Concept & Formulae

Concept & Formula

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Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember 1

FUNCTIONS



Relations : one to one, one to many, many to one, many to many.



Functions , f(x) = y,



Composite functions; fg, gf, f 2



Inverse Functions ; f



Absolute functions f(x) = y

f: x

x = object , y = image

y

f –1 (2) = x  2 = f(x)

–1

- Graf y = ax + b

if  y  = k  y = ax + b

,

y = k or y = –k

b

-b a

2

QUADRATIC EQUATIONS



ax2 + bx + c = 0



x 2 – (a + b)x + ab = 0

sum of roots =  b

a

x2 – ( sum of roots)x + (product of roots) = 0 

b2-4ac >0

2 distinct real roots

b2-4ac = 0

2 equal real roots

b2-4ac < 0

no roots

product of roots =

 

Find the roots : factorisation, completing square, using formula Form quadratic equation: given roots  and .

3

QUADRATIC FUNCTIONS



f(x) = a(x + p)2 + q  completing the square form



Equation of axis of symmetry , x   b

x

c a

 b  b2  4ac 2a

max /min point  (– p , q)

2a

 

Sketch Graph : Shape : Max/min point: Two other points seen Inequalities : if there are two inequalities solve using graph or number line.

4

SILMULTANEOUS EQUATIONS



Use SUBSTITUTION method. 5 types - simple , one line , with fraction - solutions given with different unknown. - find the intersections points



5

INDICES & LOGARITHM



Rule of Indices:



Rule of Logarithms:



Change base: x

a<0

a

, a>0

b x>b

x
a

a<x < b

check answers by substituting values into the non-linear equation.

a m x an = a m + n , am  an = a m – n , (a m) n = a m n



If y = a



Solving index and logarithms equations

log m + log n = log mn l og m – log n = log m/n log mn = n log m

loga y = x and vice versa.



log a x 

Formula not given / important concept 4

log b x log b a

b

Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember 6

STATISTICS



Ungrouped data: mean, median, mode, range, interquartile range, standard deviation, variance.



Grouped Data : (written in class) mean - use the midpoint of class median - use formula or orgive mod - use Histogram Interquartile range - use formula or orgive standard deviation & variance - use formula

x

x N ( x  x )2  N

 

x 2  x2 N

x

 fx f

 

 f ( x  x )2  f

 fx2  x2 f

1NF C m  L 2  f  m  



Effects on Changes in data .

7

COORDINATE GEOMETRY



distance between 2 points

AB =



midpoint

x  x y  y2 midpoint  ( 1 2 , 1 ) 2 2

 

see NOTES

gradient , m  y1  y 2 x1  x 2 Equation of straight line: ax + by + c = 0 (general) y = mx + c (gradient form) x y  1 a b

(intercept form)



find equation of straight line: y – y1 = m (x – x1), y = mx+c



Parallel : m1 = m2



equation of locus : use the distance formula



area of polygon

( x1  x2 )2  ( y1  y2 )2

point that divides a line segment in m: n ny1  my2   nx1  mx2 ,   mn   mn

perpendicular : m. m2 = – 1 area of triangle = 1 ( x1 y2  x2 y3  x3 y1)  ( x2 y1  x3 y2  x1 y3 ) 2

8 DIFFERENTIATION 

Idea of limit

 x = x + x



First derivative using first principle.

 Find lim

x 0

 



dy  nax n1 dx dy y = (ax+b)n ;  naax  b n 1 dx u y = uv and y  v

y = axn ;

dy dy du   dx du dx



If y = u, and x = u , thus



2 Second derivative, d y  d  dy  2

dx

dy dv du u v dx dx dx dy  dx

dx  dx 

5

v

and

y x

du dv  u dx dx v2

y = y + y

Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember [Differentiation]  APPLICATIONS: - Equation of tangent and normal use y = mx + c or y – y1 = m (x – x1) where m = gradient of tangent -

Problems on Maximum / Minimum

 dy/dx = gradient of tangent  m NORMAL  m TANGENT = –1

dy/dx = 0

d2y

max point when

dx 2 d2y

min point when -

-

Rate of change ,

0

dy/dt = dy/dx  dx/dt

small changes / approximate change , y  dy/dx . x

use negative values for decreasing changes.

9

CIRCULAR MEASURES



 radians = 180



arc length, s = r , the angle is in radian

s = r



area of sector, A = ½ R2  , angle is in radian



Area of segment = ½ r2 ( - sin )

A = ½ r2 

10

dx 2

0

90 =  , 60 =  , 45 =  , 30 =  2

3

4

PROGRESSIONS

 AP: a, a+ d, a+ 2d ….

 GP: a, ar, ar2, ar3 …





common difference,

common ratio r  T2 T1

d = T2 – T1

GP : Tn = ar n – 1

 nth term : Tn  Sum of the first n terms: Sn n( a  l )  Sum:: Sn  2  a = T1 = S1

AP : Tn = a + (n – 1)d Sn  n2 ( 2a  ( n  1 )d )

Sn 

 Sum to infinity

 Tn = Sn – Sn-1

a( 1  r n ) a( r n  1 )  , r 1 1 r r 1

S 

a 1 r

 Sa m = Sm – Sa --1

11 LINEAR LAW   

Convert to linear form, Y = mX + c draw line of best fit. find unknown from graph

always show table when drawing graphs

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6

Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember 12 INTEGRATION 

 f ( x)  g ( x)  c





(ax  b) n dx 

(ax  b) n1 c (a)( n  1)

Luas 



dy dx  Equation of curve y  dx (integrate the gradient fxn.)  Area under graph  Generated volume when area revolved 360 about



b

b

y dx atau

a



Isipadu

 x dy a

 y b

2

dx

2

dy

a



: x- axis

b

 x a

: y - axis 13 VECTORS 

   

vectors a parallel to b : a = k b, k = constant Law of triangle, parallelogram, polygon addition and substraction of vectors multiply scalar with vectors.  x vectors in Cartesan coordinate r  xi  y j     y - magnitude r,

r  x2  y2

- vector unit in the direction of r,

rˆ 

xi  yj x2  y2

14

TRIGONOMETRIC FUNCTIONS



positive and negative angles



functions: sin, cos, tan, sec, cosec, cot



Graph sine, cosine and tangent



Basic identity



Double angle: sin 2A, cos 2A and tan 2A



Addition formula: sin (A B), cos(A B), tan(AB)



Solving dan Proofing



 

sin2 A + cos2A = 1 cosec2A = 1 + cot2A sec2 A = 1 + tan2 A sin (AB) = sin AcosB  cosAsinB cos(AB) = cos AcosB  sin Asin B tan(AB) = tan A  tan B 1  tan A tan B sin 2A = 2 sin A cos A

sin A tan A  cos A 1 1 1 cos ec A  , sec A  , cot A  sin A cos A tan A

cos 2A = cos2A – sin2A = 2 cos2A – 1 = 1 – 2sin2 A tan 2A =

special angles : SEE NOTES

7

2 tan A . 1 – tan2A

Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember 15 PERMUTATION & COMBINATIONS  Permutation : choose r objects from n objects ( the order is important) - permutations under given conditions 

16

Combinations : the order of choosing is not important. - combinations under given conditions

n

Pr 

n! ( n  r )!

n

Cr 

n! ( n  r )! r!

PROBABILITY



Probability, P  A  n A



Event A or B

P(A) + P(B)



Event A and B

P(A)  P(B)



Probability of 2 combine events ( non-mutually exclusive)

nS 

P(AB) = P(A) + P(B) – P(AB)

17 PROBABILITY DISTRIBUTIONS 

Binomial Distributions:

P(X=r) = nCr pr q n – r ,

- Binomial Distributions Probability

p+q=1

Mean,  = np

- Graph of Binomial Distributions - mean, variance & standard deviation

 =  npq variance = npq



Normal Distributions:

Z

- Graph of Standard Normal Distribution

X μ σ

- Normal Distributions Probability 18 SOLUTIONS TO TRAINGLE

a b c   sin A sin B sin C



Sine Rule - Ambiguous Case



Cosine Rule

a2 = b2 + c2 – 2bc kos A



Area of triangle

A = ½ ab sin C where C is included angle

19 INDEX NUMBERS  Nombor indeks : o Indeks Harga :

I

P1  100 P0

 Indeks harga pada tahun asas, I0 = 100  Indeks gubahan

8

I

Q1  100 Q0

I

Wi I i Wi

Concept & Formula GIVEN FORMULAE / POINTS TO NOTE

CONCEPT & Formulae to remember

20 PENGATURCARAAN LINEAR 

write inequalities that satisfy the condition given.



draw and find the region that satisfies the inequalities



find the max / min values within the region.

–

21 MOTION IN A STRAIGHT LINE o

s

Displacement:

 v dt

- return to O : s = 0 ds 0 dt

- maximum displacement ;

v =0

- distant traveled in the nth second: dist = s n – s n–1 o

velocity: v  ds dt

- maximum velocity: dv  0

a=0

dt

- constant velocity : a = 0 o

acceleration: a 

dv , dt

- maximum acceleration:

a

d 2s dt 2

da 0 dt

9

Concept & Formula

EFFECTS ON CHANGES IN DATA each data is added with

each data is multiplied by

+p

k

mean

+p

k

mode

+p

k

median

+p

k

range

no change

k

interquartile range

no change

k

standard deviation

no change

k

variance

no change

 k2

The new value of

TRIGONOMETRIC SPECIAL ANGLES

0

30

45

sin A

0



1

cos A

1

3 2

1

tan A

0

1 3

2

2

1

60 3 2

 3

10

90

180

270

360

1

0

–1

0

0

1

0

1



0

–1

0