1 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 and are the roots of a quadratic equation b c + = = 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 ( + ) x + = 0 b 2 4ac 0 the roots are real or x ( SoR ) x + ( PoR ) = 0.
2 2. SoR = Sum of Roots PoR = Product of Roots 1. 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) value = q *Note that the highest power of an unknown of a (iii) 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) value = f ( ). 2a b (iii) equation of axis of symmetry, x =.
3 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. 2. 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.
4 Changing the Base log c b log a b =. log c a 1. log a b =. logb a 3. 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.
5 2 2 P = 1 , . m+n m+n . 4. 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.
6 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 5. 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.
7 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 6.
8 07 Statistics Measure of Central Tendency Grouped Data Ungrouped Data Without Class Interval With Class Interval Mean x fx fx x= x= x=. N f f x = mean x = mean x = mean x = sum of x x = 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 C. When N is an odd number. When N is an odd number. fm . 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.
9 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 =2 x2. x 2. =. 2 fx 2. x 2. =. 2 fx 2. x 2. N f f = variance = variance = variance (x x ). 2. (x x ). 2. Standard f (x x). 2. = =. Deviation N N =. f x 2 x 2. = x2 = x2 fx 2. N N = x2. f 7. 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.
10 08 Circular Measures Terminology Convert degree to radian: Convert radian to degree: . D xo = ( x )radians 180 180.. 180. x radians = ( x ) degrees . radians degrees .. 180D. Remember: 180D = rad rad O. ??? rad ??? 360 = 2 rad D. 8. Length and Area r = radius A = area s = arc length = angle l = length of chord Arc Length: Length of chord: Area of Sector: Area of Triangle: Area of Segment: s = r 1 2 1 2 1 2. l = 2r sin A= r A= r sin A= r ( sin ). 2 2 2 2. 09 Differentiation Differentiation of a Function I. Gradient of a tangent of a line (curve or straight) y = xn dy y dy = lim ( ) = nx n 1.