1 f) Find the range of values of h for the quadratic equation
2x2 + 3x +4p = 1 which has no roots
[h= ]

3

e) Given that x2 +(p- 2)x + 10 - p = 0 has two

equal roots, find the values of p .

[p= 6 ] 17

Homework : Text Book page 37 Exercise 2. 3.2. and skill Practice 2.3 [p> ]

32
32

QUADRATIC EQUATIONS – SPM QUESTIONS

1. (1997)
Given that m + 2 and n – 1 are the roots of the equation x2  5x  4 . Find the possible values of m
and n.
[3 marks]

2. (1998) [1 mark]
The equation px2  px  3q  1 2x has the roots of 1 and q.
p

(a) Find the value of p and of q.

(b) Hence, by using the value of p and q from (a) , form the quadratic equation with roots p and – 2q.
[4 marks]

3. (1999)
Given that one of the root of the equation 2x2  6x  2k 1 is twice the other root,
where k is the constant. Find the roots and the value of k.

[4 marks]

4. (1999)
(a) Given that the equation of x2  6x  7  h(2x  3) has equal roots. Find the values of h .

[4 marks]
(b) Given that  and  are the roots of the equation x2  2x  k  0, while 2 and 2 are the

roots for the equation x2  mx  9  0. Calculate the possible values of k and m.
[6 marks]

33

5. (2000) [3 marks]
The quadratic equation 2x2  px  q  0 has the roots of - 6 and 3. Find [2 marks]
(a) the value of p and of q.

(b) the range of values of k such that 2x2  px  q  k has no real roots

6. (2001)
Given that 2 and m are the roots of the equation (2x 1)(x  3)  k(x 1) where k is the constant.

Find the value of m and of k.
[4 marks]

7. (2001)
If  and  are the roots of the equation 2x 2  3x  1  0 , form the quadratic equation with roots
3  2 and 3  2 .
[5 marks]

8. (2002)
Given that the quadratic equation x2  3  k(x 1), where k is the constant, has the roots of p and q .

Find the range of values of k if the equation has two different roots. [5 marks]

34

9. (2002)
Given that  and  are the roots of the equation kx(x 1)  2m  x. If     6 and   3, find
22
the value of k and of m.
[5 marks]

10. (2003) [3 marks]
Solve the quadratic equation 2x(x  4)  (1 x)(x  2) . Give your answer correct to

four significant figures.

11. (2003)
The quadratic equation x(x 1)  px  4 has two distinct roots. Find the range of values of p.

[3 marks]

12. (2004)

Form the quadratic equation which has the roots - 3 and 1 . Give your answer in the form
2

ax2  bx  c  0, where a , b and c are constants. [2 marks]

13. (2005)
The straight line y = 5x – 1 does not intersect the curve y  2x2  x  p. Find the range of values of p.

[3 marks]

35

14. (2005)
Solve the quadratic equation x(2x  5)  2x 1. Give your answer correct to three decimal places.

[3 marks]

15. (2006)
A quadratic equation x2  px  9  2x has two equal roots. Find the possible values of p.

[3 marks]

16. (2007)
(a) Solve the following quadratic equation: 3x2  5x  2  0
(b) The quadratic equation hx2  kx  3  0, where h and k are constants, has two
equal roots. Express h in terms of k.

[4 marks]

17. (2008-P1.Q4)
It is given that - 1 is one of the roots of the quadratic equation x2  4x  p  0 . Find the value of p.

[2 marks]

ANSWERS 7. 2x2  x 19  0 13. p  1
8. k  6,k  2 14. x = 8.153 , 0.149
1. m = -3 , - 6 n = - 3 , 0
9. k   1 , m  3 15. p = 8 , p = - 4
2. (a) p  2 , q  1 2 16 16. (a) x  1 ,  2 (b) h  k 2

32 10. x = 2.591 , - 0.2573 3 12

(b) 3x2  x  2  0 11. p  3 , p  5 17. p = 5
3. k   3 12. 2x2  5x  3  0

2

4. (a) h = - 1 , - 2
(b) m  4 , k  9

4

5. (a) p = 6 , q = - 36
(b) k  40.5

6. k = 15 , m = 3

36

37

Chapter 3 – Quadratic Functions

1. Understand the concept of quadratic functions and their graphs.

1.1 Recognise quadratic functions.
1.2 Plot quadratic function graphs

a) based on given tabulated values;
b) by tabulating values based on given functions.
1.3 Recognise shapes of graphs of quadratic functions.
1.4 Relate the position of quadratic function graphs with types of roots for f (x)  0.

1. Understand the concept of quadratic functions and their graphs.

1.1 Recognise quadratic functions.
The general form of quadratic functions is f (x) = ax2 + b x + c . The highest power of variable x is 2

Example 1 .1

Determine whether the given functions is a quadratic functions . If so Express it in the general form

a) f (x) = - 3x + 2x2 – 5 b) f (x) = ( 2x – 1 )(3x +1) c) f (x) = (x-3)2 – 8

Exercise 1.1
Determine whether the given functions is a quadratic functions . If so Express it in the general form

a) f (x) = - 5x + 3x2 – 5 b) f (x) = ( 3x – 4 )(2x +1) c) f (x) = ( x -2 )2 + 8

Home work : Text book page 44 Exercise 3.1.1
1.2 Plot quadratic function graphs

a) based on given tabulated values;
b) by tabulating values based on given functions.

Example 1.2

For each of the following quadratic function, complete the table given and plot the graph of the function f(x)

x 3 2 1 0 1 2 3 x 3 2 1 0 1 2 3

y 9 0 5 6 3 4 15 y
(b) f(x) = 4 – 3x – x2
(a) f(x) = 2x2 - x- 6

16 y

x

14

12 y  2x2  x  6
0

10
x

8

6

4x

2

3 x2 1 123 4x

2 x

-4
x

6x

Home work : Text book page 45 Exercise 3.1.2

37

1.3 Recognise shapes of graphs of quadratic functions.

Note : Maximum and minimum values of ax2 + b x +c
If a > o the function has a minimum value and the shape of a graph is
If a < 0 the function has a maximum value and the shape of a graph is

Example 1.3
Identify the shape of the graph of the following quadratic function

a) f (x) = 6 + 4x – 2x2 b) f (x) = x2 – 4x + 3 c) f (x) = ( 4 – x) ( 2 x - 5)

Home work : Text book page 46 Exercise 3.1.3

1.4 Relate the position of quadratic function graphs with types of roots for f (x)  0.

Note : Quadratic functions

Position of quadratic function b2 -4ac a>0 a< 0
f(x) = ax2 + bx + c

(i) a > 0 =0
b2 -4ac < 0
b2 -4ac = 0 b2 -4ac > 0

(ii) a < 0 x - axis <0
x - axis

b2 -4ac < 0 b2 -4ac =0 b2 -4ac > 0

>0

Example 1.4(i) Determine the type of the roots of the following quadratic equations for f(x) = 0 and sketch the position

of the graph f (x) relative to the x – axis

a) f (x)  6 - x - x2 b) f (x)  4x2 + 4x + 1 c) f (x)  5x2 - x + 1

Exercise 1.4 (i) Determine the type of the roots of the following quadratic equations for f(x) = 0 and sketch the position

of the graph f (x) relative to the x – axis

a) f (x)  x2 – 3x + 5 b) f (x)  2 + 3x – 2x2 d) f (x)  x2 – 4x + 4

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Example 1.4 (ii) b) Find the range of values of k for C) Find the range of values of m
a) Find the value of p if the graph of which the quadratic function
the quadratic function which the quadratic function f(x) = x2 - 8x + 5 m - 4 does not
f(x) = x2 - 2px + 4p + 5 g(x) = 2x2 - 12x + 3 -k has two intersect the x-axis . [ m >4 ]
touches the x axis at one point.
[ p = 5 or p=-1] x – intercept

[ k > -15]

Exercise 1.4(ii) b) ) Find the range of values of k for a) Show that the graph of quadratic
a) Find the value of p if the graph function
of the quadratic which the quadratic function g(x) =
f(x) = px2 + (2p + 6 )x + 5p-3 (p+5)x2 - 8x + 8 has two f(x) = t x2 + (3 - 2t)x - 5 + t intercept the
touches the x axis at one point.
[ p = -3/4, 3] x – intercept [ p <-3] 9

x-axis in two distinct point if t > -

8

Home work : Text book page 49 Exercise 3.1.4 and Skill Practice 3.1 page 50.
2. Find the maximum and minimum values of quadratic functions.
2.1 Determine the maximum or minimum value of a quadratic function by completing the
square.

Finding the maximum and minimum values of quadratic functions.
Note : A maximum or a minimum value of a quadratic function can be expressed in the form

a ( x + p )2 + q , where a, p, and q are constants by completing the square .
This can be done as follows

f (x) = ax2 + b x + c
=

39

2.1 Determine the maximum or minimum value of a quadratic function by completing the square.

Example 2.1 (i)

In each of the following, state the maximum or minimum value of f ( x) and the corresponding value of x

a) f(x) = 3 (x + 5)2 + 6 b) g(x) = 8 - 2(x-9)2 c) h(x) = 16 - x2

Class exercises
In each of the following, state the maximum or minimum value of f ( x) and the corresponding value of x

a) f(x) = 15 - 3(x + 4)2 b) h(x) = (x - 5)2 + 8 c) m(x) = 4 - x2

Example 2.1(ii)

Find the maximum or minimum values of the following quadratic function and the values of x when these

occur by completing the square

a) f(x) = 2x2 - 6x + 7 b) g(x) = 4 + 12x - 3x2

Class exercises

Find the maximum or minimum values of the following quadratic function and the values of x when these

occur by completing the square

a) h(x) = 2x - 1 - 3x2 [ max = -2/3 , x = 1/3] b) s(x) = 3x2 - 4x - 2 [ min = -10/3 , x = 2/3]

Home work : Text book page 52 Skill Practice 3.2.

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3. Sketch graphs of quadratic functions
3.1 Sketch quadratic function graphs by determining the maximum or minimum point

and two other points

A quadratic function can be sketched by the following step :

Identify the value of a

 the shape of the graph a  0 : 
a 0 : 

Find the value of b2 – 4ac  position of the graph

Determine max or min point by a > 0, f (x) minimum
completing the square minimum value is q when x = - p
f (x) = a (x + p)2 + q a < 0, f (x) maximum
maximum value is q when x = -p

Find intersect of

a) x – axis , f(x) = 0
b) y – axis , x = 0

Example 3.1 b) Find the maximum value of the quadratic function
a) Sketch the graph of quadratic function f(x) = 2x - 3 - x2 , then state the axis of symmetry and
sketch the graph.
y = 3x2 + 5x + 2 .State the axis of symmetry and
also the coordinate of the maximum or the minimum
point.

Class exercises b) Sketch the graph of quadratic function
a) Find the maximum value of the quadratic function f (x) = 2 + 3x - 2x2 . .State the axis of symmetry and also
y = 2x2 + 2x + 2, then state the axis of symmetry and the coordinate of the maximum or the minimum point.
sketch the graph.

Home work : Text book page 55 Skill Practice 3.3.

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4. Understand and use the concept of quadratic inequalities.
4.1 Determine the ranges of values of x that satisfies quadratic inequalities

QUADRATIC INEQUALITIES.

The range of values of x which satisfies this inequality can be found from this step
i. Find the intersects of x - axis
ii. Determine the shape of the graph
iii Sketch the graph
iv From the graph the range of values of x which satisfies this inequality can be found

Alternative method of finding the range of values of x to draw a number line. Expressions involving more
than two factors can be similarly treated .

Example 4.1 Method 2: Number line
Find the range of value of x if 2x2 < 5x + 3
Method 1: Graph

2. Find the range of value of x for the following quadratic inequalities.

a) (x  4)(x  2)  0 b) (2x + 1)(x + 3) > 7
x3

c) Find the range of values of p which the quadratic d) Given that y = tx2 + 8x + 10 - t, Find the range of values
equations 3x2 – p x +2p = 0 does not has a real
of t where y always positive . [ 2 < t <8 ]

roots [ 0 < p < 24]

Exercise 4.1

a) Find the smallest of k with the condition that the b) Find the range of p if y = 2x - 2p does not intersect
equation kx2 + (2k – 15)x + k = 0 does not have at the curve x2 + 2y2 = 8.
[ p < - 3, p > 3 ]

a real root. [ k = 4]

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c. Given that the quadratic equation d. . Find the range of values of x for which
2x2  2mx  5m  12 has two roots, find the − x2 + 2x + 15 < 0
range of value m.

Ans: m  4, m  6 Ans: x  3, x  5

Find the range of value of x for which 2x2 + x ≥ 3 Find the range of value of x for which x8  x  12

x 1 , x  3 Ans: 2  x  6
2
[3 marks] 99P1
Home work : Text book page 56 Skill Practice 3.4 Review Exercise page 40

QUADRATIC FUNCTIONS – SPM QUESTIONS

1. Find the range of values of x if given (x  2)(2x + 3) > (x  2)(x + 2).

2. Without using the method of differentiation or drawing the graph, find the maximum [4 marks]00P.1.5]
or minimum value for the function y = 1 + 2x  3x2. Hence, find the equation of the

axis of symmetry of the graph.

3. Find the range of values of k so that the straight line y = 2x + k does not intersects the curve

x2 + y2  6 = 0. [5 marks] 00P.1.13(a)

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4. Given y = p + qx  x2 = k  (x + h)2. [3 marks]
(a) Find, [1 marks]
(i) h, (ii) k, in terms of p and/or q.
(b) If q = 2, state the axis of symmetry of the curve. [6 marks] 02P1.14
(c) The straight line y = 3 touches the curve y = p + qx  x2.
(i) Express p in terms q.

(ii) Hence, sketch the graph of the curve.

5. The function f(x) = x2  4kx + 5k2 + 1 has a minimum value r2 + 2k, where r and k [4 marks]
are constant. [4 marks] 03P.2.2

(a) With the method of completing the square, shows that r = k  1.
(b) Hence, or otherwise, find the value of k and value of r if the graph of the function

is symmetry at x = r2  1.

6. The straight line y = 5x  1 does not intersects the curve y = 2x2 + x + p. Find the range of values of p.
[3 marks]

05P.1.4

7. Diagram shows the graph of quadratic function f(x) = 3(x + p)2 + 2, where p is a
constant.
y
y = f(x)

 [3 marks] 05P.1.6
(1, q)
Ox
The graph y = f(x) has minimum point (1, q), where q is a constant.
State,
(a) the value of p,
(b) the value of q,
(c) the equation of axis of symmetry.

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8. Diagram shows the graph of a quadratic function y = f(x). The straight line y = –4 is tangent to the curve
y = f(x).

y

y = f(x)

O1 5 x

y = –4

(a) Write the equation of the axis of symmetry of the curve. [3 marks]
(b) Express f(x) in the form of (x + b)2 + c, where b and c are constants. 06P.1.4

9. Find the range of the values of x for (2x – 1)(x + 4) > 4 + x. [2 marks]
06P.1.5

10. Find the range of values of x for which 2x2  1 + x . [3 marks]
07P1.5

11. The quadratic function f(x) = x2 + 2x – 4 can be expressed in the form
f(x) = (x + m)2 – n , where m and n are constants. Find the value of m and of n.

[3 marks]
07P1.6

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12. The quadratic function f(x) = p(x + q)2 + r, where p, q and r are constants, has a
minimum value of – 4. The equation of the axis of symmetry is x = 3. State
(a) the ranges of values of p,

(b) the value of q,

(c) the value of r.

[3 marks]
08P1.5

13. Find the ranges of the values of x for (x – 3)2 < 5 – x . [3 marks]
08P1.6

14. Diagram 2 shows the curve of a quadratic function f(x) = - x2 +kx – 5. The curve has
a maximum point at B(2,p) and intersects the f(x)-axis at point A.
f(x)

x

O B(2,p)

A

(a) States the coordinates of A. [1 mark]
(b) By using the method of completing the square, find the value of k and of p.
[4 marks]
(c) Determine the ranges of values of x, if f(x)  - 5. [2 marks]

08P2.2

Answers.

1. x < 1, x > 2 q 7.(a) 1 (b) 2 11. m = 1 and n = 5
(c) x = 1
41 . 4.(a)(i) h =  12. (a) p > 0 (b) - 3
8. (a) x = 3 (c) - 4
2. y max. = , x = 2 (b) (x – 3)2 – 4
13. 1 < x < 4
33 q2 9. x < –4, x > 1
3. k <  30 or 14. (a) A(0, - 5)
(ii) k = p + 1 (b) k = 4 , p = - 1
k > 30 (c) 0  x  4
4 10. -  x  1
5. (b) r = 1, k = 0;
r = 3, k = 4. (b) x = 1 2

(c)(i) p = 12  q 2 . 46

4

6. p > 1

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Chapter 4 – Simultaneous Equations

1. Solve simultaneous equations in two unknowns: one linear equation and one non -linear equation.

1.1 Solve simultaneous equations using the substitution method.
1.2 Solve simultaneous equations involving real-life situations.

1.1 Simultaneous linear and non-linear equations in two variables

Step to solve the simultaneous equations
i. Identify the linear equation
ii. Make one of the variables the subject of the equation

iii. Substitute this variable in the second equation, giving a quadratic equation in one
variable.

iv. Solve as a quadratic equation.

Acti vi ty 1
State whether each of the following equation are linear or non- linear equation.

Equation Linear/non- linear

1. x – 2y = 7
2. x2 + 5y2 = 49

3. x(y – 2x) = 1

4. 2x + 3y = 14
5. x2 + xy + y2 = 7

6. 4  15  5
xy

7. x – 2y = 2

8. xy = 4

9. y = 2x + 3
10. y – x2 = 2x

Example 1 Example 2 [ x = 12.19 , y = 0.55 , x = 5.14 , y = - 1.22
Solve the following simultaneous equations :
x+y =6 Solve the following simultaneous equations and give the
2x2 + y2 = 27 answer correct to two decimal places.

x  3y2  7
2

x – 4y = 10

Exercise 1 [Ans x = -2 y = 5 , x=5 y = -2 ]

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