special functions. piece-wise functions a function is piece-wise, if it is defined over a union of...

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 domain.

Example: y x

0

0

x xy

x x

y

x

0

y x

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

Can you think of other examples of even and odd functions?

Can you think of other examples of even and odd functions?

Even Function Odd Function y

x0

y x

Step functions

y

x

This is an example of a step function, named for its various horizontal ‘steps’

Standard Graphs

Recall:

y ax0a 0a

Recall:

y ax0a 0a

Recall:

0a 0a

2y ax

Recall:

0a 0a

2y ax

Recall:

0a 0a

3y ax

Recall:

0a 0a

3y ax

What do you think this graph looks like?

0a 0a

ay

x

What do you think this graph looks like?

0a 0a

ay

x

What do you think this graph looks like?

0a 0a 2

ay

x

What do you think this graph looks like?

0a 0a 2

ay

x

What do you think this graph looks like?

1a xy ka xy ka

What do you think this graph looks like?

1a xy ka xy ka

What do you think this graph looks like?

0k 0k

2y kx

What do you think this graph looks like?

0k 0k

2y kx

y x

y = x 2

y =

lgy xWhat do you think this graph looks like?

http://www.uncwil.edu/courses/mat111hb/EandL/log/log.html

What do you think this graph looks like?

lgy x

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.

Transformation of Graphs

(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

, 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

Translation along the x-axis

Compare y = x2 and y = (x−1)2

y = x 2 y = (x–1)2

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

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

? __________

? __________

(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

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

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

(3) Reflection in the x-axis

)(xfy

( )y f x

( )y f x

y

x

Reflect the whole graph in the x-axis

(4) Reflection in the y-axis

( )y f x Reflect the whole graph in the y-axisy

x

( )y f x ( )y f x

(5) Modulus (Type 1) ( )y f x

x

y

( )y f x

( )y f x( )y f x

( )y f x

y

x

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

Analysis: Whether it is the positive or negative x-value, they will have the same y-value.

Observe that sin(90 ) sin( 90 )

y = |sin x| y = sin |x|

Difference between ( ) and ( )y f x y f x

(7) Stretch( )y f ax Under the transformation,

the graph is compressed horizontally / vertically

( )y af xOR

Difference between “stretched” and “compressed”

2 22y x y x

compressed

Compressed narrower

Difference between “stretched” and “compressed”

2 21

2y x y x

stretched

Stretched wider

Stretch/Compressed which way?

( )y f ax ( )y af xOR

( )y af x

Stretched / compressed parallel to y-axis

Stretch/Compressed?

( )y f ax ( )y af xOR

( )y f ax

Stretched / compressed parallel to x-axis

sin 3siny x y x Stretched along the y-axis with scale factor of 3

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

Combining Transformations

( ) ( )y f x y f x

Describe the transformation(s)y x y x

Reflection in the x-axis

Describe the transformation(s)

Translate vertically upwards by 1 unit

y = 5x3 y = 5x3 + 1

Describe the transformation(s)

Translate horizontally to the right by 3 units

2y x 2( 3)y x

Describe the transformation(s)

Reflection in the y-axis

3 5y x 3( ) 5y x

Describe the transformation(s)

Translate vertically downwards by 4 units

y = lg x y = lg x 4

Describe the transformation(s)

Translate horizontally to the left by 2 units

4xy 24xy

Describe the transformation(s)

Translate right by 2 units then translate up by 3 units

1y

x 1

32

yx

Describe the transformation(s)

Reflection in the x-axis

y = (x 6)(x+4) y = (x 6)(x+4)

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