03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

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1 A-PDF Wtermrk DEMO: Purchse from to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f ( x ), then f ( y ) x Rememer: Oject the vlue of x Imge the vlue of y or f(x) f(x) mp onto itself mens f(x) x f ( x), if f ( x ) < 0 0 Qudrtic Equtions Qudrtic Formul Generl Form x + x + c 0 ± 4c x where,, nd c re constnts nd 0. *Note tht the highest power of n unknown of qudrtic eqution is. When the eqution cn not e fctorized. Nture of Roots Forming Qudrtic Eqution From its Roots: If α nd β re the roots of qudrtic eqution α +β αβ c 4c 4c 4c 4c The Qudrtic Eqution x (α + β ) x + αβ 0 or x ( SoR ) x + ( PoR ) 0 SoR Sum of Roots PoR Product of Roots >0 0 <0 0 two rel nd different roots two rel nd equl roots no rel roots the roots re rel

2 03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note tht the highest power of n unknown of qudrtic function is. the vlue of x, x p min./mx. vlue q min./mx. point ( p, q) eqution of xis of symmetry, x p Alterntive method: > 0 minimum (smiling fce) f ( x) x + x + c < 0 mximum (sd fce) Qudrtic Inequlities > 0 nd f ( x) > 0 the vlue of x, x (ii) min./mx. vlue f ( (iii) eqution of xis of symmetry, x x < or x > ) Nture of Roots > 0 nd f ( x) < 0 intersects two different points t x-xis 4c 0 touch one point t x-xis 4c < 0 does not meet x-xis 4c > 0 (i) < x< 04 Simultneous Equtions To find the intersection point solves simultneous eqution. Rememer: sustitute liner eqution into non- liner eqution.

3 05 Indices nd Logrithm Lws of Indices Fundmentl if Indices Zero Index, 0 m n m+n Negtive Index, m n m n ( m ) n m n ( ) Frctionl Index n m n ( ) n n n n n n n ( ) n m Fundmentl of Logrithm Lw of Logrithm log y x x y log mn log m + log n log log log x x log mn n log m log 0 m log m log n n Chnging the Bse 3 log log c log c log log

4 06 Coordinte Geometry Distnce nd Grdient Distnce Between Point A nd C (x x ) + (x x ) Grdient of line AC, m y y x x Or y int ercept Grdient of line, m x int ercept Prllel Lines Perpendiculr Lines When lines re prllel, When lines re perpendiculr to ech other, m m. m m m grdient of line m grdient of line Midpoint A point dividing segment of line A point dividing segment of line x + x y + y, Midpoint, M nx + mx ny + my P, m+n m+n 4

5 Are of tringle: Are of Tringle A ( x y + x y3 + x3 y ) ( x y + x3 y + x y3 ) Form of Eqution of Stright Line Generl form x + y + c 0 Grdient form Intercept form y mx + c x y + m grdient c y-intercept Eqution of Stright Line Grdient (m) nd point (x, y) points, (x, y) nd (x, y) given given y y y y y y m( x x ) x x x x x-intercept y-intercept m x-intercept nd y-intercept given x y + Eqution of perpendiculr isector gets midpoint nd grdient of perpendiculr line. Informtion in rhomus: A D B C sme length AB BC CD AD prllel lines mab mcd or mad mbc digonls (perpendiculr) mac mbd shre sme midpoint midpoint AC midpoint BD ny point solve the simultneous equtions (i) (ii) (iii) (iv) (v) 5

6 Rememer: y-intercept x 0 cut y-xis x 0 x-intercept y 0 cut x-xis y 0 **point lies on the line stisfy the eqution sustitute the vlue of x nd of y of the point into the eqution. Eqution of Locus ( use the formul of distnce) The eqution of the locus of moving point P ( x, y ) which is lwys t constnt distnce (r) from fixed point A ( x, y ) is The eqution of the locus of moving point P ( x, y ) which is lwys t constnt distnce from two fixed points A ( x, y ) nd B ( x, y ) with rtio m : n is PA r PA m PB n ( x x ) + ( y y ) r PA PB ( x x ) + ( y y ) ( x x ) + ( y y ) ( x x ) + ( y y ) m ( x x ) + ( y y ) n More Formule nd Eqution List: At The eqution of the locus of moving point P ( x, y ) which is lwys equidistnt from two fixed points A nd B is the perpendiculr isector of the stright line AB. 6

7 07 Sttistics Mesure of Centrl Tendency Ungrouped Dt Men x Without Clss Intervl Σx N x x men Σx sum of x x vlue of the dt N totl numer of the dt Medin Σ fx Σf TN + TN When N is n odd numer. TN + T N + m When N is n even numer. Σ fx Σf x men f frequency x clss mrk m TN + When N is n odd numer. m x x men Σx sum of x f frequency x vlue of the dt m TN + Grouped Dt With Clss Intervl + When N is n even numer. (lower limit+upper limit) N F C m L + fm m medin L Lower oundry of medin clss N Numer of dt F Totl frequency efore medin clss fm Totl frequency in medin clss c Size clss (Upper oundry lower oundry) Mesure of Dispersion Ungrouped Dt vrince x σ N x σ vrince Stndrd Devition Σ(x x ) σ N σ Σx x N Grouped Dt Without Clss Intervl With Clss Intervl fx σ f x σ vrince σ σ Σ(x x ) fx σ f Σx x N x σ vrince N 7 σ σ Σ f (x x) Σf Σ fx x Σf

8 The vrince is mesure of the men for the squre of the devitions from the men. The stndrd devition refers to the squre root for the vrince. Effects of dt chnges on Mesures of Centrl Tendency nd Mesures of dispersion Dt re chnged uniformly with +k k k k +k k k k Mesures of Men, medin, mode Centrl Tendency Rnge, Interqurtile Rnge Mesures of Stndrd Devition dispersion Vrince k k k No chnges No chnges No chnges 08 Circulr Mesures Terminology Convert degree to rdin: Convert rdin to degree: xo ( x 80 π rdins π )rdins x rdins ( x ) degrees D π degrees π 80D Rememer: 80D π rd??? 360 π rd D O??? 0.7 rd 8. rd k k k

9 Length nd Are r rdius A re s rc length θ ngle l length of chord Arc Length: s rθ Length of chord: l r sin Are of Sector: θ A Are of Tringle: rθ A r sin θ Are of Segment: A 09 Differentition Differentition of Function I Grdient of tngent of line (curve or stright) y xn nx n δy lim ( ) δ x 0 δ x Exmple y x3 Differentition of Algeric Function Differentition of Constnt y 0 3x is constnt Differentition of Function II y x x x 0 Exmple y 0 Exmple y 3x 3 9 r (θ sin θ )

10 Differentition of Function III Chin Rule y x n y un nx n du du Exmple y x3 Exmple y ( x + 3)5 (3) x 6 x u x + 3, y u5, Differentition of Frctionl Function u nd v re functions in x therefore therefore du 4x 5u 4 du du du 5u 4 4 x xn Rewrite y 5( x + 3) 4 4 x 0 x( x + 3) 4 y x n Or differentite directly y (x + ) n n nx n n+ x n..(x + ) n Exmple y x y x x x y ( x + 3)5 5( x + 3) 4 4 x 0 x( x + 3) 4 Lw of Differentition Sum nd Difference Rule y u±v u nd v re functions in x du dv ± Exmple y x3 + 5 x (3) x + 5() x 6 x + 0 x 0

11 Product Rule Quotient Rule y uv u nd v re functions in x du dv v +u y u v Exmple y ( x + 3)(3 x 3 x x) u nd v re functions in x v du dv u v Exmple x y x + u x v x + du dv x du dv u v v ( x + )( x) x () ( x + ) u x + 3 v 3x3 x x du dv 9 x 4x du dv v +u 3 (3 x x x)() + ( x + 3)(9 x 4 x ) Or differentite directly y ( x + 3)(3x3 x x) (3x3 x x)() + ( x + 3)(9 x 4 x ) 4 x + x x x + x ( x + ) ( x + ) Or differentite directly x y x + ( x + )( x) x () ( x + ) 4 x + x x x + x ( x + ) ( x + )

12 Grdients of tngents, Eqution of tngent nd Norml Grdient of tngent t A(x, y): grdient of tngent Eqution of tngent: y y m( x x ) Grdient of norml t A(x, y): mnorml If A(x, y) is point on line y f(x), the grdient of the line (for stright line) or the grdient of the tngent of the line (for curve) is the vlue of when x x. mtngent grdient of norml Eqution of norml : y y m( x x ) Mximum nd Minimum Point Turning point At mximum point, 0 d y <0 0 At minimum point, 0 >0

13 Rtes of Chnge Chin rule Smll Chnges nd Approximtion Smll Chnge: da da dr dt dr dt 5 dt Decreses/leks/reduces NEGATIVES vlues!!! If x chnges t the rte of 5 cms - δ y δ y δ x δ x Approximtion: ynew yoriginl + δ y yoriginl + δ x δ x smll chnges in x δ y smll chnges in y If x ecomes smller δ x NEGATIVE 3

14 0 Solution of Tringle Sine Rule: Cosine Rule: Use, when given sides nd non included ngle ngles nd side A B + c c cosa + c c cosb c + cosc c sin A sin B sin C Are of tringle: cos A C + c c A Use, when given sides nd included ngle 3 sides A 80 (A+B) 4 C is the included ngle of sides nd. c A sin C

15 If C, the length AC nd length AB remin unchnged, the point B cn lso e t point B where ABC cute nd A B C otuse. If ABC θ, thus AB C 80 θ. Cse of AMBIGUITY A 80 - θ Rememer : sinθ sin (80 θ) θ C B B Cse : When < sin A CB is too short to rech the side opposite to C. Cse : When sin A CB just touch the side opposite to C Outcome: No solution Cse 3: When > sin A ut <. CB cuts the side opposite to C t points Outcome: solution Cse 4: When > sin A nd >. CB cuts the side opposite to C t points Outcome: solution Outcome: solution Useful informtion: c θ In right ngled tringle, you my use the following to solve the prolems. (i) Phythgors Theorem: c + (ii) Trigonometry rtio: sin θ c, cos θ c, tn θ (iii) Are ½ (se)(height) 5

16 Index Numer Price Index Composite index I P 00 P0 I I Price index / Index numer I Composite Index W Weightge I Price index P0 Price t the se time P Price t specific time I A, B I B,C I A,C 00 Σ Wi I i Σ Wi 6

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

Add Maths Formulae List: Form 4 (Update 18/9/08) Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Fuctios Asolute Vlue Fuctio f ( ) f( ), if f( ) 0 f( ), if f( ) < 0 Iverse Fuctio If y f( ), the Rememer: Oject the vlue of Imge the vlue of y or f() f()