Transcript facts of mathematics

Nature of Roots
Nature of Roots
Quadratic Equation: ax2 + bx + c = 0 ; a  0
Discriminant =  = b2 – 4ac
>0
Two unequal real roots
=0
One double real root
(Two equal real roots)
<0
No real roots
Note:
0
Real roots
Transformation of a graph
Translation
The original graph is y = f(x) . Let h, k > 0 .
Graph
Transformation
Description
y = f(x) + k
Translation
The graph y = f(x) + k is obtained by translating the graph
along the y-axis of y = f(x) k units upwards.
y = f(x)  k
Translation
The graph y = f(x)  k is obtained by translating the graph
along the y-axis of y = f(x) k units downwards.
y = f(x – h)
Translation
The graph y = f(x – h) is obtained by translating the graph
along the x-axis of y = f(x) h units to the right.
y = f(x + h)
Translation
The graph y = f(x + h) is obtained by translating the graph
along the x-axis of y = f(x) h units to the left.
Translation
Examples
Graph
Transformation
Description
y = f(x) + 3
Translation
The graph y = f(x) + 3 is obtained by translating the graph
along the y-axis of y = f(x) 3 units upwards.
y = f(x)  3
Translation
The graph y = f(x)  3 is obtained by translating the graph
along the y-axis of y = f(x) 3 units downwards.
y = f(x – 2)
Translation
The graph y = f(x – 2) is obtained by translating the graph
along the x-axis of y = f(x) 2 units to the right.
y = f(x + 2)
Translation
The graph y = f(x + 2) is obtained by translating the graph
along the x-axis of y = f(x) 2 units to the left.
Reflection
The original graph is y = f(x) .
Graph
Transformation
Description
y = f(x)
Reflection
The graph y = f(x) is obtained by reflecting the graph of
about the y-axis y = f(x) about the y-axis.
y = f(x)
Reflection
The graph y = f(x) is obtained by reflecting the graph of
about the x-axis y = f(x) about the x-axis.
Reflection
Examples
Graph
Transformation
Description
y = 2x
Reflection
The graph y = 2x is obtained by reflecting the graph of
about the y-axis y = 2x about the y-axis.
y = 2x
Reflection
The graph y = 2x is obtained by reflecting the graph of
about the x-axis y = 2x about the x-axis.
Enlargement and Reduction
The original graph is y = f(x) .
Graph
Transformation
Description
y = kf(x) ,
k>1
Enlargement
The graph of y = kf(x) is obtained by enlarging to k times
along the y-axis the graph of y = f(x) along the y-axis.
y = kf(x) ,
k<1
Reduction along The graph of y = kf(x) is obtained by reducing to k of the
the y-axis
graph of y = f(x) along the y-axis.
y = f(kx) ,
k>1
Reduction along The graph of y = f(kx) is obtained by reducing to 1/k of
the x-axis
the graph of y = f(x) along the x-axis.
y = f(kx) ,
k<1
Enlargement
The graph of y = f(kx) is obtained by enlarging to 1/k
along the x-axis times of the graph of y = f(x) along the x-axis.
Enlargement and Reduction
Examples
Graph
Transformation
y = 2f(x)
Enlargement
The graph of y = 2f(x) is obtained by enlarging to 2 times
along the y-axis the graph of y = f(x) along the y-axis.
1
Description
1
y = 2 f(x)
Reduction along The graph of y = 2 f(x) is obtained by reducing to 1/2 of
the y-axis
the graph of y = f(x) along the y-axis.
y = f(2x)
Reduction along The graph of y = f(2x) is obtained by reducing to 1/2 of
the x-axis
the graph of y = f(x) along the x-axis.
1
y = f(2x)
1
Enlargement
The graph of y = f( 2 x) is obtained by enlarging to 2 times
along the x-axis of the graph of y = f(x) along the x-axis.
Trigonometric Functions
Trigonometric Functions of Special Angles (I)
sin
0
30
45
60
90
0
1
2
2
2
3
2
1
30
2
2
3
60
cos
tan
1
3
2
0
3
3
2
2
1
1
2
3
1
0
undefined
2
1
45
1
Trigonometric Functions of Special Angles (II)
(0, 1)
(1, 0)
(1, 0)
(0, 1)
1
0
sin
1
0
0
1
cos
0
undefined
1
0
tan
undefined
0
Trigonometric Functions of General Angles (I)
A
S
II I
III IV
T
C
Trigonometric Functions of General Angles (II)
90
180
180+
360
360+
sin
cos
sin
sin
sin
sin
cos
sin
cos
cos
cos
cos
tan
1
tan θ
tan
tan
 tan
tan
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.
Nets of a cube
There are a total of 11 different nets of a cube as shown.
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 = 4
order of rotational symmetry = 3
order of rotational symmetry = 2
Axes of Rotation of a Regular Tetrahedron
order of rotational symmetry = 3
order of rotational symmetry = 2
Compare Slopes of
Different Lines
undefined slope
m1 < m2 < m3 < m4 < m5 < m6 < m7
m7 > 1
m6 = 1 ( = 45)
0 < m5 < 1
m4 = 0
1 < m3 < 0
m2 = 1 ( = 135)
m1 < 1
x