<< Chapter < Page Chapter >> Page >

Access the following online resource for additional instruction and practice with graphing polynomial functions.

Key concepts

  • Polynomial functions of degree 2 or more are smooth, continuous functions. See [link] .
  • To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See [link] , [link] , and [link] .
  • Another way to find the x - intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x - axis. See [link] .
  • The multiplicity of a zero determines how the graph behaves at the x - intercepts. See [link] .
  • The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
  • The end behavior of a polynomial function depends on the leading term.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree n has at most n 1 turning points. See [link] .
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n 1 turning points. See [link] and [link] .
  • Graphing a polynomial function helps to estimate local and global extremas. See [link] .
  • The Intermediate Value Theorem tells us that if f ( a )   and   f ( b ) have opposite signs, then there exists at least one value c between a and b for which f ( c ) = 0. See [link] .

Section exercises

Verbal

What is the difference between an x - intercept and a zero of a polynomial function f ?

The x - intercept is where the graph of the function crosses the x - axis, and the zero of the function is the input value for which f ( x ) = 0.

Got questions? Get instant answers now!

If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function?

Got questions? Get instant answers now!

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

If we evaluate the function at a and at b and the sign of the function value changes, then we know a zero exists between a and b .

Got questions? Get instant answers now!

Explain how the factored form of the polynomial helps us in graphing it.

Got questions? Get instant answers now!

If the graph of a polynomial just touches the x -axis and then changes direction, what can we conclude about the factored form of the polynomial?

There will be a factor raised to an even power.

Got questions? Get instant answers now!

Algebraic

For the following exercises, find the x - or t -intercepts of the polynomial functions.

C ( t ) = 2 ( t 4 ) ( t + 1 ) ( t 6 )

Got questions? Get instant answers now!

C ( t ) = 3 ( t + 2 ) ( t 3 ) ( t + 5 )

( 2 , 0 ) , ( 3 , 0 ) , ( 5 , 0 )

Got questions? Get instant answers now!

C ( t ) = 4 t ( t 2 ) 2 ( t + 1 )

Got questions? Get instant answers now!

C ( t ) = 2 t ( t 3 ) ( t + 1 ) 2

( 3 , 0 ) , ( 1 , 0 ) , ( 0 , 0 )

Got questions? Get instant answers now!

C ( t ) = 2 t 4 8 t 3 + 6 t 2

Got questions? Get instant answers now!

C ( t ) = 4 t 4 + 12 t 3 40 t 2

( 0 , 0 ) ,   ( 5 , 0 ) ,   ( 2 , 0 )

Got questions? Get instant answers now!

f ( x ) = x 4 x 2

Got questions? Get instant answers now!

f ( x ) = x 3 + x 2 20 x

( 0 , 0 ) ,   ( 5 , 0 ) ,   ( 4 , 0 )

Got questions? Get instant answers now!

f ( x ) = x 3 + 6 x 2 7 x

Got questions? Get instant answers now!

f ( x ) = x 3 + x 2 4 x 4

( 2 , 0 ) ,   ( 2 , 0 ) ,   ( 1 , 0 )

Got questions? Get instant answers now!

f ( x ) = x 3 + 2 x 2 9 x 18

Got questions? Get instant answers now!

f ( x ) = 2 x 3 x 2 8 x + 4

( 2 , 0 ) , ( 2 , 0 ) , ( 1 2 , 0 )

Got questions? Get instant answers now!

f ( x ) = x 6 7 x 3 8

Got questions? Get instant answers now!

f ( x ) = 2 x 4 + 6 x 2 8

( 1 , 0 ) ,   ( 1 , 0 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = x 6 2 x 4 3 x 2

( 0 , 0 ) , ( 3 , 0 ) , ( 3 , 0 )

Got questions? Get instant answers now!

f ( x ) = x 6 3 x 4 4 x 2

Got questions? Get instant answers now!

f ( x ) = x 5 5 x 3 + 4 x

( 0 , 0 ) ,   ( 1 , 0 ) ( 1 , 0 ) ,   ( 2 , 0 ) ,   ( 2 , 0 )

Got questions? Get instant answers now!

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

f ( x ) = x 3 9 x , between x = −4 and x = −2.

Got questions? Get instant answers now!

f ( x ) = x 3 9 x , between x = 2 and x = 4.

f ( 2 ) = 10 and f ( 4 ) = 28. Sign change confirms.

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask