Today in Pre-Calculus

• Go over homework• Notes: Symmetry

– Need a calculator

• Homework

1. Vertical: x = -3 Horizontal: none

2. Vertical: x =-2, x = 2 Horizontal: y=0

3. Vertical: x = 3 Horizontal: y =0

4. lim ( ) 5, lim ( ) 5x x

f x f x

9. lim ( ) 4, lim ( ) 4x x

f x f x

8. lim ( ) , lim ( ) 0x x

f x f x

7. lim ( ) 7, lim ( ) 0x x

f x f x

6. lim ( ) 1, lim ( ) 1x x

f x f x

Symmetry• Symmetry with respect to the y-axis.

• These are EVEN functions

• Their graphs appear the same on both sides of the y-axis.

• Algebraically for every x in the domain of f , f(-x)=f(x)

Symmetry• Symmetry with respect to the x-axis.

• Note: These are NOT functions

• Their graphs appear the same on both sides of the x-axis.

• Algebraically for every (x,y) on the graph (x,-y) is also on the graph.

Symmetry• Symmetry with respect to the origin

• These are ODD functions

• Their graphs appear the same on both sides of the origin.

• Algebraically for every x in the domain of f , f(-x)=-f(x)

Example a

When graphed, appears

EVEN

Algebraically:

f(-x)=4(-x)2-1

= 4x2-1

=f(x)

x

y

Example bWhen graphed, appears

ODD

Algebraically:

3

2

3

2

5

5( )

x

x

x

xf x

x

y

3

2

( )( )

( ) 5

xf x

x

Example cWhen graphed, appears

Neither

Algebraically:

f(-x) = (-x)3+2(-x)2

= -x3+2x2

≠ f(x) or –f(x)

x

y

Example dWhen graphed, appears

ODD

Algebraically:

f(-x) = 2(-x)3-3(-x)

= -2x3+3x =-(2x3-3x)

=-f(x)

x

y

Example eWhen graphed, appears

Neither

Algebraically:

f(-x) = -(-x)2+0.03(-x)+5

= -x2-0.03x+5 ≠ f(x) or –f(x)

x

y

Example fWhen graphed, appears

Even

Algebraically:

x

y

2

2

( ) ( ) 2

2

( )

f x x

x

f x

Homework

• Wkst.

• Quiz: Thursday, September 18