Add Maths Formulae List: Form 4 (Update 18/9/08)

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. 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.

2 (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 = . 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 =.

3 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. 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 . 2 2 P = 1 , . m+n m+n . 4. Area of triangle: Area of Triangle 1.

4 =. 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. 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.

5 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. 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. 07 Statistics Measure of Central Tendency Grouped Data Ungrouped Data Without Class Interval With Class Interval Mean x fx fx x= x= x=.

6 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. 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.

7 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. 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.

8 Dx x 0 x dx Example y = x3. Differentiation of Algebraic Function dy Differentiation of a Constant = 3x 2. dx y=a a is a constant dy =0 Differentiation of a Function II. dx y = ax Example dy y=2 = ax1 1 = ax 0 = a dx dy =0. dx Example y = 3x dy =3. dx 9. Differentiation of a Function III Chain Rule y = ax n y = un u and v are functions in x dy dy dy du = anx n 1 = . dx dx du dx Example Example y = 2 x3 y = (2 x 2 + 3)5. dy du = 2(3) x 2 = 6 x 2 u = 2 x 2 + 3, therefore = 4x dx dx dy y = u5 , therefore = 5u 4. du Differentiation of a Fractional Function dy dy du = . 1 dx du dx y=. xn = 5u 4 4 x Rewrite = 5(2 x 2 + 3) 4 4 x = 20 x(2 x 2 + 3) 4. y = x n dy n Or differentiate directly = nx n 1 = n+1 y = (ax + b) n dx x dy = (ax + b) n 1. Example dx 1. y=. x y = (2 x 2 + 3)5. y = x 1 dy = 5(2 x 2 + 3) 4 4 x = 20 x(2 x 2 + 3) 4. dy 1 dx = 1x 2 = 2. dx x Law of Differentiation Sum and Difference Rule y =u v u and v are functions in x dy du dv =.

9 Dx dx dx Example y = 2 x3 + 5 x 2. dy = 2(3) x 2 + 5(2) x = 6 x 2 + 10 x dx 10. Product Rule Quotient Rule y = uv u and v are functions in x u y= u and v are functions in x dy du dv v = v +u dx dx dx du dv v u dy dx dx =. Example dx v2. y = (2 x + 3)(3 x 3 2 x 2 x). Example u = 2x + 3 v = 3x3 2 x 2 x x2. du dv y=. =2 = 9 x2 4 x 1 2x +1. dx dx u = x2 v = 2x +1. dy du dv =v +u du dv dx dx dx = 2x =2. dx dx =(3 x 2 x x)(2) + (2 x + 3)(9 x 2 4 x 1). 3 2. du dv v u dy = dx 2 dx Or differentiate directly dx v y = (2 x + 3)(3x3 2 x 2 x) dy (2 x + 1)(2 x) x 2 (2). =. dy dx (2 x + 1) 2. = (3x3 2 x 2 x)(2) + (2 x + 3)(9 x 2 4 x 1). dx 4 x2 + 2 x 2 x2 2 x2 + 2 x = =. (2 x + 1) 2 (2 x + 1) 2. Or differentiate directly x2. y=. 2x +1. dy (2 x + 1)(2 x) x 2 (2). =. dx (2 x + 1) 2. 4 x2 + 2 x 2 x2 2 x2 + 2 x = =. (2 x + 1) 2 (2 x + 1) 2. 11. Gradients of tangents, Equation of tangent and Normal Gradient of tangent at A(x1, y1): dy = gradient of tangent dx Equation of tangent: y y1 = m( x x1 ).

10 Gradient of normal at A(x1, y1): 1. mnormal = . mtangent If A(x1, y1) is a point on a line y = f(x), the gradient 1. of the line (for a straight line) or the gradient of the = gradient of normal dy dy dx tangent of the line (for a curve) is the value of dx Equation of normal : y y1 = m( x x1 ). when x = x1. Maximum and Minimum Point dy Turning point =0. dx At maximum point, At minimum point , 2. dy d y dy d2y =0 <0 =0 >0. dx dx 2 dx dx 2. 12. Rates of Change Small Changes and Approximation Small Change: dA dA dr Chain rule = . dt dr dt y dy dy y x x dx dx dx If x changes at the rate of 5 cms -1 . =5 Approximation: dt Decreases/leaks/reduces NEGATIVES values!!! ynew = yoriginal + y dy = yoriginal + x dx x = small changes in x y = small changes in y If x becomes smaller x = NEGATIVE. 13. 10 Solution of Triangle Sine Rule: Cosine Rule: Area of triangle: a a b c a2 = b2 + c2 2bc cosA. = = b2 = a2 + c2 2ac cosB.