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:
Jan 05, 2016
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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
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