Name
LESSON
6-7
Date
Class
Reteach
Investigating Graphs of Polynomial Functions
Examine the sign and the exponent of the leading term (term of greatest degree) of a
polynomial P! x " to determine the end behavior of the function.
Even degree functions: Exponent of leading term is even.
Read:
As x approaches
positive infinity, P! x "
approaches negative
infinity.
Negative leading coefficient
As x ! , P! x " ! .
As x " , P! x " ! .
Positive leading coefficient
As x ! , P! x " " .
As x " , P! x " " .
Y
Y
X
X
Example: P! x " # 3x " 2x ! 5 Leading term: 3x 4
End behavior: As x ! , P! x " " .
As x " , P! x " " .
4
3
Sign: positive
Degree: 4, even
Odd degree functions: Exponent of leading term is odd.
Negative leading coefficient
As x ! , P! x " " .
As x " , P! x " ! .
Positive leading coefficient
As x " , P! x " " .
As x ! , P! x " ! .
Y
Y
X
X
Example: P! x " # !2x 5 ! 6x 2 " x Leading term: !2x 5
End behavior: As x ! , P! x " " .
As x " , P! x " ! .
Sign: negative
Degree: 5, odd
Identify the end behavior of each function.
1. P! x " # 4x 3 " 8x 2 ! 5
Leading term: 4x
as x
6
3
as x
!$, P!x"
"$ , P ! x "
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c06-7_rt.indd 54
3
!9x 6
Leading term:
Positive, 3, odd
Sign and degree:
End behavior:
2. P! x " # !9x " 2x ! x " 7
Process Black
Sign and degree:
!$
End behavior:
as x
"$
54
negative, 6, even
as x
!$, P!x"
"$, P ! x "
!$
!$
Holt Algebra 2
12/29/05 8:29:20 PM
Name
LESSON
6-7
Date
Class
Reteach
Investigating Graphs of Polynomial Functions (continued)
You can use the graph of a polynomial function to analyze the function.
P ! x ": odd degree and
positive leading coefficient if:
Y
Notice the graph
increases, then
decreases, and then
increases again.
As x
as x
! , P ! x " ! and
" , P !x " x " .
X
P ! x ": odd degree and
negative leading coefficient if:
Y
Notice this graph is the
reverse. It decreases,
then increases, and
then decreases again.
As x
as x
! , P !x " x
" , P !x " x
" and
! .
X
P ! x ": even degree and
positive leading coefficient if:
Y
Look at the end
behavior. The graph
increases at both ends.
As x
as x
! , P !x " x
" , P !x " x
" and
" .
X
P ! x ": even degree and
negative leading coefficient if:
Y
This is the reverse. The
graph decreases at
both ends.
As x
as x
! , P !x "
" , P !x "
! and
! .
X
Identify whether each function has an odd or even degree and a
positive or negative leading coefficient.
3.
Y
4.
Y
5.
X
Y
X
X
Even; positive
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c06-7_rt.indd 55
Process Black
Odd; negative
55
Odd positive
Holt Algebra 2
12/29/05 8:29:24 PM