special functions. piece-wise functions a function is piece-wise, if it is defined over a union of...
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![Page 1: Special functions. Piece-wise functions A function is piece-wise, if it is defined over a union of domains which have different rules for each set of](https://reader035.vdocuments.net/reader035/viewer/2022062713/56649f415503460f94c60639/html5/thumbnails/1.jpg)
Special functions
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Piece-wise functions
A function is piece-wise, if it is defined over a union of domains which have different rules for each set of domain.
Example: y x
0
0
x xy
x x
y
x
0
y x
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Even/Odd functions
( ) ( )f x f x
( ) ( )f x f x
A function is said to be even if
Similarly, a function is odd if
In other words, the graph of an even function is symmetric about the y-axis while the graph of an odd function is symmetrical about the origin.
Example of Even Function Example of Odd Function
2y x
y
x
3 ?y ax
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Can you think of other examples of even and odd functions?
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Can you think of other examples of even and odd functions?
Even Function Odd Function y
x0
y x
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Step functions
y
x
This is an example of a step function, named for its various horizontal ‘steps’
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Standard Graphs
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Recall:
y ax0a 0a
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Recall:
y ax0a 0a
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Recall:
0a 0a
2y ax
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Recall:
0a 0a
2y ax
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Recall:
0a 0a
3y ax
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Recall:
0a 0a
3y ax
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What do you think this graph looks like?
0a 0a
ay
x
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What do you think this graph looks like?
0a 0a
ay
x
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What do you think this graph looks like?
0a 0a 2
ay
x
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What do you think this graph looks like?
0a 0a 2
ay
x
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What do you think this graph looks like?
1a xy ka xy ka
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What do you think this graph looks like?
1a xy ka xy ka
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What do you think this graph looks like?
0k 0k
2y kx
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What do you think this graph looks like?
0k 0k
2y kx
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y x
y = x 2
y =
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lgy xWhat do you think this graph looks like?
http://www.uncwil.edu/courses/mat111hb/EandL/log/log.html
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What do you think this graph looks like?
lgy x
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In general
Red is log(x) base 2. Green is ln(x) (log(x) base e). Blue is log(x) base 10. Cyan is log(x) base 0.5.
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Transformation of Graphs
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(1) Translation along the y-axis
x
y
+a units
( )y f x a
( )y f x
x
y
−a units
( )y f x a
( )y f x
( )y f x a
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, the graph is translated along the x-axis by a units to the left.
For
(2) Translation along the x-axis
( )y f x a 0a
( )y f x a 0a
For , where
, where
, the graph is translated along the x-axis by a units to the right.
x
y
a units
( )y f x( )y f x a
x
y
a units
( )y f x ( )y f x a
( )y f x a
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Translation along the x-axis
Compare y = x2 and y = (x−1)2
y = x 2 y = (x–1)2
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Q1y
x
3
5 2
( ) 1f x (3,1) ( )f x
( ) 2f x
(a) Sketch the graph of on the same pair of axes.
lies on the graph of
, determine, under the transformation
the new coordinate of the point.
(b) If the point
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Q2
x
y
(2,5)A
( 3, 3)B
( 2)y f x
( 4)y f x
(a) What is the coordinates of A under the transformation
(b) What is the coordinates of B under the transformation
? __________
? __________
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(3) Modulus
( )y f x ( )y f xThe graph of is derived from that of
by reflecting the portion of the graph which lies below the x-axis
y
x
0
y x
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y f x ( )y f xThe graph of is derived from that of
by reflecting the portion of the graph in the y-axis
(3) Modulus
x
y
x
y
( )y f x( )y f x
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Example: 22( 1) 3y x
22( 1) 3y x
22( 1) 3y x
Sketch the graph of
Hence or otherwise, draw the graphs of
and
22( 1) 3y x
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(3) Reflection in the x-axis
)(xfy
( )y f x
( )y f x
y
x
Reflect the whole graph in the x-axis
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(4) Reflection in the y-axis
( )y f x Reflect the whole graph in the y-axisy
x
( )y f x ( )y f x
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(5) Modulus (Type 1) ( )y f x
x
y
( )y f x
( )y f x( )y f x
( )y f x
y
x
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1. remove left half of the graph
2. take the mirror image of right half of the graph in y-axis
(6) *Modulus (Type 2) ( )y f x
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Analysis: Whether it is the positive or negative x-value, they will have the same y-value.
Observe that sin(90 ) sin( 90 )
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y = |sin x| y = sin |x|
Difference between ( ) and ( )y f x y f x
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(7) Stretch( )y f ax Under the transformation,
the graph is compressed horizontally / vertically
( )y af xOR
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Difference between “stretched” and “compressed”
2 22y x y x
compressed
Compressed narrower
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Difference between “stretched” and “compressed”
2 21
2y x y x
stretched
Stretched wider
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Stretch/Compressed which way?
( )y f ax ( )y af xOR
( )y af x
Stretched / compressed parallel to y-axis
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Stretch/Compressed?
( )y f ax ( )y af xOR
( )y f ax
Stretched / compressed parallel to x-axis
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sin 3siny x y x Stretched along the y-axis with scale factor of 3
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sin sin 3y x y x Compressed along x-axis with scale factor of 3
Stretched along x-axis with scale factor of 1/3
OR
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Combining Transformations
( ) ( )y f x y f x
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Describe the transformation(s)y x y x
Reflection in the x-axis
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Describe the transformation(s)
Translate vertically upwards by 1 unit
y = 5x3 y = 5x3 + 1
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Describe the transformation(s)
Translate horizontally to the right by 3 units
2y x 2( 3)y x
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Describe the transformation(s)
Reflection in the y-axis
3 5y x 3( ) 5y x
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Describe the transformation(s)
Translate vertically downwards by 4 units
y = lg x y = lg x 4
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Describe the transformation(s)
Translate horizontally to the left by 2 units
4xy 24xy
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Describe the transformation(s)
Translate right by 2 units then translate up by 3 units
1y
x 1
32
yx
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Describe the transformation(s)
Reflection in the x-axis
y = (x 6)(x+4) y = (x 6)(x+4)
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Q1
Describe following transformations step by step.
( ) 3 (2 1)y f x y f x
Q2 ( ) 1 (1 )y f x y f x
Q3
Q4
( ) ( 2)y f x y f x
( ) 2 2 ( 1) 3y f x y f x