Nature of Roots. > 0 Two unequal real roots = 0 One double real root (Two equal real roots) < 0 No real roots Note: 0 Real roots Quadratic Equation:

Nature of Roots

Nature of Roots

> 0 Two unequal real roots

= 0 One double real root

(Two equal real roots)

< 0 No real roots

Note:

0 Real roots

Quadratic Equation: ax2 + bx + c = 0 ; a 0

Discriminant = = b2 – 4ac

Transformation of a graph

TranslationThe original graph is y = f(x) . Let h, k > 0 .

Graph Transformation Description

y = f(x) + k Translation along the y-axis

The graph y = f(x) + k is obtained by translating the graph of y = f(x) k units upwards.

y = f(x) k Translation along the y-axis

The graph y = f(x) k is obtained by translating the graph of y = f(x) k units downwards.

y = f(x – h) Translation along the x-axis

The graph y = f(x – h) is obtained by translating the graph of y = f(x) h units to the right.

y = f(x + h) Translation along the x-axis

The graph y = f(x + h) is obtained by translating the graph of y = f(x) h units to the left.

TranslationExamples

Graph Transformation Description

y = f(x) + 3 Translation along the y-axis

The graph y = f(x) + 3 is obtained by translating the graph of y = f(x) 3 units upwards.

y = f(x) 3 Translation along the y-axis

The graph y = f(x) 3 is obtained by translating the graph of y = f(x) 3 units downwards.

y = f(x – 2) Translation along the x-axis

The graph y = f(x – 2) is obtained by translating the graph of y = f(x) 2 units to the right.

y = f(x + 2) Translation along the x-axis

The graph y = f(x + 2) is obtained by translating the graph of y = f(x) 2 units to the left.

ReflectionThe original graph is y = f(x) .

Graph Transformation Description

y = f(x) Reflection about the y-axis

The graph y = f(x) is obtained by reflecting the graph of

y = f(x) about the y-axis.

y = f(x) Reflection about the x-axis

The graph y = f(x) is obtained by reflecting the graph of

y = f(x) about the x-axis.

ReflectionExamples

Graph Transformation Description

y = 2x Reflection about the y-axis

The graph y = 2x is obtained by reflecting the graph of

y = 2x about the y-axis.

y = 2x Reflection about the x-axis

The graph y = 2x is obtained by reflecting the graph of

y = 2x about the x-axis.

Enlargement and ReductionThe original graph is y = f(x) .

Graph Transformation Description

y = kf(x) ,

k > 1

Enlargement along the y-axis

The graph of y = kf(x) is obtained by enlarging to k times the graph of y = f(x) along the y-axis.

y = kf(x) ,

k < 1

Reduction along the y-axis

The graph of y = kf(x) is obtained by reducing to k of the graph of y = f(x) along the y-axis.

y = f(kx) ,

k > 1

Reduction along the x-axis

The graph of y = f(kx) is obtained by reducing to 1/k of the graph of y = f(x) along the x-axis.

y = f(kx) ,

k < 1

Enlargement along the x-axis

The graph of y = f(kx) is obtained by enlarging to 1/k times of the graph of y = f(x) along the x-axis.

Enlargement and ReductionExamples

Graph Transformation Description

y = 2f(x) Enlargement along the y-axis

The graph of y = 2f(x) is obtained by enlarging to 2 times the graph of y = f(x) along the y-axis.

y = f(x) Reduction along the y-axis

The graph of y = f(x) is obtained by reducing to 1/2 of the graph of y = f(x) along the y-axis.

y = f(2x) Reduction along the x-axis

The graph of y = f(2x) is obtained by reducing to 1/2 of the graph of y = f(x) along the x-axis.

y = f( x) Enlargement along the x-axis

The graph of y = f( x) is obtained by enlarging to 2 times of the graph of y = f(x) along the x-axis.

2

12

1

2

12

1

Trigonometric Functions

Trigonometric Functions of Special Angles (I)

0 30 45 60 90

sin 0 1

cos 1 0

tan 0 1 undefined

2

1

2

1

2

3

2

3

3

33

2

2

2

2

2 230

1

60

3

12

45

1

Trigonometric Functions of Special Angles (II)

(1, 0)

(0, 1)

(1, 0)

(0, 1)

sin 00

1

1

cos 11

0

0

tan 00

undefined

undefined

Trigonometric Functions of General Angles (I)

A

CT

SIVIIIIII

Trigonometric Functions of General Angles (II)

90 180 180+ 360 360+

sin cos sin sin sin sin

cos sin cos cos cos cos

tan tan tan tan tanθtan

1

Nets of a cube

Nets of a cube

Two nets are identical if one can be obtained from the other from rotation (turn it round) or/and reflection (turn it over).

An example of identical nets.

There are a total of 11 different nets of a cube as shown. Nets of a cube

Planes of Reflection

Planes of Reflection of a Cube

Planes of Reflection of a Regular Tetrahedron

Axes of Rotation

Axes of Rotation of a Cube

order of rotational symmetry = 2

order of rotational symmetry = 3

order of rotational symmetry = 4

Axes of Rotation of a Regular Tetrahedron

order of rotational symmetry = 3

order of rotational symmetry = 2

Compare Slopes of Different Lines

0 < m5 < 1

1 < m3 < 0

x

m6 = 1 ( = 45)

undefined slope

m4 = 0

m1 < 1

m2 = 1 ( = 135)

m7 > 1

m1 < m2 < m3 < m4 < m5 < m6 < m7