S4 Math Revision Formula Chapter 1 : Number System (p.1 of 2)

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1 Integer 整數 are numbers for counting. S4 Math Revision Formula Chapter : Number Sstem (p. of ) include all positive integers, zero and negative integers. eample :, 0, 05, etc. Rational Number 有理數 numbers which can be epressed as p, where p, q are integers and q 0. q include all integers 整數, fractions 分數, terminating decimal 有盡小數 and recurring decimal 循環小數. 5 eample :, 0, 05,, 0., 0., etc. 6 Irrational Number 無理數 numbers whose decimal form is neither repeated or terminated eample : π, surd form like, 5, 5, etc. Real Number 實數 the union of rational and irrational numbers. 5 eample :, 0, 05,, 0., 0., π,, 5, 5, etc. 6 Purel Imaginar Number 純虛數 number in the form bi, where b is a non-zero real number and i =. eample : i, 4i, etc. Comple Number 複數 number in the form a + bi, where a, b are non-zero real numbers and i =. in a + bi, a is the real part 實部. b is the imaginar part 虛部. i is the imaginar unit 虛數單位. besides the number with a + bi form, comple number also includes all real numbers (when b = 0) and all purel imaginar numbers (when a = 0 and b 0 ). 5 eample :, 0, 05,, 0., 0., π,, 5, 5, i, 4i, 6 + i 5i, etc. For a comple number a + bi, if the imaginar part b = 0, then the comple number is a real number. if the real part a = 0 and the imaginar part b 0, then the comple number is a purel imaginar number.

2 Chapter : Number Sstem (p. of ) Eample of perations in Comple Numbers Addition / Subtraction (Same as that of polnomial) ( + i ) + (4 7i) = i 7i = 6 4i ( + i ) (4 7i) = + i 4 + 7i = 6 + 0i Multiplication (Same as that of polnomial, note that i = ) ( + i )(4 7i) = (4 7i) + i(4 7i) = 8 4i + i + i = 8 i + () = i Division (Note the aim is to eliminate i in the denominator) 9 + i 9 i = + i i i = + i ( i ) = + (use i = ) i = i + = i 4 7i + i 4 7i i = + i i (4 7i)( i) = (i) = = 8 i 4i + i 4 9( ) 6i = i (use (a + b)(a b) = a b ) Equalit of Comple Numbers If a + bi = c + di, then a = c, b = d. e.g. If + i = i, then = and =.

3 Chapter : Equations of Straight Lines (p. of ) S topic (I) Distance Formula 距離公式 : distance between A and B, length of AB AB = ( ) + ( ) (II) Slope 斜率 : slope of AB m = 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. Inclination 傾角 : In the figure, if the inclination of L is θ, where 0 o θ < 80 o, then m = tan θ. L θ m = m. (III) If A, B and C are collinear, then AB BC If two lines are parallel, their slope are equal. m = m If two lines are perpendicular to each others, the product of slopes is. m m = B(, ) + + (IV) mid-point 中點 M =, M A(, ) (V) Point of Division 分點 Given that the coordinates of A and B are (, ) and (, ). If P is a point on the line segment AB such that AP : PB = r : s, s + r s + r then P =,. r + s r + s B(, ) P A(, )

4 Chapter : Equations of Straight Lines (p. of ) S4 topic (I) Equation of Straight Line. Two Point Form 兩點式 =. Point Slope Form 點斜式 = m( ) ( or. Slope Intercept Form 斜截式 = m + b 4. Intercepts Form 截距式 + = a b 5. General Form 一般式 A + B + C = 0 = m ) (II) Importance of Equation The coordinates of an point on the line must satisf the equation of the line. The coordinates of an point not on the line will never satisf the equation of the line. (III) Use General Form A + B + C = 0 to find the slope and intercepts slope = A B -intercept = C A -intercept = C B and To find -intercept, put = 0 in the equation. To find -intercept, put = 0 in the equation. To find the slope, make the subject of the equation (slope-intercept form), the coefficient of 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 L and L have no intersecting point, then the are parallel lines A B C and thus =. A B C If L and L have one intersecting point, then A B. If L and L have infinitel man intersecting points, then the represent the same A B C straight line and thus = =. A B C A B 4

5 Chapter : Quadratic Equations in ne Variable (p. of ). Solving Quadratic Equations (a) B Factorization Theor : If mn = 0, then m = 0 or n = 0. e.g = = 0 (4 )( + ) = 0 = or = 4 (b) B Taking Square Root Theor : If = k, then (c) Graphical Method Theor : = ± k. The -intercepts of the graph of = a + b + c are the roots of the quadratics equation a + b + c = 0. e.g. = 0 + From the graph, the roots of 0 + = 0 8 are and 8. (d) B Formula Quadratic Formula = b ± b 4ac a 首先睇定 a b c, 負 b 加減開方根, b 二次減 4 a c, 除埋 a 好 eas! 5

6 Chapter : Quadratic Equations in ne Variable (p. of ). Discriminant 判別式 : = b 4ac and the nature of roots : If b 4ac > 0, then the equation has two unequal / distinct real roots. If b 4ac = 0, then the equation has a double (real) root. If b 4ac < 0, then the equation has no real root. If the equation has two unequal / distinct real roots, then b 4ac > 0. If the equation has a double (real) root, then b 4ac = 0. If the equation has no real root, then b 4ac < 0.. Let α and β be the roots of the equation a + b + c = 0. Then Sum of roots α + β = b a Product of roots αβ = c a 4. Constructing Quadratic Equations Method : The reverse process of factorization e.g. Construct a quadratic equation whose roots are and. 4 Soln : = or = 4 4 = 0 or + = 0 (4 )( + ) = = 0 Method : Using (sum of roots) + (product of roots) = 0 6

7 Chapter 4 : Introduction to Functions Representation of Functions:. b table graphical method = f () = f (). algebraic method e.g. = + 4 f () = + 4 The notation f () can make substitution easier to write. e.g. Let f () = + 4. Then f () = () + 4 = 0 f () = () + 4 = 5 f (k) = k + 4 f ( + ) = ( + ) + 4 = + 7 7

8 Chapter 5 : Quadratic Functions (p. of ) Quadratic Equation:a + b + c = 0 (from which ou can find, or in other words, solve equation) The equation of a quadratic function: = a + b + c (from which ou can draw the graph on the plane) Important Words = 5 8 (5, 8) n the left, -intercepts are and 8. -intercept is. ais of smmetr is = 5. The verte is (5, 8). It is also the minimum point. The minimum value of is 8. To find the -intercept, substitute = 0. The -intercepts are the roots of a + b + c = 0. To find the -intercept, substitute = 0. The -intercepts is c in = a + b + c. 8

9 Chapter 5 : Quadratic Functions (p. of ) What a, b, c and will change the graph of = a + b + c a : a > 0 open upward a < 0 open downward The bigger the value of a (ignore + or ), the thinner the graph is. b : If a and b have same sign, then the verte is on the left of the -ais. If a and b have different signs, then the verte is on the right of the -ais. If b = 0, then the verte is on the -ais. c : -intercept (Note : The -intercepts are the roots of a + b + c = 0. ) : = b 4ac > 0:the graph cuts -ais at points ( -intercepts) = 0:the graph cuts -ais at point ( -intercept) < 0:the graph does not cut the -ais (no -intercept) Coordinates of the Verte If a quadratic function is written as = a( h) + k, then the ais of smmetr is = h, the verte is (h, k), If a > 0, then = h gives a minimum value and the minimum of is k. If a < 0, then = h gives a maimum value and the maimum of is k. The method of completing the square is used to change = a + b + c into = a( h) + k. 9

10 Chapter 6 : More About Polnomials Division Algorithm In f () g(), where Q() is the quotient and R () is the remainder, we have f () = g() Q() + R (). Remainder Theorem If f () is a polnomial, then. In f () ( a), the remainder R = f (a).. In f () (m n), the remainder R = f ( n m ). Factor Theorem Given a polnomial f ().. If f (a) = 0, then f () is divisible b a. ( a is a factor of f (). ). If f ( n ) = 0,,then f () is divisible b m n. (m n is a factor of f (). ) m The Converse of Factor Theorem Given a polnomial f ().. If f () is divisible b a ( a is a factor of f (). ), then f (a) = 0.. If f () is divisible b m n (m n is a factor of f (). ), then f ( n m ) = 0. Factor theorem is useful in () factorizing polnomial f (), () solving equations f () = 0. Equations and Identit Equation Equalit holds for some values of. e.g. + = 5 (onl hold for = ) Identit Equalit holds for all value of. e.g. ( + ) = + In a identit, we use to replace =. e.g. ( + ) + 0

11 Chapter 7 : Eponential Function S topic Law of indices. a m a n = a m+n. a m a n = a mn for a 0.. (a m ) n = (a n ) m = a mn 4. (ab) n = a n b n 5. n a n a ( ) = n b b for b a 0 = for a n a = n a for a 0. S4 topic Law of indices n 8. a n = a 9. for a > 0, n > 0. m n n m = a n = ( a m for a > 0, n > 0. a ) Graph of Eponential Function = a = a = a a > 0 < a <

12 Chapter 8 : Logarithmic Function Definition of Logarithm with base a (where a > 0, a ) If a =, then = log a. Properties of Logarithm with base a (where a > 0, a ). log a M + log a N = log a MN. log a M log a N = log a M N. log a M n = n log a M 4. log a a = 5. log a = 0 6. change of base formula log = a log log b b a Graph of logarithmic Function = log a = a = a = log a = log a a > 0 < a <

13 Chapter 0 : Rational Function Highest Common Factor (HCF) (It is Factor, thus a smaller one) Take the factor onl all epressions have, and take the smallest degree. Least Common Multiple (LCM) (It is Multiple, thus a bigger one) Take the factor for an epression has, and take the highest degree. The HCF of a,a b,a 4 is a. The LCM of a,a b,a 4 is a 4 b. Identities for revision. a + ab + b = (a + b). a ab + b = (a b). a b = (a + b)(a b) 4. a + b = (a + b)(a ab + b ) 5. a b = (a b)(a + ab + b ) Chapter : Basic Properties of Circle Chapter : More Basic Properties of Circle Refer to Plane-Geo-note-v

14 Chapter : Elementar Trigonometr (p. of ) S topic 對邊 opp sin θ = 斜邊 hp 鄰邊 adj cos θ = 斜邊 hp 對邊 opp tan θ = 鄰邊 adj 對邊 opp 斜邊 hp 鄰邊 adj θ 對斜鄰斜對鄰 H A H A S Topic Trigonometric ratio of special angles 特殊角的三角比 θ 0 o 45 o 60 o sin θ cos θ tan θ ( ) ( ) ) ( ) ( ( ) Trigonometric identities sinθ. tan θ = cosθ cosθ sinθ (can get =, cos θ =,cos θ tan θ = sin θ, etc. ) tanθ sinθ tanθ. sin θ + cos θ = (can get cos θ = sin θ, sin θ = cos θ, etc. ). sin (90 o θ ) = cos θ,cos (90 o θ ) = sin θ, tan( 90 θ) = tan θ 4

15 Chapter : Elementar Trigonometr (p. of ) S4 Topic Angle of rotation : measuring from the positive -ais anticlockwise gives positive angle, while clockwise gives negative angle 象限 II 象限 I Positive angle Negative angle 象限 III 象限 IV Definition of trigonometric ratio for all angles. P (, ) cos θ = r sin θ = r Note that r = + tan θ = r θ Reduction Formula ( 蝴蝶 4 兄弟 : 不必轉三角比 ) sin 0 o sin cos ( 80 o ± θ ) = ± cos θ tan 60 o tan The ± on the right side is based on CAST rule. ( 沙漏 4 姊妹 : 要轉三角比 ) sin 90 o cos θ 70 o cos ( ± θ ) = ± sin θ tan tanθ The ± on the right side is based on CAST rule. 80 o Sin Tan 90 o 70 o All Cos 0 o, 60 o 5

16 Chapter : Elementar Trigonometr (p. of ) S4 Topic (cont.) Graph of Trigonometric Functions = sin, 0 o 60 o = cos, 0 o 60 o o o 60 o o 70 o 60 o = tan, 0 o 60 o o 70 o 60 o From the graph, we have The maimum values and minimum values of the trigonometric functions are cos sin < tan < 6