Advance Mathematics

Objectives:

•Define Even and Odd functions algebraically and graphically

•Sketch graphs of functions using shifting, and reflection

Section 3.5

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

f is an

odd function

f(-x) = - f(x) y = f(x ) = x3 w.r.t

Origin

Example 9

Determine whether a function is even, odd or neither.

a) f ( x ) = 3x4 + 5x2 –4 b) f( x ) = -2x5 +4x3 +7x c) f( x ) = x3 +x2

Solution:a) 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.

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

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.

Continue…Example 10

Check whether the following graphs represent an even or odd functions or neither.

The graph represents an even function The graph represents neither

The graph represents an odd functionThe graph represents neither

a)

b)

c)

d)

Continue… Example 11Complete the graph of the following if

a) Symmetric w.r.t y-axisb) Symmetric w.r.t origin

c) Function is even d) Function is odd

e) Symmetric w.r.t x-axis

Graph of (e ) does not represent a function

Example 12

y = f( x ) + c Up c units

y = f ( x ) - c Down c units

Vertical Shifting

Below is the graph of a function y = f ( x ). Sketch the graphs of

a) y = f ( x ) + 1

b) y = f ( x ) - 2 y = f(x) + 1

y = f (x)

y = f (x)-2

Horizontal Shifting

y = f( x + c ) Left c units

y = f ( x - c ) Right c units

Continued…

Example 13.

Given the graph of a function

y = f ( x ). Sketch the graphs of

a) y = f ( x + 3 )

b) y = f ( x – 4 )

y = f ( x )

y = f ( x )

y = f ( x ) +

3

y =

f ( x

) - 4

Horizontal Shift 3 units to the left

Horizontal Shift 4 units to the right

Continued… Example 14Can 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

Example 15Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1

y = f( x )

y = f( x + 2 ) - 1

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

Continued…

Example 16

Below is the graph of a function . Sketch the graph of xxfy

3232 xxfy

y = f( x )

y =f(x-2)-3

The graph of the absolute value is shifted 2 units to the right and 3 units down

Solution:

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 17Below is the graph of a function y = x2 . Sketch the graphs of

1. y = 5 x2

2. y = (1/5)x2

x y = x2 y=5x2 y=1/5x2

2 10 .4

1 1 5 .2

0 0 0 0

-1 1 5 .2

2 10 .42

2

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.

1) P ( 0, 5 ) y = f( x + 2 ) – 1

2) P ( 3, -1 ) y = 2f(x) +4

3) P( -2,4) y = (1/2) f( x-3) + 3

Solution: 1) 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 ).

3) 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 ).

Reflecting a graph

through the x-axis

y = -f( x) Reflection through the x-axis

(x-axis acts as a plane mirror)

Example 19

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

Below is the graph of a function y = x2 . Sketch the graph of

1. y = - x2

x y = x2 y = -x2

2 4 -4

1 1 -1

0 0 0

-1 1 -1

-2 4 -4

Sketching a piece-wise function

Example 20

Definition: Piece-wise function is a function that can be described in more than one expression.

Sketch the graph of the function f if

12

1

152

)( 2

xif

xifx

xifx

xf

Solution:

1 x IfGraph y = 2x + 5 and take only the portion to the left of the line x = -1. The point (-1, 3 ) is included.

1xIf

Graph y = x2 and take only the portion where –1 < x < 1. Note: the points ( -1,1) and ( 1, 1 ) are not included

1xIf

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.

Sketching the graph of an equation containing an absolute

Example 21

Sketch the graph of y = g ( x ) = 92 x

Note: To sketch an absolute value function . xfy

We have to remember that .0yAnd 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.

Strategy:

1. Graph y = f(x) = x2.

Solution:

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.

92 xy

4. Delete the unwanted portion

Example 22

Below is the graph of y = f(x). Graph xfy

Let the animation talk about itself

Solution:

A picture can replace 1000 words

Do all homework exercises in the syllabus