VIEWS: 315 PAGES: 16 CATEGORY: Algebra POSTED ON: 2/26/2012 Public Domain
ONE-SCHOOL.NET 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 α and β are the roots of a quadratic equation b c α +β =− αβ = 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 − (α + β ) x + αβ = 0 b 2 − 4ac ≥0 ⇔ the roots are real or x − ( SoR ) x + ( PoR ) = 0 2 SoR = Sum of Roots PoR = Product of Roots http://www.one-school.net/notes.html 1 ONE-SCHOOL.NET 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) min./max. value = q *Note that the highest power of an unknown of a (iii) min./max. 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) min./max. 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. http://www.one-school.net/notes.html 2 ONE-SCHOOL.NET 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 Changing the Base log c b log a b = log c a 1 log a b = logb a http://www.one-school.net/notes.html 3 ONE-SCHOOL.NET 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 ⎠ http://www.one-school.net/notes.html 4 ONE-SCHOOL.NET 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. 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 http://www.one-school.net/notes.html 5 ONE-SCHOOL.NET 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 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 2 ( x − x1 ) + ( y − y1 ) 2 = ( x − x2 ) 2 + ( y − y2 ) 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 http://www.one-school.net/notes.html 6 ONE-SCHOOL.NET 07 Statistics Measure of Central Tendency Grouped Data Ungrouped Data Without Class Interval With Class Interval Mean Σx Σ fx Σ fx x= x= x= N Σf Σf x = mean x = mean x = mean Σx = sum of x Σx = 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 ⎜ f ⎟C ⎟ When N is an odd number. When N is an odd number. ⎝ m ⎠ 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 σ =2∑ x2 −x 2 σ = 2 ∑ fx 2 −x 2 σ = 2 ∑ fx 2 −x 2 N ∑f ∑f σ = variance σ = variance σ = variance Σ(x − x ) 2 Σ(x − x ) 2 Standard Σ f (x − x) 2 σ= σ= Deviation N N σ= Σf Σx 2 Σx 2 σ= − x2 σ= − x2 Σ fx 2 N N σ= − x2 Σf http://www.one-school.net/notes.html 7 ONE-SCHOOL.NET 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 08 Circular Measures Terminology Convert degree to radian: Convert radian to degree: π xo = ( x × )radians 180 180 × π 180 x radians = ( x × ) degrees π radians degrees π × 180 Remember: 180 = π rad 1.2 rad O ??? 0.7 rad ??? 360 = 2π rad http://www.one-school.net/notes.html 8 ONE-SCHOOL.NET Length and Area r = radius A = area s = arc length θ = angle l = length of chord Arc Length: Length of chord: Area of Sector: Area of Triangle: Area of Segment: s = rθ θ 1 2 1 2 1 2 l = 2r sin A= rθ A= r sin θ A= r (θ − sin θ ) 2 2 2 2 09 Differentiation Differentiation of a Function I Gradient of a tangent of a line (curve or straight) y = xn dy δy dy = lim ( ) = nx n−1 dx δ x →0 δ 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 http://www.one-school.net/notes.html 9 ONE-SCHOOL.NET 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 = n.a.(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 = ± dx dx dx Example y = 2 x3 + 5 x 2 dy = 2(3) x 2 + 5(2) x = 6 x 2 + 10 x dx http://www.one-school.net/notes.html 10 ONE-SCHOOL.NET 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 http://www.one-school.net/notes.html 11 ONE-SCHOOL.NET 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 ) 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 http://www.one-school.net/notes.html 12 ONE-SCHOOL.NET Rates of Change Small Changes and Approximation Small Change: dA dA dr Chain rule = × dt dr dt δ y dy dy ≈ ⇒ δ y ≈ ×δ x δ x dx dx dx If x changes at the rate of 5 cms -1 ⇒ =5 Approximation: dt Decreases/leaks/reduces ⇒ NEGATIVES values!!! ynew = yoriginal + δ y dy = yoriginal + ×δ x dx δ x = small changes in x δ y = small changes in y If x becomes smaller ⇒ δ x = NEGATIVE http://www.one-school.net/notes.html 13 ONE-SCHOOL.NET 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 sin A sin B sin C C c2 = a2 + b2 – 2ab cosC b Use, when given 2 sides and 1 non included b2 + c2 − a 2 cos A = angle 2bc 1 2 angles and 1 side A= a b sin C 2 Use, when given 2 sides and 1 included angle C is the included angle of sides a a a B 3 sides and b. A A a a c b 180 – (A+B) A b b http://www.one-school.net/notes.html 14 ONE-SCHOOL.NET Case of AMBIGUITY If ∠C, the length AC and length AB remain unchanged, the point B can also be at point B′ where ∠ABC = acute A and ∠A B′ C = obtuse. If ∠ABC = θ, thus ∠AB′C = 180 – θ . 180 - θ θ Remember : sinθ = sin (180° – θ) C B′ B Case 1: When a < b sin A Case 2: When a = b sin A CB is too short to reach the side opposite to C. CB just touch the side opposite to C Outcome: Outcome: No solution 1 solution Case 3: When a > b sin A but a < b. Case 4: When a > b sin A and a > b. CB cuts the side opposite to C at 2 points CB cuts the side opposite to C at 1 points Outcome: Outcome: 2 solution 1 solution Useful information: In a right angled triangle, you may use the following to solve the c problems. b (i) Phythagoras Theorem: c = a 2 + b2 θ a Trigonometry ratio: (ii) sin θ = b , cos θ = a , tan θ = c c b a (iii) Area = ½ (base)(height) http://www.one-school.net/notes.html 15 ONE-SCHOOL.NET 11 Index Number Price Index Composite index P Σ Wi I i I = 1 × 100 I= P0 Σ Wi I = Price index / Index number I = Composite Index P0 = Price at the base time W = Weightage P1 = Price at a specific time I = Price index I A, B × I B ,C = I A,C ×100 http://www.one-school.net/notes.html 16