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End Behavior of Polynomial Functions: Identifying Form, Degree, and End Behavior, Study notes of Elementary Mathematics

Western States Chiropractic CollegeElementary Mathematics

In this document, students are introduced to the concept of end behavior of polynomial functions. They are asked to identify the form and degree of each polynomial function given, and then determine the end behavior as x approaches positive and negative infinity. several examples and practice problems for students to work on.

Typology: Study notes

2021/2022

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NOTES: END BEHAVIOR DAY 5
Textbook Chapter 5.3
OBJECTIVE: Today you will learn about the end behavior of functions!
A polynomial function is in STANDARD FORM if its terms are written in
descending order of exponents
from left to right.
Standard Form Example:
f(x) =
2x3 โ€“ 5x2 โ€“ 4x + 7
Leading Coefficient
_____
Degree
_____
Factored Form Example:
f(x) =
x (x + 2) (x โ€“ 5)3
Leading Coefficient
_____
Degree
_____
Circle all polynomial functions. For each polynomial function, state the form and the degree.
1.
f(x)=1
2x4โˆ’3x2โˆ’7
Form: __________________________
Degree: ________ LC: __________!
!
2.!!
f(x)=x2x+3
( )
!
!
Form: __________________________
Degree: ________ LC: __________!
!
3.
f(x)=6x2+2xโˆ’1+x
!
Form: __________________________
Degree: ________ LC: __________!
!
4.!!
f(x)=3
5x4+2x+9
Form: __________________________
Degree: ________ LC: __________!
!
5.
f(x)=โˆ’5x+12
Form: __________________________
Degree: ________ LC: __________!
!
6.
f(x)=22 โˆ’19x+2x
Form: __________________________
Degree: ________ LC: __________!
!
7.
Form: __________________________
Degree: ________ LC: __________!
!
8.
f(x)=36x4โˆ’x3+x2
Form: __________________________
Degree: ________ LC: __________!
!
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Partial preview of the text

Download End Behavior of Polynomial Functions: Identifying Form, Degree, and End Behavior and more Study notes Elementary Mathematics in PDF only on Docsity!

NOTES: END BEHAVIOR DAY 5

Textbook Chapter 5.

OBJECTIVE : Today you will learn about the end behavior of functions!

A polynomial function is in STANDARD FORM if its terms are written in descending order of exponents from left to right. Standard Form Example : f(x) = 2x^3 โ€“ 5x^2 โ€“ 4x + 7 Leading Coefficient _____ Degree _____ Factored Form Example : f(x) = x (x + 2) (x โ€“ 5)^3 Leading Coefficient _____ Degree _____

Circle all polynomial functions. For each polynomial function, state the form and the degree.

1. f ( x ) =

x^4 โˆ’ 3 x^2 โˆ’ 7

Form: __________________________

Degree: ________ LC: __________

2. f ( x ) = x

2

( x^ +^3 )

Form: __________________________

Degree: ________ LC: __________

3. f ( x ) = 6 x

2

+ 2 x

โˆ’ 1

+ x

Form: __________________________

Degree: ________ LC: __________

4. f ( x ) =

x

4

+ 2 x + 9

Form: __________________________

Degree: ________ LC: __________

5. f ( x ) = โˆ’ 5 x + 12

Form: __________________________

Degree: ________ LC: __________

6. f ( x ) = 22 โˆ’ 19 x + 2

x

Form: __________________________

Degree: ________ LC: __________

7. f ( x ) = x ( x + 3 )

2

( x โˆ’ 1 )

3

Form: __________________________

Degree: ________ LC: __________

8. f ( x ) = 36 x

4

โˆ’ x

3

+ x

2

Form: __________________________

Degree: ________ LC: __________

Describe the end behavior of the graph of the function.

    As x โ†’ + โˆž then f ( x ) โ†’ _____ As x โ†’ โˆ’ โˆž then f ( x ) โ†’ _____ As x โ†’ + โˆž then f ( x ) โ†’ _____ As x โ†’ โˆ’ โˆž then f ( x ) โ†’ _____ Use your calculator the graph the following and determine the end behavior.
  2. y = - 3x^5 โ€“ 6x^2 + 3x โ€“ 8 4. h(x) = 6x^8 โ€“ 7x^5 + 4x As x โ†’ + โˆž then f ( x ) โ†’ _____ As x โ†’ โˆ’ โˆž then f ( x ) โ†’ _____ As x โ†’ + โˆž then f ( x ) โ†’ _____ As x โ†’ โˆ’ โˆž then f ( x ) โ†’ _____
  1. Describe the end behavior of the following functions: a. b.

PRACTICE: END BEHAVIOR

Determine the end behavior of each of the graphs

x โ†’ โˆž, f ( x ) โ†’ _____

x โ†’ โˆ’โˆž, f ( x ) โ†’ _____

x โ†’ โˆž, f ( x ) โ†’ _____

x โ†’ โˆ’โˆž, f ( x ) โ†’ _____

x โ†’ โˆž, f ( x ) โ†’ _____

x โ†’ โˆ’โˆž, f ( x ) โ†’ _____

2. Graph x^3 โ€“ 9x^2 + 8x + 60 using your calculator. Sketch its graph below.

a. How many zeros of the polynomial __________________________________

b. Find the real zeros of the polynomial ________________________________

c. Determine the number of turning points ______________________________

d. Where are the relative minimums or maximums? ______________________

e. Where are the absolute minimums or maximums? ______________________

f. Describe the end behavior of the graph:

As x โ†’ +โˆž , f ( x ) โ†’ _____ as x โ†’ โˆ’โˆž , f ( x ) โ†’ _____

3. Graph 3x^4 + x^3 - 10x^2 + 2x + 7 using your calculator. Sketch its graph below.

a. How many zeros of the polynomial __________________________________

b. Find the real zeros of the polynomial ______________________________

c. Determine the number of turning points ____________________________

d. Where are the relative minimums or maximums? ______________

e. Where are the absolute minimums or maximums? ______________

f. Describe the end behavior of the graph:

As x โ†’ +โˆž , f ( x ) โ†’ _____ as x โ†’ โˆ’โˆž , f ( x ) โ†’ _____