1. Advance Mathematics Section 3.5 Objectives:
Define Even and Odd functions algebraically and graphically
Sketch graphs of functions using shifting, and reflection

2. f(-x) = -2 (-x)5 + 4( -x )3 +7(-x) f(-x) = ( -x )5 + ( -x )2
Even and Odd Functions
Terminology
Definition
Example
Type of Symmetry
f is an
even function
f(-x) = f(x)
y = f(x) = x2
w.r.t
y-axis
odd function
f(-x) = - f(x)
y = f(x ) = x3
Origin
Example 9
Determine whether a function is even, odd or neither.
a) f ( x ) = 3x4 + 5x2 – b) f( x ) = -2x5 +4x3 +7x c) f( x ) = x3 +x2
Substitute x by –x
f(-x) = -2 (-x)5 + 4( -x )3 +7(-x)
= 2x5 – 4x3 – 7x
= - (-2x5 +4x3 +7 )
= - f(x) f(-x) = - f(x)
f is odd
Solution:
Substitute x by –x
f(-x) = ( -x )5 + ( -x )2
= - x5 + x2
f(-x) is not equal to f( x) nor –f(x). Therefore, f is neither.
Substitute x by –x
f( -x) = 3( -x )4 + 5 ( -x )2 - 4
= 3x4 + 5x2 – 4
= f(x)
f( -x) = f(x)
f is even.

3. Example 10 a) c) b) d) Continue…
Check whether the following graphs represent an even or odd functions or neither. a) c)
The graph represents an even function
The graph represents neither b) d)
The graph represents neither
The graph represents an odd function

4. Example 11 Continue… Complete the graph of the following if
b) Symmetric w.r.t origin
a) Symmetric w.r.t y-axis
c) Function is even
d) Function is odd
e) Symmetric w.r.t x-axis
Graph of (e ) does not represent a function

5. Vertical Shifting Example 12 y = f( x ) + c Up c units y = f ( x ) - c
Down c units
Example 12
Below is the graph of a function y = f ( x ). Sketch the graphs of
y = f ( x ) + 1
b) y = f ( x ) - 2
y = f(x) + 1
y = f (x)
y = f (x)-2

6. Horizontal Shifting Continued… y = f( x + c ) Left c units
Right c units
Example 13.
Given the graph of a function
y = f ( x ). Sketch the graphs of
y = f ( x + 3 )
b) y = f ( x – 4 )
y = f ( x )
y = f ( x ) + 3
Horizontal Shift 3 units to the left
y = f ( x )
y = f ( x ) - 4
Horizontal Shift 4 units to the right

7. Left h units and Up k units Left h units and Down k units
Example 14
Continued…
Can you tell the effects on the graph of y = f ( x )
y = f( x + h ) + k
y = f( x + h ) - k
y = f( x - h ) + k
y = f( x - h ) - k
Left h units and Up k units
Left h units and Down k units
Right h units and Up k units
Right h units and Down k units
Example 15
Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1
y = f( x + 2 ) - 1
y = f( x )

8. Example 16
Continued…
Below is the graph of a function Sketch the graph of
Solution:
The graph of the absolute value is shifted 2 units to the right and 3 units down
y = f( x )
y =f(x-2)-3

9. Vertical Compress by a factor 1/c
Vertical Stretching
y = cf( x) ( c> 1 )
Vertical Stretch by a
factor c
y = (1/c)f ( x) ( c > 1 )
Vertical Compress by a factor 1/c
Note1 :When c > 1. Then 0 < 1/c < 1
Note 2 : c effects the value of y only.
Example 17
Below is the graph of a function y = x2 . Sketch the graphs of
y = 5 x2
2. y = (1/5)x2 x
y = x2
y=5x2
y=1/5x2 2 10 .4 1 5 .2 -1

10. Example 18
If the point P is on the graph of a function f. Find the corresponding point on the graph of the given function.
P ( 0, 5 ) y = f( x + 2 ) – 1
P ( 3, -1 ) y = 2f(x) +4
3) P( -2,4) y = (1/2) f( x-3) + 3
Solution:
P ( 0,5). y = f( x + 2 ) – 1 shifts x two units to the left and shifts y one unit down. The new x =0 – 2 = -2, and the new y = 5 – 1 = 4. The corresponding point is ( -2, 4 ).
2) P(3,-1). y = 2f(x) +4 has no effect on x. But it doubles the value of y and shifts it 4 units vertically up. Therefore the new x = 3(same as before ), and the new value of y = 2 (-1 ) + 4 = 2. Therefore, the corresponding point is ( 3,2 ).
P(-2, 4 ). y = (1/2) f( x-3) + 3 shifts x 3 units to the right and splits the value of y in half and then shifts it 3 units up. That is, the new value of
y = (1/2)(4) + 3 = 5. Therefore, the corresponding point is ( 1, 5 ).

11. Reflecting a graph through the x-axis
y = -f( x)
Reflection through the x-axis
(x-axis acts as a plane mirror)
Note: For any point P(x,y) on the graph of y = f(x), The graph of y = - f(x) does not effect the value of x, but changes the value of y into - y
Example 19
Below is the graph of a function y = x2 . Sketch the graph of
y = - x2 x
y = x2
y = -x2 2 4 -4 1 -1 -2

12. Sketching a piece-wise function
Definition: Piece-wise function is a function that can be described in more than one expression.
Example 20
Sketch the graph of the function f if
Solution:
Graph y = 2x + 5 and take only the portion to the left of the line x = -1. The point (-1, 3 ) is included.
Graph y = x2 and take only the portion where –1 < x < 1. Note: the points ( -1,1) and ( 1, 1 ) are not included
Graph y = 2 and take only the portion to the right of x = 1. Note: y = 2 represents a horizontal line. The point (1, 2 ) is included.

13. Sketching the graph of an equation containing an absolute
Note: To sketch an absolute value function
We have to remember that
And hence, the graph is always above the x-axis. The part of the graph that is below the x-axis will be reflected above the x-axis.
Example 21
Sketch the graph of y = g ( x ) =
Solution:
Strategy:
1. Graph y = f(x) = x2.
2. Graph y = f( x ) - 9 = x2 – 9 by shifting the graph of f 9 units down
3. Graph g(x) by keeping the portion of the graph y = f( x ) - 9 = x2 – 9 which is above the x-axis the same, and reflecting the portion where y < 0 with respect to the x-axis.
4. Delete the unwanted portion

14. Example 22 Solution: A picture can replace 1000 words
Below is the graph of y = f(x). Graph
Solution:
A picture can replace 1000 words
Let the animation talk about itself

15. Do all homework exercises in the syllabus