QUADRATICSy = ax2 + bx+ c

Solving Quadratic Equation

Factorisation QuadraticFormula

Completing the Square

Completing the Square

y = x2 − 6x+ 3

= ( )2 +3x− 3 −32

= (x− 3)2 − 9 + 3

= (x− 3)2 − 6

2

y = 2x2 − 3x+ 5

= 2

�x2 − 3

2x+

5

2

�

2

= 2

�( )2 +

5

2

�x− 3

4 −�3

4

�2

= 2

�(x− 3

4)2 − 9

16+

5

2

�

= 2

�(x− 3

4)2 − 31

16

�

= 2

�x− 3

4

�2

− 31

8

Completing the Square

y = −x2 − 5x+ 2

= −(x2 + 5x− 2)

x+5

2−�5

2

�2

= −�( )2 − 2

�2

= −��

x+5

2

�2

− 25

4− 2

�

= −��

x+5

2

�2

− 33

4

�

= −�x+

5

2

�2

+33

4

y = −2x2 + 4x− 1

= −2

�x2 + 2x− 1

2

�

= −2

�( )2 − 1

2

�2

x+ 1 −12

= −2

�(x+ 1)2 − 1− 1

2

�

= −2

�(x+ 1)2 − 3

2

�

= −2(x+ 1)2 + 3

y = x2 − 6x+ 3

= (x− 3)2 − 6

y = 2x2 − 3x+ 5

= 2

�x− 3

4

�2

− 31

8

What you get from completing square

• U shaped because a is positive

• stationary point : min point

• line of symmetry

• domain :

• range :

• U shaped because a is positive

• stationary point : min point

• line of symmetry

• domain :

• range :

(3,−6)�3

4,−31

8

�

x = 3 x =3

4

ROOTS

b2 − 4ac = 0 b2 − 4ac > 0 b2 − 4ac < 0

• equal roots

• line is tangent to the curve

• two distinct roots

• line intersects the curve at two distinct points

• no roots

• does not cut the x-axis

• line does not intersect the curve

SIMULTANEOUS EQUATION

ROOTS

• two real roots

• line intersects the curve at two points

SIMULTANEOUS EQUATION

b2 − 4ac ≤ 0

REDUCIBLE EQUATIONS

FUNCTIONS

COMPOSITE FUNCTIONf(x) = 3x− 2

g(x) = x2 + 1

fg(x) = f(g(x))

= f(x2 + 1)

= 3(x2 + 1)− 2

= 3x2 + 3− 2

= 3x2 + 1

gf(x) = g(f(x))

= g(3x− 2)

= (3x− 2)2 + 1

= 9x2 − 12x+ 4 + 1

= 9x2 − 12x+ 5

ONE TO ONE FUNCTION

Straight LineQuadratic Equation

• Inverse Function exist

1. Complete the square

2. The x-coordinate of the stationary point makes it into a one to one function

3. Inverse Function then exist. You must choose the correct sign. + or - according to the inequality above.

x < A x > Aor

RANGE AND DOMAIN

f(x) f−1(x)

domain range

range domain

x > 3 y > 3

y < 5 x < 5

SKETCH INVERSE FUNCTION

Straight LineQuadratic Equation

• one to one function.

• Calculate x when y = 0

• Calculate y when x =0

1. Complete the square

2. Sketch the graph (one to one function) i.e. either

x < A x > Aor

Show the relationship between and by clearly showing the dotted line

f(x) f−1(x)

y = x

must be a one to one function with the condition given

left side...... or .....right side

COORDINATE GEOMETRY

FORMULAE

Length of line

Midpoint of line

Gradient of line

�(x2 − x1)2 + (y2 − y1)2

�x1 + x2

2,y1 + y2

2

�

m =y2 − y1x2 − x1

PERPENDICULAR

Line AB is perpendicular to the line CD. Find the equation of the perpendicular line.

A

B

C

D

(ii) Find the gradient of ABm

− 1

m(i) the point on the line

A or B− 1

m

y − y1 = − 1

m(x− x1)

PERPENDICULAR BISECTOR

Line AB is perpendicular bisector to the line CD. Find the equation of the perpendicular bisector.

A

B

C

D (ii) Find the gradient of ABm

− 1

m(i) the midpoint of the line CD

− 1

m

y − y1 = − 1

m(x− x1)

M

CIRCULAR MEASURE

FORMULAE

Length of arc

Area of sector

CALCULATOR SET IN RADIANS!

s = rθ

A =1

2r2θ

=1

2rs

o

r

r

s

θ

Chord

OTHER FORMULAE

Cosine Rule

Sine Rule

a b

c

AB

C

c2 = a2 + b2 − 2ab cosC

sinA

a=

sinB

b=

sinC

c

Area of non-right angle triangle

A =1

2ab sinC

SPECIAL ANGLES

π

4

π

4

1

1

√2

π

3

π

6

1

√3

2

when the answer required to be in exact form

TRIGONOMETRY

SIMPLE IDENTITIES

sin2 θ + cos2 θ = 1

tan θ =sin θ

cos θ

use the following identities for proving

GRAPH

• make sure you know how to sketch the original graphs of sin, cos and tan from or

• understand the change in the graph when a, b, c varies

0◦ ≤ θ ≤ 360◦ 0 ≤ θ ≤ 2π

a trig(bx) + c

a amplitude

b cyclec positive shift up

negative shift down

PROPERTIES of TRIG GRAPH

• Sin graph

• Cos graph

−1 ≤ sinx ≤ 1

−1 ≤ cosx ≤ 1

• Tan graph - Make sure you remember where the asymptotes are

SOLVE TRIGONOMETRY EQUATION

1. Put the equation into the form

2. Fix the limit (if necessary)

3. Find basic angle

4. Use ASTC

5. Solve the values of x

sin(ax+ b) = kcos(ax+ b) = ktan(ax+ b) = k

sinx = k

cosx = k

tanx = k

α

SERIESARITHMETIC AND GEOMETRIC PROGRESSION

SERIES

Arithmetic Progression

(AP)

Geometric Progression

(GP)

• nth term

• sum of the first n terms

• nth term

• sum of the first n terms

• sum to infinitySn =

1

2n[2a+ (n− 1)d]

Sn =1

2n[a+ l]

Sn = a(1− rn)

1− r

S∞ =a

1− r

an = a+ (n− 1)d an = arn−1

METHOD

• List down all the information given to you in ‘series’ form

SERIESBINOMIAL EXPANSION

FORMULA

(a+ b)n = an + nC1 an−1b1 + nC2 an−2b2 + nC3 an−3b3 + ....... + nCn−1 abn−1 + bn

OR you can use what you did in P3.

EXPAND UNTIL WHEN?

• Expand..............including and up to the term

• Expand.............. up to the term

• Expand..............where x is sufficiently small for

• Expand.............. the first three non-zero terms

x3

x4

x3

1st term + 2nd term + 3rd term

.....+ ..... x+ ...... x2 + ...... x3

.....+ ..... x+ ...... x2 + ...... x3

.....+ ..... x+ ...... x2

VECTORS

FORMULA

Magnitude of vector

Unit vector

Scalar Product

u = x+ iy

|u| = |x+ iy| =�

x2 + y2

u

|u|

321

−120

= −3 + 4 + 0

= 1

UNDERSTAND THE DIAGRAM

Finding the vectors

O P

QR

S T

UV

−−→V Q =

−−→V R +

−−→RQ

=−→V S +

−→SO +

−−→OR

+−−→RO +

−−→OP +

−−→PQ

DIFFERENTIATION

RULES

EQUATION OF TANGENT

• find the gradient of the tangent

• the point on the line

y − y1 = mT (x− x1)

dy

dx

EQUATION OF NORMAL

• find the gradient of the tangent

• gradient of normal

• the point on the line

dy

dx

y − y1 = − 1

mT(x− x1)

− 1

mT

STATIONARY POINTS

• Calculate the

• Substitute the value of the x-coordinate into and answer > 0

Minimum Point Maximum Point

dy

dx= 0

• Find the x-coordinate and the y-coordinate

d2y

dx2

d2y

dx2

• Calculate the

• Substitute the value of the x-coordinate into and answer < 0

d2y

dx2

d2y

dx2

APPLICATION

• Find the differential equation based on what given in the question

• Solve the differentiation

• Answer the question

INTEGRATION

RULES

AREA BETWEEN THE CURVE AND AXIS

x-axis y-axis

� b

af(x) dy

� b

af(y) dx

AREA BETWEEN TWO CURVES

x-axis y-axis� b

af(x) dy −

� b

ag(x) dy

� b

af(y) dx −

� b

ag(y) dx

top bottom top bottom

VOLUME BETWEEN THE CURVE AND AXIS

x-axis y-axis

V = π

� b

ay2 dx V = π

� b

ax2 dy

VOLUME BETWEEN TWO CURVES

x-axis y-axis

top bottom top bottom

V =

� b

ay2 dx −

� b

ay2 dx V =

� b

ax2 dy −

� b

ax2 dy