Download S4 Math Revision Formula

S4 Math Revision Formula
Chapter 1 : Number System (p.1 of 2)
Integer 整數
-
are numbers for counting.
-
include all positive integers, zero and negative integers.
-
example : 3 , 0 , -2015 , etc.
Rational Number 有理數
p
, where p , q are integers and q ¹ 0 .
q
-
numbers which can be expressed as
-
include all integers 整數 , fractions 分數 , terminating decimal 有盡小數 and
-
recurring decimal 循環小數.
·
5
, 0.12 , 0.12 , etc.
example : 3 , 0 , -2015 ,
6
Irrational Number 無理數
-
numbers whose decimal form is neither repeated or terminated
example : p , surd form like 2 , 3 5 , 3 5 , etc.
Real Number 實數
-
the union of rational and irrational numbers.
·
5
example : 3 , 0 , -2015 ,
, 0.12 , 0.12 , p ,
6
2 , 3 5 ,
3
5 , etc.
Purely Imaginary Number 純虛數
-
number in the form bi , where b is a non-zero real number and
-
example : 3i , -4i , etc.
i = -1 .
Complex Number 複數
-
number in the form a + bi , where a , b are non-zero real numbers and
i = -1 .
- in a + bi , a is the real part 實部 . b is the imaginary part 虛部 . i is the
imaginary unit 虛數單位 .
-
besides the number with a + bi form, complex number also includes all real
-
numbers (when b = 0) and all purely imaginary numbers (when a = 0 and b ¹ 0 ).
·
5
example : 3 , 0 , -2015 ,
, 0.12 , 0.12 , p , 2 , 3 5 , 3 5 , 3i , -4i ,
6
2 + 3i、 2 - 5i , etc.
For a complex number a + bi ,
if the imaginary part b = 0 , then the complex number is a real number.
if the real part a = 0 and the imaginary part b ¹ 0 , then the complex number is a
purely imaginary number.
1
Chapter 1 : Number System (p.1 of 2)
Example of Operations in Complex Numbers
Addition / Subtraction (Same as that of polynomial)
(2 + 3i ) + (4 - 7i) = 2 + 4 + 3i - 7i
= 6 - 4i
(2 + 3i ) - (4 - 7i) = 2 + 3i - 4 + 7i
= -6 + 10i
Multiplication (Same as that of polynomial , note that i 2 = -1 )
(2 + 3i )(4 - 7i)
= 2(4 - 7i) + 3i(4 - 7i)
= 8 - 14i + 12i + 21 i 2
= 8 - 2i + 21(-1)
= -13 - 2i
Division (Note the aim is to eliminate i in the denominator)
9 + 2i
3i
=
=
=
=
=
4 - 7i
2 + 3i
9 2i
+
3i 3i
3 2
+
i 3
3(-i 2 ) 2
+
i
3
2
- 3i +
3
2
- 3i
3
æ 4 - 7i öæ 2 - 3i ö
= ç
֍
÷
è 2 + 3i øè 2 - 3i ø
(4 - 7i )(2 - 3i)
=
2 2 - (3i ) 2
(use -i 2 = 1)
(use (a + b)(a - b) = a2 - b2 )
8 - 12i - 14i + 21i 2
4 - 9(-1)
- 13 - 26i
=
13
= -1 - 2i
=
Equality of Complex Numbers
If a + bi = c + di , then a = c , b = d .
e.g.
If x + yi = 2 - i , then x = 2 and y = -1 .
2
Chapter 2 : Equations of Straight Lines (p.1 of 2)
S3 topic
(I) Distance Formula 距離公式: distance between A and B , length of AB
AB = ( x 2 - x1 ) 2 + ( y 2 - y1 ) 2
m=
(II) Slope 斜率: slope of AB
y 2 - y1
x2 - x1
m > 0：the straight line is going upward from left to right
m < 0：the straight line is going downward from left to right
m = 0：horizontal lines
Vertical Line : The slope is undefined.
y
Inclination 傾角：
In the figure, if the inclination of L is q ,
where 0o £ q < 180o , then m = tan q .
L
q
O
(III)
If A , B and C are collinear,
then
m AB = m BC
.
If two lines are parallel, their slope are equal.
m1 = m2
If two lines are perpendicular to each others, the product of slopes
is -1 . m1 ´ m2 = -1
(IV)
mid-point 中點
x
æ x + x 2 y1 + y 2 ö
M =ç 1
,
÷
2 ø
è 2
B(x2 , y2 )
M
A(x1 , y1 )
(V) Point of Division 分點
Given that the coordinates of A and B are (x1 , y1 ) and (x2 , y2 ) .
If P is a point on the line segment AB such that AP : PB = r : s ,
then
æ sx + rx 2 sy1 + ry 2 ö
P=ç 1
,
÷ .
r+s ø
è r+s
P
A(x1 , y1 )
3
B(x2 , y2 )
Chapter 2 : Equations of Straight Lines (p.2 of 2)
S4 topic
(I) Equation of Straight Line
1.
Two Point Form 兩點式
y - y1 y 2 - y1
=
x - x1 x 2 - x1
2.
Point Slope Form 點斜式
y - y1 = m(x - x1)
3.
Slope Intercept Form 斜截式
y = mx + b
x y
Intercepts Form 截距式
+ =1
a b
General Form 一般式
Ax + By + C = 0
4.
5.
( or
y - y1
=m )
x - x1
(II) Importance of Equation
The coordinates of any point on the line must satisfy the equation of the line.
The coordinates of any point not on the line will never satisfy the equation of the
line.
(III) Use General Form Ax + By + C = 0 to find the slope and intercepts
A
C
C
slope = x-intercept = y-intercept = B
A
B
To find x-intercept , put y = 0 in the equation.
To find y-intercept , put x = 0 in the equation.
To find the slope , make y the subject of the equation (slope-intercept form) ,
and
the coefficient of x is the slope of the line.
(IV) To find the point of intersection of two lines : Solve simultaneous equation.
(V) Condition for number of intersecting points of two lines :
If L1 and L2 have no intersecting point, then they are parallel lines
A
B
C
and thus 1 = 1 ¹ 1 .
A2 B2 C 2
A
B
If L1 and L2 have one intersecting point, then 1 ¹ 1 .
A2 B2
If L1 and L2 have infinitely many intersecting points, then they represent the same
A
B
C
straight line and thus 1 = 1 = 1 .
A2 B2 C 2
4
Chapter 3 : Quadratic Equations in One Variable (p.1 of 2)
1.
Solving Quadratic Equations
(a) By Factorization
Theory : If mn = 0 , then
m=0
or
n=0.
2
e.g.
4x + 5x = 6
4x2 + 5x - 6 = 0
(4x - 3)(x + 2) = 0
3
x=
or x = -2
4
(b) By Taking Square Root
Theory : If x2 = k , then x = ± k .
(c) Graphical Method
Theory :
The x-intercepts of the graph of y = ax2 + bx + c are the roots of the
quadratics equation ax2 + bx + c = 0 .
e.g.
y
y = 2x2 - 20x + 32
From the graph, the roots
of 2x2 - 20x + 32 = 0
O
2
x
8
are 2 and 8 .
(d) By Formula
Quadratic Formula
- b ± b 2 - 4ac
x=
2a
首先睇定 a、b、c，
負 b 加減開方根，
b 二次減 4 a c，
除埋 2 a 好 easy！
5
Chapter 3 : Quadratic Equations in One Variable (p.2 of 2)
2.
Discriminant 判別式：D = b2 - 4ac and the nature of roots :
If b2 - 4ac > 0 , then the equation has two unequal / distinct real roots.
If b2 - 4ac = 0 , then the equation has a double (real) root.
If b2 - 4ac < 0 , then the equation has no real root.
If the equation has two unequal / distinct real roots, then b2 - 4ac > 0 .
If the equation has a double (real) root, then b2 - 4ac = 0 .
If the equation has no real root, then b2 - 4ac < 0 .
3.
Let a and b be the roots of the equation ax2 + bx + c = 0 .
b
Then
Sum of roots
a+b =a
c
Product of roots ab =
a
4.
Constructing Quadratic Equations
Method 1 :
The reverse process of factorization
e.g.
Construct a quadratic equation whose roots are
Soln :
3
and -2 .
4
3
or x = -2
4
4x - 3 = 0 or x + 2 = 0
x=
(4x - 3)(x + 2) = 0
4x2 + 5x - 6 = 0
Method 2 :
Using
x2 - (sum of roots) x + (product of roots) = 0
6
Chapter 4 : Introduction to Functions
Representation of Functions：
1. by table
x
1
2
y
4
7
3
10
4
13
2. graphical method
y
y
y = f (x)
y = f (x)
x
3. algebraic method
e.g.
x
y = 3x + 4
f (x) = 3x + 4
The notation f (x) can make substitution easier to write.
e.g.
Let f (x) = 3x + 4 .
Then f (2) = 3(2) + 4 = 10
f (-3) = 3(-3) + 4 = -5
f (k) = 3k + 4
f (x + 1) = 3(x + 1) + 4
= 3x + 7
7
Chapter 5 : Quadratic Functions (p.1 of 2)
Quadratic Equation：ax2 + bx + c = 0
(from which you can find x , or in other words, solve equation)
The equation of a quadratic function：y = ax2 + bx + c
(from which you can draw the graph on the xy plane)
Important Words
x=5
y
On the left,
32
x-intercepts are 2 and 8 .
y-intercept is 32 .
2
8
x
axis of symmetry is x = 5 .
The vertex is (5 , -18) .
It is also
the minimum point.
(5 , -18)
The minimum value of y is -18 .
To find the x-intercept, substitute y = 0 .
The x-intercepts are the roots of ax2 + bx + c = 0 .
To find the y-intercept, substitute x = 0 .
The y-intercepts is c in y = ax2 + bx + c .
8
Chapter 5 : Quadratic Functions (p.2 of 2)
What a , b , c and D will change the graph of
a: a>0
a<0
open upward
È
open downward
Ç
y = ax2 + bx + c
The bigger the value of a (ignore + or -) , the “thinner” the graph is.
b : If a and b have same sign, then the vertex is on the left of the y-axis.
If a and b have different signs, then the vertex is on the right of the y-axis.
If b = 0, then the vertex is on the y-axis.
c:
y-intercept
(Note : The x-intercepts are the roots of ax2 + bx + c = 0 . )
D : D = b2 - 4ac
D > 0：the graph cuts x-axis at 2 points (2 x-intercepts)
D = 0：the graph cuts x-axis at 1 point (1 x-intercept)
D < 0：the graph does not cut the x-axis (no x-intercept)
Coordinates of the Vertex
If a quadratic function is written as y = a(x - h)2 + k , then
-
the axis of symmetry is x = h ,
-
the vertex is (h, k) ,
-
If a > 0 , then x = h gives y a minimum value and the minimum of y is k .
If a < 0 , then x = h gives y a maximum value and the maximum of y is k .
The method of completing the square is used to change y = ax2 + bx + c
into y = a(x - h)2 + k .
9
Chapter 6 : More About Polynomials
Division Algorithm
In f (x) ¸ g(x) , where Q(x) is the quotient and R (x) is the remainder, we have
f (x) = g(x) × Q(x) + R (x) .
Remainder Theorem
If f (x) is a polynomial, then
1.
2.
In f (x) ¸ (x - a) , the remainder R = f (a) .
n
In f (x) ¸ (mx - n) , the remainder R = f ( ) .
m
Factor Theorem
Given a polynomial f (x) .
1.
2.
If f (a) = 0 , then f (x) is divisible by x - a . (x - a is a factor of f (x) . )
n
If f ( ) = 0 ,，then f (x) is divisible by mx - n . (mx - n is a factor of f (x) . )
m
The Converse of Factor Theorem
Given a polynomial f (x) .
1.
2.
If f (x) is divisible by x - a (x - a is a factor of f (x) . ) , then f (a) = 0 .
n
If f (x) is divisible by mx - n (mx - n is a factor of f (x) . ) , then f ( ) = 0 .
m
Factor theorem is useful in
(1) factorizing polynomial f (x) ,
(2) solving equations f (x) = 0 .
Equations and Identity
Equation - Equality holds for some values of x . e.g. x + 3 = 5 (only hold for x = 2)
Identity - Equality holds for all value of x . e.g. 2(x + 1) = 2x + 2
In a identity, we use “º” to replace “=” . e.g. 2(x + 1) º 2x + 2。
10
Chapter 7 : Exponential Function
S3 topic
Law of indices
1. am ´ an = am+n
2.
am ¸ an = am-n
m n
n m
3.
(a ) = (a ) = a
4.
(ab)n = an ´ bn
a
an
( )n = n
b
b
5.
6.
7.
for a ¹ 0 .
mn
a0 = 1
for a ¹ 0 .
1
= n
a
a -n
for b ¹ 0 .
for a ¹ 0 .
S4 topic
Law of indices
1
8.
9.
an = n a
a
m
n
for a > 0 , n > 0 .
= n a m = (n a ) m
for a > 0 , n > 0 .
Graph of Exponential Function
y
y = ax
y = ax
y = ax
y
1
1
x
x
a>1
0<a<1
11
Chapter 8 : Logarithmic Function
Definition of Logarithm with base a (where a > 0 , a ¹ 1)
If
ax = y , then
x = loga y .
Properties of Logarithm with base a (where a > 0 , a ¹ 1)
1.
2.
loga M + loga N = loga MN
M
loga M - loga N = loga
N
3.
loga Mn = n loga M
4.
loga a = 1
5.
loga 1 = 0
6.
change of base formula
log a x =
Graph of logarithmic Function
y
logb x
log b a
y = log a x
y = ax
y = ax
y
y = log a x
x
x
y = log a x
a>1
0<a<1
12
Chapter 10 : Rational Function
Highest Common Factor (HCF)
(It is Factor, thus a smaller one)
Take the factor only all expressions have, and take the smallest degree.
Least Common Multiple (LCM) (It is Multiple, thus a bigger one)
Take the factor for any expression has, and take the highest degree.
The HCF of a2，a3b，a4 is a2 .
The LCM of a2，a3b，a4 is a4b .
Identities for revision
1.
a2 + 2ab + b2 = (a + b)2
2.
a2 - 2ab + b2 = (a - b)2
3.
a2 - b2 = (a + b)(a - b)
4.
a3 + b3 = (a + b)(a2 - ab + b2)
5.
a3 - b3 = (a - b)(a2 + ab + b2)
Chapter 11 : Basic Properties of Circle
Chapter 12 : More Basic Properties of Circle
Refer to “Plane-Geo-note-v2”
13
Chapter 12 : Elementary Trigonometry (p.1 of 3)
S2 topic
sin q =
對邊opp
斜邊hyp
斜邊 hyp
對邊 opp
鄰邊adj
cos q =
斜邊hyp
對
斜
鄰
斜
對
鄰
q
鄰邊 adj
tan q =
對邊opp
鄰邊adj
S3 Topic
Trigonometric ratio of special angles 特殊角的三角比
q
sin q
cos q
tan q
30o
1
2
(
3
2
1
3
45o
1
)
2
(
1
3
60o
2
2
2
2
)
1
3
2
1
2
(
2
2
)
3
(
1
)
2
3
( )
1
Trigonometric identities
sin q
1. tan q =
cosq
1
cosq
sin q
(can get
=
， cos q =
，cos q tan q = sin q , etc. )
tan q sin q
tan q
2.
3.
sin 2 q + cos 2 q = 1
(can get 1 - cos 2 q = sin 2 q，1 - sin 2 q = cos 2 q
, etc. )
sin (90o - q ) = cos q ，cos (90o - q ) = sin q， tan(90° - q ) =
14
1
tan q
O
H
A
H
O
A
Chapter 12 : Elementary Trigonometry (p.2 of 3)
S4 Topic
Angle of rotation : measuring from the positive x-axis anticlockwise gives positive
y
angle, while clockwise gives negative angle
y
y
象限 I
象限 II
x
O
x
O
Positive angle
x
Negative angle
O
象限 III
象限 IV
y
Definition of trigonometric ratio for all angles.
x
y
sin q =
r
r
2
2
2
Note that r = x + y 。
cos q =
tan q =
P (x , y)
y
x
r
q
x
O
Reduction Formula
(蝴蝶 4 兄弟：不必轉三角比)
0o
sin
cos
sin
o
( 180
±q) = ±
360o
tan
90o
cos q
tan
All
Sin
The “ ±” on the right side is based on “CAST” rule .
0o , 360o
180o
(沙漏 4 姊妹：要轉三角比)
cos
tan
cos q
90o
sin
±q) = ±
(
o
270
sin q
1
tan q
The “ ±” on the right side is based on “CAST” rule .
15
Cos
Tan
270o
Chapter 12 : Elementary Trigonometry (p.3 of 3)
S4 Topic (cont.)
Graph of Trigonometric Functions
y
y
y = sin x , 0o £ x £ 360o
1
1
270o
O
y = cos x , 0o £ x £ 360o
90
180
360o
o
180o
x
O
-1
270o
90
-1
y = tan x , 0o £ x £ 360o
y
O
90
180o
270o
360o
x
From the graph, we have
The maximum values and minimum values of the trigonometric functions are
-1 £ cos x £ 1
-1 £ sin x £ 1
-¥ < tan x < ¥
16
360o
x