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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 ÷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