The document provides a summary of mathematics formulae for Form 4 students. It includes: 1) Common functions and their derivatives such as absolute value, inverse, quadratic, and fractional functions. 2) Key concepts in algebra including the quadratic formula, nature of roots, and forming quadratic equations from roots. 3) Essential statistics measures like mean, median, variance, and standard deviation. 4) Formulas for coordinate geometry topics like distance, gradient, parallel and perpendicular lines, and locus equations. 5) Rules for differentiation including algebraic, fractional, and chain rule.

1. ONE-SCHOOL.NET
Add Maths Formulae List: Form 4 (Update 18/9/08)
01 Functions
Absolute Value Function Inverse Function
If y = f ( x ) , then f −1 ( y ) = x
f ( x ), if f ( x ) ≥ 0
f ( x) Remember:
− f ( x), if f ( x ) < 0 Object = the value of x
Image = the value of y or f(x)
f(x) map onto itself means f(x) = x
02 Quadratic Equations
General Form Quadratic Formula
ax 2 + bx + c = 0
−b ± b 2 − 4ac
where a, b, and c are constants and a ≠ 0. x=
2a
*Note that the highest power of an unknown of a
quadratic equation is 2. When the equation can not be factorized.
Forming Quadratic Equation From its Roots: Nature of Roots
If α and β are the roots of a quadratic equation
b c
α +β =− αβ = b 2 − 4ac >0 ⇔ two real and different roots
a a
b 2 − 4ac =0 ⇔ two real and equal roots
The Quadratic Equation b 2 − 4ac <0 ⇔ no real roots
x 2 − (α + β ) x + αβ = 0 b 2 − 4ac ≥0 ⇔ the roots are real
or
x − ( SoR ) x + ( PoR ) = 0
2
SoR = Sum of Roots
PoR = Product of Roots
http://www.one-school.net/notes.html 1

2. ONE-SCHOOL.NET
03 Quadratic Functions
General Form Completing the square:
f ( x) = ax 2 + bx + c f ( x) = a ( x + p)2 + q
where a, b, and c are constants and a ≠ 0. (i) the value of x, x = − p
(ii) min./max. value = q
*Note that the highest power of an unknown of a (iii) min./max. point = (− p, q)
quadratic function is 2. (iv) equation of axis of symmetry, x = − p
Alternative method:
a > 0 ⇒ minimum ⇒ ∪ (smiling face)
f ( x) = ax 2 + bx + c
a < 0 ⇒ maximum ⇒ ∩ (sad face)
b
(i) the value of x, x = −
2a
b
(ii) min./max. value = f (− )
2a
b
(iii) equation of axis of symmetry, x = −
2a
Quadratic Inequalities Nature of Roots
a > 0 and f ( x) > 0 a > 0 and f ( x) < 0
b 2 − 4ac > 0 ⇔ intersects two different points
at x-axis
a b a b b − 4ac = 0 ⇔ touch one point at x-axis
2
b 2 − 4ac < 0 ⇔ does not meet x-axis
x < a or x > b a< x<b
04 Simultaneous Equations
To find the intersection point ⇒ solves simultaneous equation.
Remember: substitute linear equation into non- linear equation.
http://www.one-school.net/notes.html 2

3. ONE-SCHOOL.NET
05 Indices and Logarithm
Fundamental if Indices Laws of Indices
Zero Index, a0 = 1 a m × a n = a m+n
1
Negative Index, a −1 =
a a m ÷ a n = a m−n
a
( ) −1 =
b ( a m ) n = a m× n
b a
1 ( ab) n = a n b n
Fractional Index an = a n
m a n an
an = a n m
( ) = n
b b
Fundamental of Logarithm Law of Logarithm
log a y = x ⇔ a x = y log a mn = log a m + log a n
log a a = 1 log a
m
= log a m − log a n
n
log a a x = x
log a mn = n log a m
log a 1 = 0
Changing the Base
log c b
log a b =
log c a
1
log a b =
logb a
http://www.one-school.net/notes.html 3

4. ONE-SCHOOL.NET
06 Coordinate Geometry
Distance and Gradient
Distance Between Point A and C =
(x1 − x2 )2 + (x1 − x2 )2
y2 − y1
Gradient of line AC, m =
x2 − x1
Or
⎛ y − int ercept ⎞
Gradient of a line, m = − ⎜ ⎟
⎝ x − int ercept ⎠
Parallel Lines Perpendicular Lines
When 2 lines are parallel, When 2 lines are perpendicular to each other,
m1 = m2 . m1 × m2 = −1
m1 = gradient of line 1
m2 = gradient of line 2
Midpoint A point dividing a segment of a line
⎛ x1 + x2 y1 + y2 ⎞ A point dividing a segment of a line
Midpoint, M = ⎜ , ⎟ ⎛ nx + mx2 ny1 + my2 ⎞
⎝ 2 2 ⎠ P =⎜ 1 , ⎟
⎝ m+n m+n ⎠
http://www.one-school.net/notes.html 4

5. ONE-SCHOOL.NET
Area of triangle:
Area of Triangle
1
=
2
1
A=
2
( x1 y2 + x2 y3 + x3 y1 ) − ( x2 y1 + x3 y2 + x1 y3 )
Form of Equation of Straight Line
General form Gradient form Intercept form
ax + by + c = 0 y = mx + c x y
+ =1
a b
m = gradient
c = y-intercept b
a = x-intercept m=−
b = y-intercept a
Equation of Straight Line
Gradient (m) and 1 point (x1, y1) 2 points, (x1, y1) and (x2, y2) given x-intercept and y-intercept given
given
y − y1 = m( x − x1 ) y − y1 y2 − y1 x y
= + =1
x − x1 x2 − x1 a b
Equation of perpendicular bisector ⇒ gets midpoint and gradient of perpendicular line.
Information in a rhombus:
A B
(i) same length ⇒ AB = BC = CD = AD
(ii) parallel lines ⇒ mAB = mCD or mAD = mBC
(iii) diagonals (perpendicular) ⇒ mAC × mBD = −1
(iv) share same midpoint ⇒ midpoint AC = midpoint
D BD
C (v) any point ⇒ solve the simultaneous equations
http://www.one-school.net/notes.html 5

6. ONE-SCHOOL.NET
Remember:
y-intercept ⇒ x = 0
cut y-axis ⇒ x = 0
x-intercept ⇒ y = 0
cut x-axis ⇒ y = 0
**point lies on the line ⇒ satisfy the equation ⇒ substitute the value of x and of y of the point into the
equation.
Equation of Locus
( use the formula of The equation of the locus of a The equation of the locus of a moving
distance) moving point P ( x, y ) which is point P ( x, y ) which is always
The equation of the locus of a always at a constant distance equidistant from two fixed points A and B
moving point P ( x, y ) which from two fixed points is the perpendicular bisector of the
is always at a constant A ( x1 , y1 ) and B ( x2 , y 2 ) with straight line AB.
distance (r) from a fixed point a ratio m : n is
A ( x1 , y1 ) is PA = PB
PA m ( x − x1 ) + ( y − y1 ) 2 = ( x − x2 ) 2 + ( y − y2 ) 2
2
=
PA = r PB n
( x − x1 ) 2 + ( y − y1 ) 2 = r 2 ( x − x1 ) 2 + ( y − y1 ) 2 m 2
=
( x − x2 ) + ( y − y 2 ) 2 n 2
More Formulae and Equation List:
SPM Form 4 Physics - Formulae List
SPM Form 5 Physics - Formulae List
SPM Form 4 Chemistry - List of Chemical Reactions
SPM Form 5 Chemistry - List of Chemical Reactions
All at
http://www.one-school.net/notes.html 6

7. ONE-SCHOOL.NET
07 Statistics
Measure of Central Tendency
Grouped Data
Ungrouped Data
Without Class Interval With Class Interval
Mean
Σx Σ fx Σ fx
x= x= x=
N Σf Σf
x = mean x = mean x = mean
Σx = sum of x Σx = sum of x f = frequency
x = value of the data f = frequency x = class mark
N = total number of the x = value of the data
data (lower limit+upper limit)
=
2
Median
m = TN +1 m = TN +1 ⎛ 1N −F⎞
2 2 m = L+ ⎜ 2
⎜ f ⎟C
⎟
When N is an odd number. When N is an odd number. ⎝ m ⎠
m = median
TN + TN TN + T N L = Lower boundary of median class
+1 +1
m= 2 2
m= 2 2 N = Number of data
2 2 F = Total frequency before median class
When N is an even When N is an even number. fm = Total frequency in median class
number. c = Size class
= (Upper boundary – lower boundary)
Measure of Dispersion
Grouped Data
Ungrouped Data
Without Class Interval With Class Interval
variance
σ =2∑ x2 −x
2
σ =
2 ∑ fx 2 −x
2
σ =
2 ∑ fx 2 −x
2
N ∑f ∑f
σ = variance σ = variance σ = variance
Σ(x − x )
2
Σ(x − x )
2
Standard Σ f (x − x)
2
σ= σ=
Deviation N N σ=
Σf
Σx 2 Σx 2
σ= − x2 σ= − x2 Σ fx 2
N N σ= − x2
Σf
http://www.one-school.net/notes.html 7

8. ONE-SCHOOL.NET
The variance is a measure of the mean for the square of the deviations from the mean.
The standard deviation refers to the square root for the variance.
Effects of data changes on Measures of Central Tendency and Measures of dispersion
Data are changed uniformly with
+k −k ×k ÷k
Measures of Mean, median, mode +k −k ×k ÷k
Central Tendency
Range , Interquartile Range No changes ×k ÷k
Measures of
Standard Deviation No changes ×k ÷k
dispersion
Variance No changes × k2 ÷ k2
08 Circular Measures
Terminology
Convert degree to radian:
Convert radian to degree:
π
xo = ( x × )radians
180 180
×
π 180
x radians = ( x × ) degrees
π
radians degrees
π
×
180
Remember:
180 = π rad 1.2 rad
O
??? 0.7 rad ???
360 = 2π rad
http://www.one-school.net/notes.html 8

9. ONE-SCHOOL.NET
Length and Area
r = radius
A = area
s = arc length
θ = angle
l = length of chord
Arc Length: Length of chord: Area of Sector: Area of Triangle: Area of Segment:
s = rθ θ 1 2 1 2 1 2
l = 2r sin A= rθ A= r sin θ A= r (θ − sin θ )
2 2 2 2
09 Differentiation
Differentiation of a Function I
Gradient of a tangent of a line (curve or
straight) y = xn
dy δy dy
= lim ( ) = nx n−1
dx δ x →0 δ x dx
Example
y = x3
Differentiation of Algebraic Function
dy
Differentiation of a Constant = 3x 2
dx
y=a a is a constant
dy
=0 Differentiation of a Function II
dx
y = ax
Example dy
y=2 = ax1−1 = ax 0 = a
dx
dy
=0
dx Example
y = 3x
dy
=3
dx
http://www.one-school.net/notes.html 9

10. ONE-SCHOOL.NET
Differentiation of a Function III Chain Rule
y = ax n y = un u and v are functions in x
dy dy dy du
= anx n−1 = ×
dx dx du dx
Example Example
y = 2 x3 y = (2 x 2 + 3)5
dy du
= 2(3) x 2 = 6 x 2 u = 2 x 2 + 3, therefore = 4x
dx dx
dy
y = u5 , therefore = 5u 4
du
Differentiation of a Fractional Function
dy dy du
= ×
1 dx du dx
y=
xn = 5u 4 × 4 x
Rewrite = 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4
y = x−n
dy −n Or differentiate directly
= − nx − n−1 = n+1 y = (ax + b) n
dx x
dy
= n.a.(ax + b) n −1
Example dx
1
y=
x y = (2 x 2 + 3)5
y = x −1 dy
= 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4
dy −1 dx
= −1x −2 = 2
dx x
Law of Differentiation
Sum and Difference Rule
y =u±v u and v are functions in x
dy du dv
= ±
dx dx dx
Example
y = 2 x3 + 5 x 2
dy
= 2(3) x 2 + 5(2) x = 6 x 2 + 10 x
dx
http://www.one-school.net/notes.html 10

11. ONE-SCHOOL.NET
Product Rule Quotient Rule
y = uv u and v are functions in x u
y= u and v are functions in x
dy du dv v
= v +u
dx dx dx du dv
v −u
dy dx dx
=
Example dx v2
y = (2 x + 3)(3 x 3 − 2 x 2 − x)
Example
u = 2x + 3 v = 3x3 − 2 x 2 − x
x2
du dv y=
=2 = 9 x2 − 4 x − 1 2x +1
dx dx
u = x2 v = 2x +1
dy du dv
=v +u du dv
dx dx dx = 2x =2
dx dx
=(3 x − 2 x − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1)
3 2
du dv
v −u
dy
= dx 2 dx
Or differentiate directly dx v
y = (2 x + 3)(3x3 − 2 x 2 − x) dy (2 x + 1)(2 x) − x 2 (2)
=
dy dx (2 x + 1) 2
= (3x3 − 2 x 2 − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1)
dx 4 x2 + 2 x − 2 x2 2 x2 + 2 x
= =
(2 x + 1) 2 (2 x + 1) 2
Or differentiate directly
x2
y=
2x +1
dy (2 x + 1)(2 x) − x 2 (2)
=
dx (2 x + 1) 2
4 x2 + 2 x − 2 x2 2 x2 + 2 x
= =
(2 x + 1) 2 (2 x + 1) 2
http://www.one-school.net/notes.html 11

12. ONE-SCHOOL.NET
Gradients of tangents, Equation of tangent and Normal
Gradient of tangent at A(x1, y1):
dy
= gradient of tangent
dx
Equation of tangent: y − y1 = m( x − x1 )
Gradient of normal at A(x1, y1):
1
mnormal = −
mtangent
If A(x1, y1) is a point on a line y = f(x), the gradient 1
of the line (for a straight line) or the gradient of the = gradient of normal
− dy
dy dx
tangent of the line (for a curve) is the value of
dx Equation of normal : y − y1 = m( x − x1 )
when x = x1.
Maximum and Minimum Point
dy
Turning point ⇒ =0
dx
At maximum point, At minimum point ,
2
dy d y dy d2y
=0 <0 =0 >0
dx dx 2 dx dx 2
http://www.one-school.net/notes.html 12

13. ONE-SCHOOL.NET
Rates of Change Small Changes and Approximation
Small Change:
dA dA dr
Chain rule = ×
dt dr dt δ y dy dy
≈ ⇒ δ y ≈ ×δ x
δ x dx dx
dx
If x changes at the rate of 5 cms -1 ⇒
=5 Approximation:
dt
Decreases/leaks/reduces ⇒ NEGATIVES values!!! ynew = yoriginal + δ y
dy
= yoriginal + ×δ x
dx
δ x = small changes in x
δ y = small changes in y
If x becomes smaller ⇒ δ x = NEGATIVE
http://www.one-school.net/notes.html 13

14. ONE-SCHOOL.NET
10 Solution of Triangle
Sine Rule: Cosine Rule: Area of triangle:
a
a b c a2 = b2 + c2 – 2bc cosA
= = b2 = a2 + c2 – 2ac cosB
sin A sin B sin C C
c2 = a2 + b2 – 2ab cosC
b
Use, when given
2 sides and 1 non included b2 + c2 − a 2
cos A =
angle 2bc 1
2 angles and 1 side A= a b sin C
2
Use, when given
2 sides and 1 included angle C is the included angle of sides a
a a B 3 sides and b.
A
A a a c
b 180 – (A+B)
A
b b
http://www.one-school.net/notes.html 14

15. ONE-SCHOOL.NET
Case of AMBIGUITY If ∠C, the length AC and length AB remain unchanged,
the point B can also be at point B′ where ∠ABC = acute
A
and ∠A B′ C = obtuse.
If ∠ABC = θ, thus ∠AB′C = 180 – θ .
180 - θ
θ Remember : sinθ = sin (180° – θ)
C B′ B
Case 1: When a < b sin A Case 2: When a = b sin A
CB is too short to reach the side opposite to C. CB just touch the side opposite to C
Outcome: Outcome:
No solution 1 solution
Case 3: When a > b sin A but a < b. Case 4: When a > b sin A and a > b.
CB cuts the side opposite to C at 2 points CB cuts the side opposite to C at 1 points
Outcome: Outcome:
2 solution 1 solution
Useful information:
In a right angled triangle, you may use the following to solve the
c problems.
b (i) Phythagoras Theorem: c = a 2 + b2
θ
a Trigonometry ratio:
(ii)
sin θ = b , cos θ = a , tan θ =
c c
b
a
(iii) Area = ½ (base)(height)
http://www.one-school.net/notes.html 15

16. ONE-SCHOOL.NET
11 Index Number
Price Index Composite index
P Σ Wi I i
I = 1
× 100 I=
P0 Σ Wi
I = Price index / Index number I = Composite Index
P0 = Price at the base time W = Weightage
P1 = Price at a specific time
I = Price index
I A, B × I B ,C = I A,C ×100
http://www.one-school.net/notes.html 16