1. NOTES AND FORMULAE ADDITIONAL MATHEMATICS FORM 4
1. FUNCTIONS 2 6 The axis of symmetry is x – p = 0 or
(a) Arrow Diagram fg(x) = f( )= 2 x=p
x x
(c) Quadratic inequalities
2. QUADRATIC EQUATION (x – a)(x – b) 0
2
(a) ax + bx + c = 0 Range x a, x b
2
x = b b 4ac (x – a)(x – b) 0
2a Range a x b
Sum of roots: 4. INDICES AND LOGARITHM
Domain = set A = {−2, 1, 2, 3}
b
Codomain = set B = {1, 4, 8, 9} n
Range = set of images = {1, 4, 9} a (a) x=a loga x = n
Object of 4 = −2, 2 Product of roots: Index Logarithm
Image of 3 = 9 c Form Form
2
In function notation f : x x a
(b) Types of Relation (b) Logrithm Law
(b) Equation from the roots: 1. logaxy = logax + logay
2
x - (sum of roots)x + product of 2. loga x = logax – logay
roots = 0 y
n
One-to-one One-to-many 3. loga x = n logax
3. QUADRATIC FUNCTION 4. loga a = 1
(a) Types of roots 5. loga 1 = 0
2
b - 4ac > 0 2 real and
6. loga b = log c b
distinct/different roots.
Many-to-one Many-to-many 2 log c a
b - 4ac = 0 2 real and equal
roots/two same roots. 7. loga b = 1
(c) Inverse Function 2
b - 4ac < 0 no real root. log b a
Function f maps set A to set B 2
-1 b - 4ac 0 got real roots.
Inverse function f maps set B to y y y
set A. 5. COORDINATE GEOMETRY
Given f(x) = ax + b, let y = ax + b (a) Distance between A(x1, y1) and
y b -1 xb B(x2, y2)
x= f :x 0 x
a a 0 AB = ( x 2 x1 ) 2 ( y 2 y1 ) 2
x
(d) Composite Function 0 x (b) Mid point AB
2 2 2
fg(x) means the function g followed b - 4ac > 0 b - 4ac = 0 b - 4ac < 0 M x1 x 2 , y1 y 2
by the function f.
2 2
2 (b) Completing the square
E.g. f : x 3x – 2, g : x 2 (c) P which divides AB in the ratio m : n
x y = a(x - p) + q
a +ve minimum point (p, q)
a –ve maximum point(p, q)
Prepared by Mr. Sim Kwang Yaw 1
2. m : n By Histogram :
x
fx Frequency
P B(x2 , y2 )
f
A(x 1 , y1 )
For ungrouped data with frequency.
P nx1 mx 2 ny1 my 2 fx
, x
i
nm nm f
(d) Gradient AB For grouped data, xi = mid-point
m = y 2 y1
x 2 x1 (b) Median
The data in the centre when 0 Mode Class boundary
m = y-intercept arranged in order (ascending or
x-intercept descending).
(e) Equation of straight line Measurement of Dispersion
(a) Interquartile Range
Formula Formula :
(i) Given m and A(x1, y1) 1
nF 1
M=L+ 2
C Q1 = L 4 n F1 C
y – y1 = m(x – x1)
fm 1
f Q1
(ii) Given A(x1, y1) and L = Lower boundary of median 3
class. Q3 = L 4 n F3 C
B(x2, y2) 3
f Q3
n = Total frequency
y y1 y 2 y1
F = cumulative frequency before the
x x1 x 2 x1 median class Ogive :
fm = frequency of median class Cumulative frequency
(a) Area of polygon C = class interval size
L = 1 x1 x2 x3 ......... x1
By Ogive 3
__
2 y1 y 2 y 3 y1 Cumulative Frequency 4
n
(g) Parallel lines
m1 = m2 n
1
__ n
(h) Perpendicular lines 4
m1 m2 = -1. n
__
2
0 Q Q 3 Upper boundary
6. STATISTICS 1
Measurement of Central Tendency Interquartile range = Q3 – Q1
(a) Mean
0 Median Upper boundary (b) Variance, Standard Deviation
x
x
n 2
(c) Mode Variance = (standard deviation)
For ungrouped data
Data with the highest frequency
Prepared by Mr. Sim Kwang Yaw 2
3. 2 d (xn) = nxn-1 9. SOLUTIONS OF TRIANGLES
( x x) (c) (a) The Sine Rule
n dx
sin A sin B sin C
d (axn) = anxn-1 , or,
= x2 x
2 (d) a b c
n dx
a b c
For ungrouped data
(e) Differentiation of product sin A sin B sin C
2 d (uv) = u dv + v du The Ambiguous Case
f ( x x)
dx dx dx
f
2
= fx x 2
(f) Differentiation of Quotient
f d u v du u dv
For grouped data dx 2 dx
dx v v
Two triangles of the same
7. CIRCULAR MEASURE (g) Differentiation of Composite measurements can be drawn given
(b) Radian Degree Function C, AC and AB where AB < AC.
0 d (ax+b)n = n(ax+b)n-1 × a
= 180
r
dx (b) The Cosine Rue
2 2 2
a = b + c – 2bc cos A
(c) Degree Radian (h) Stationary point
dy = 0
b 2 c2 a 2
= rad cos A =
o
dx
180 Maximum point: 2bc
(d) Length of arc dy = 0 and d 2 y < 0
s = r 10. INDEX NUMBER
dx dx 2 (a) Price Index
(e) Area of sector p1
2
Minimum point: I 100
A = 1 r = 1 rs 2 po
2 2 dy = 0 and d y > 0
p0 is the price in the base year.
(f) Area of segment dx dx 2
2
A = 1 r ( – sin ) (b) Composite Index
2 (i) Rate of Change
dy dy dx
I
IW
8. DIFFERENTIATION
(a) Differentiation by First Principle
dt dx dt W
dy had y I = price index
(j) Small changes: W = weightage
dx x 0 x dy
d (a) = 0 y . x
(b) dx
dx
Prepared by Mr. Sim Kwang Yaw 3