Warm-Up
Answer the following in your notes.
1. f ( x)  x2  3 will result in a shift of the parent graph up/down/left/right
_?_ units.
2
2. f ( x)  ( x  3) will result in a shift of the parent graph up/down/left/right
_?_ units.
3. The dotted function is best represented by
2
a. f ( x)  ( x  2)
3
b. f ( x)  x  2
3
c. f ( x)  2 x
1
d. f ( x)  x3
2
4. The dotted function is best represented by
a. f ( x)  x 1  2
b. f ( x)  x  2 1
c. f ( x)  x  2
d. f ( x)  x 1
Take out your notebook/binder…
It’s NOTE-TAKING time!!
Today’s Date => 9/13/10
Today’s 1st Topic => Reflections
2nd Topic => Even/Odd Functions
Reflection Symmetry

Reflection Symmetry (sometimes called Line
Symmetry or Mirror Symmetry) is easy to
recognize, because one half is the reflection of
the other half.


Here is a dog. Her face
made perfectly
symmetrical with a bit
of photo magic.
The white line down
the center is the Line
of Symmetry.
Reflection Symmetry



The reflection in this lake also has
symmetry, but in this case:
the Line of Symmetry is the horizon
it is not perfect symmetry, because the
image is changed a little by the lake surface.
Line of Symmetry

The Line of Symmetry (also called the
Mirror Line) does not have to be updown or left-right, it can be in any
direction.
~But there are four
common directions, and
they are named for the
line they make on the
standard XY graph.
Examples of Lines of Symmetry
Line of Symmetry
Sample Artwork
Example Shape
Examples of Lines of Symmetry
Line of Symmetry
Sample Artwork
Example Shape
In-Class Assignment (Part 1)
Take out one sheet of paper.
1. Write all of the capital letters of the
alphabet. Decide which letters have
symmetry and write the type of
symmetry next to the letter.
(Example: A has y-axis symmetry)
2. Reflect numbers 0, 1, 2, 3, & 4 across
the x-axis.
3. Reflect numbers 5, 6, 7, 8, & 9 across
the y-axis.
Even & Odd Functions



Degree: highest exponent of the
function
Constants are considered to be even!
Even degrees:
f ( x)  5 x

2
f ( x)  4 x  4*1  4
0
Odd degrees:
f ( x)  x
f ( x)  2 x
3
Even Functions

EVEN => All exponents are EVEN


Example:
f ( x)  x  7
2
y-axis symmetry
f ( x)  f ( x)
Odd Functions

ODD => All exponents are ODD


Example:
f ( x)  x  3x
3
origin symmetry
f (  x)   f ( x)
NEITHER even nor odd

NEITHER => Mix of even and odd
exponents

Examples:
2 3
f ( x)  5 x  x
3
4
f ( x)  6 x  2
3
Leading Coefficient (LC)


The coefficient of the term with the
highest exponent
2 Cases:



LC > 0
LC < 0
Agree?!?!
End Behavior


What happens to f(x) or y as x
approaches -∞ and +∞
We can figure this out quickly by
the two things we’ve already
discussed



Degree of function (even or odd)
Leading coefficient (LC)
Let’s look at our 4 cases…jot these
down in your graphic organizer!
Case #1: Even Degree, LC > 0
f ( x)  x
2

Example:

Both ends go toward +∞
Case #2: Even Degree, LC < 0


Example:
f ( x)   x
2
Both ends go toward -∞
Case #3: Odd Degree, LC > 0

Example:
f ( x)  x
x  , f ( x)  
x  , f ( x)  
“match”
3
Case #4: Odd Degree, LC < 0

Example:
f ( x)   x
x  , f ( x)  
x  , f ( x)  
“opposites”
3
In-Class Assignment (Part 2)
1.
2.
3.
Determine if the following functions
are even, odd, or neither by
analyzing their graphs.
Explain why you chose your answer.
Write the equation of the function.
(You have 2 minutes for each graph)
#1
#2
#3
#4
Determine if the following are even, odd, or
neither. Then describe the end behavior.
5.
f ( x)  3x2  4
3
6. f ( x)  2 x  4 x
7. f ( x)  3x  2 x  4 x  4
2
3
2 2
3
8. f ( x)   x  4 x
3
9.
10.
f ( x)  5x  9
2
f ( x)  2 x  x
3
Answer the following:
11. Explain how you know a function
is even, odd, or neither.
12. Write an even function.
13. Write an odd function.
14. Write a function that is neither.
15. What are the four cases for
determining the end behavior of a
function?