Solving quadratic equation by Quadratic Formula Simultaneous Equations More complex quadratic equations Solving quadratic equation by factorisation Solving quadratic equation by Completing The Square (CTS) QUADRATICS Solving Quadratic Equations linear (y=2x-1) quadratic (y=x^2-4) Method substitution factorisation x = -1, y = -3 x = 3, y = 5 solution substitution factorisation
COMPOSITE FUNCTIONS -provide structured evidence that leads to a given result of f. -written as f^-1(x). domain of f(x)=range of f^-1(x) range of f(x)=domain of f^-1(x) FUNCTIONS MAPPINGS INVERSE FUCTION A composite function uses the output of one function as the input of another. fh (x) means that we substitute the inner function h (a) into the outer function f(x). One to one If h(x)= x^2 and f(x)=x-5 we can find an expression for fh(x): Many to one One to many Steps: 1.let f(x)=y 2.change x to y and y to x 3.rearrange to get y in terms of x 4.write in correct form Many to many Comparison between graph f(x) and its inverse
FUNCTIONS TRANSLATIONS REFLECTIONS f(x) f(x+3) f(x)+3 -f(x) f(-x) STRETCHES f(x) -2 2 -2 f(2x) -1 1 -2 2f(x) -2 2 -4 f(ax) when A increase, graph compress horizontally, without changing y coords multiply all x coords by normal Af(x) -when A increase, graph stretches Vertically, without changing X coords -multiply all y coords by a Group Members: -Zakir -Aidel -Yusof -Aqil -Syahmil f(x)
COORDINATES COORDINATES GEOMETRY GEOMETRY Length of a line segment & midpoint Parallel Perpendicular Formula of area when calculating = 1) midpoint, m = 2) PQ =
quadrants (+) = ANTI-CLOCKWISE (-) = CLOCKWISE trigonometry y=A sinBx ± C y= A cosBx ± C A= amplitude B= length of one complete wave Period=2π/B (sin,cos) ; π/B (tan) C= vertical translation >0 move upward <0 move downward Sketching trigonometry graph special angles
y=cosx y=sinx y=tanx y= a sinx y= sin ax y= a + sin x y= sin (x+a) Transformation of trigonometric function Graph of trigonometric function
Sn= n[2a+(n-1)d] 2 Sn=n(a+l) 2 BINOMIAL means TWO TERM The first term is 4 a and then the power of a decreases by 1 while the power of b increases by 1 in each successive term. All of the terms have a total index of 4 ( a^4 , a^3 b, a^2 b^2 , ab^3 and b^4) There is a similar pattern in the other expansions. The coefficients also form a pattern that is known as Pascal’s triangle. Ex: (a+b)^4 — -1<r<1 S = a 1-r l=last term d= common difference a= first term ( )= SERIES S = a(1-r ) —— n 1 - r n= number of term a= first term r= common ratio n S = a(r -1) n—r-—1 n r>1 BINOMIAL EXPANSION ( A+B) ARITHMETIC PROGRESSION (a+b) BINOMIAL COEFFICIENT GEOMETRIC PROGRESSION PASCAL'S TRIANGLE nth term = arn-1 n =( 0 ) n a nth term= a+(n-1)d nth term= Sn-Sn-1 n n 1 n + ( )a n-1 b 1 + -1<r<1 ... ( n ) n b 0 ( ) 0 n = 1 ( ) n n = 1 0 a b n RECURRING DECIMAL Example: INFINITE GEOMETRIC SERIES 4 n n(n-1)(n-2)(n-3) 4 x 3 x 2 x 1 0.6= 0.6+0.06+0.006+… a=0.6 r=0.06 = 1 0.6 10 1-1/10 S= 0.6 3 = 2 *Using sum to infinity formula:
DIFFERENTIATION Second derivatives d²y dx² Denoted by d²y/dx² or f''(x) & is used to determine the nature of stationary points on a curve d²y dx² =2(^3 ) =2u =^3 /=2 /=3^2 /=/×/=2(3^2) f’(x)=2x -5 Power function CHAIN RULE The chain rule is a formula that allows you to differentiate composite functions /=/×/ reassemble the pieces differentiate separately define y and u DeRIVATIVE & gradient function normals normal line is the line perpendicular to the tangent Mnormal x Mtangent = -1 normal equation for point (x,y) can be obtained using y= Mn(x) + c tangents dy = gradient function, where dy/dx is the dx for point (x,y) , gradient of the tangent is x substituted into the gradient function equation of tangent for (x,y) can be obtained using y=Mt(x) + c gradient > 0 , minimum point < 0 , maximum point NotatION y=x², then — = 2x dx dy f(x)=x², then f’(x)=2x —dx (x²)=2x dy — (x ) = nx What are the special cases of the chain rule? d n-1 dx It allows us to differentiate a function raised to a power: /((f〖()〗^)=n(f〖())〗^(−1)f’(x) y=(x^2 -5x +7)^7 f(x)=x^2 -5x +7 / = 7(x^2 -5 +7)^6 (2x-5) Diffrentiate in the bracket : x^2 -5x +7 We can write the coordinates of P and A as (x, x ) and (x+δx, (x+δx) ). 2 2 2 Power function Power function n is not only for positive integer — [kf(x)] = k — [f(x)] d dx d dx —[f(x)_g(x)]=—[f(x)]_—[g(x)] d dx d dx d dx + +
d²y DIFFERENTIATION Practical MAX AND MIN problems Stationary point DECREASING FUNCTION Application INcreasing function increasing function is the function where gradient is always positive this means that as x inrease f(x) also increase dy > 0 dx is a function where the gradient is always a negative value as x increase, f(x) will decrease dy < 0 dx - Maximum point,as x increased dy/dx decreased - Minimum point, as x increased dy/dx increased - point of inflexion (turning point) example situation where we need to know the max and min value; the manufacturers of canned food and drinks often need to minimise the cost of their manufacturing. To do this they need to find the minimum amount of metal required to make a container for a given volume. rate of change
If the result is negative the region lies below xaxis However, if the curve is rotated 360 degree for y-axis If the result is positive the region lies above x-axis The area, A, bounded by the curve and Y-axis EQUATIONOF THE CURVE N I T EGRATI O N INDEFINITE INTEGRAL DEFINITE INTEGRAL AREA UNDER A CURVE VOLUME IMPROPER INTERGALS The area, A, bounded by the curve and X-axis If the result is positive the region lies at the right of yaxis Vice versa, if the result is negative the region lies at left of y-axis The Reverse Process Of Differentiations The curve is rotated 360 degree for xaxis Group Hairee , Faiz , Farhan , Asri , Furqon