4.4 polynomial dnahhspreapalgebra2.weebly.com/uploads/1/7/0/2/17020642/4.4_notes_key.pdf3 2600 22...

© C

arn

eg

ie L

earn

ing

4.4 Key Characteristics of Polynomial Functions 349

4

LEARNING GOALS

349

KEY TERMS

absolute maximum

absolute minimum

extrema

In this lesson, you will:

Interpret polynomial key characteristics

in the context of a problem situation.

Generalize the key characteristics

of polynomials.

Sketch the graph of any polynomial given

certain key characteristics.

Polynomial DNAKey Characteristics of Polynomial Functions

4.4

Children typically resemble their parents because of the inheritance of genes from

parent to offspring. Scientists know of over 200 hereditary traits that are

transmitted across generations of families. The genes that carry these traits are in

specific strands of DNA. You can witness these traits by crossing your hands. Is your

left thumb over your right thumb? If it is, you have the dominant trait. People with the

recessive trait will cross their right thumb over their left thumb. Try it the opposite

way, it feels awkward doesn’t it?

Did you ever work with Punnett squares in biology to determine the probability of an

offspring having a particular characteristic like blue eyes versus brown eyes or

eyelash length? Being able to roll your tongue is actually a dominant genetic feature.

Some other dominant genetic human traits are non-cleft chins, widow’s peaks, broad

eyebrows, freckles, dimples, and unattached ear lobes to name a few. When you look

at the specific genotype of a species you can determine or predict what the offspring

may look like.

The same thing is true for polynomials! If you know certain characteristics about

the polynomial, you can predict what the graph will look like, as well as other

key characteristics.

© C

arn

eg

ie L

earn

ing

350 Chapter 4 Polynomial Functions

4

Problem 1

Students are given tables of

data containing an endangered

species of frog population over

a period of 44 months. A graph

and equation shows the best

!t quartic function using the

data points from the tables. The

graph and regression equation

are used to answer questions

related to the problem situation.

Grouping

Ask a student to read the

introduction. Discuss as

a class.

PROBLEM 1 Math World vs. Real World

The data shown represents the population of a rare, endangered species of frog called the

glass frog. In order to better understand the glass frog’s fertilization habits, scientists

performed a study and recorded the average number of frog eggs over the span of 44 months.

Month of

Study

Average Number

of Glass Frog

Eggs

Month of

Study

Average Number

of Glass Frog

Eggs

0 10,534 19 14,330.5

1 5500 20 13,845.1

2 5033 21 13,893.1

3 2600 22 14,546.3

4 239.4 23 11,815.8

6 137.3 23 13,086.2

7 108.4 24 15,966.9

8 667.1 29 9904.4

9 387.4 29 8257.3

12 4813.1 31 5297.5

14 9539.5 32 2494.1

15 11,318.6 33 1805.4

16 8953.3 34 665

18 15,402.5 43 4813

The data has been plotted for you and a quartic regression was used to generate the

polynomial function to best represent the data. The quartic regression option calculates the

best-!t equation of the form y 5 ax4 1 bx3 1 cx2 1 dx 1 e.

Series1

Poly. (Series1)

y = 0.2251x4 – 19.747x3 + 528.95x2 – 4292x + 10445

R2 = 0.9515

20,000

15,000

10,000

Avera

ge N

um

ber

of

Eg

gs O

bserv

ed

5000

0

–5000

0 5 10 15 20 25

Number of Months

Glass Frog Eggs Recorded

30 35 40 45 50

© C

arn

eg

ie L

earn

ing


4

Grouping

Have students complete

Questions 1 through 5 with a

partner. Then have students

share their responses as

a class.

Guiding Questions for Share Phase, Questions 1 and 2

Is the length of the study the

same as the domain of the

problem situation?

What is the length of

the study?

Is the average number of

frogs the same as the range

of the problem situation?

What was the lowest

average number of frog

eggs observed?

What was the highest

average number of frog

eggs observed?

Is the domain of a

quartic function the same

as the domain of a

quadratic function?

Where is the lowest

point on the graph of the

quartic function?

What are the coordinates of

the lowest point on the graph

of the quartic function?

Is the domain of the problem

situation the same as the

domain of the quartic

function? Why not?

Is the range of the problem

situation the same as the

domain of the quartic

function? Why not?

How did you determine the number of frog eggs at 50 months?

Why would there be an in!nite number of frog eggs if the study lasted forever?

Is it possible to have a negative number of frog eggs?

At which points on the graph of the quartic function are there 0 frog eggs?

What is the signi!cance of the x-intercepts with respect to this

problem situation?

1. Consider the graph and equation to answer each question.

a. What is the domain and range of the study?

The domain is [0, 45]. The range is [108.4, 15,966.9]

b. Explain what the domain and range represent in the context of this problem.

The domain is the length of the study, the range is the average number of frog eggs.

c. What is the domain and range of the function?

The domain is (2`, `). The range is [23098.26, `).

d. At what month in the study were the most frog eggs observed? How many eggs

were recorded?

Month 24 recorded the most frog eggs. There were 15,966.9 eggs.

e. At what month in the study were the least frog eggs observed? How many eggs

were recorded?

Month 7 recorded the least frog eggs. There were 108.4 eggs.

f. If the study lasted for 50 months, how many frog eggs would there be according to

the function?

There would be 56,720 eggs at month 50.

g. If the study lasted forever, how many eggs would there be according to the function?

There would be an infinite amount of eggs if the study lasted forever.

h. How many frog eggs appeared between months 35 and 40?

There appeared to be a negative amount of eggs.

i. At what month(s) of the study were there approximately 4800 glass frog

eggs observed?

Using the given graph, the approximate months when there were 4813 glass frog

eggs happens approximately at months 2, 10, 30, and 45.

2. Use a graphing calculator to determine the x-intercepts of the function. What do the

x-intercepts mean in the context of this problem situation?

The x-intercepts are 41.69, 34.48, 6.82, and 4.73.

These are the months that there are no frog eggs.

600005_A2_TX_Ch04_293-402.indd 351 16/04/15 9:38 AM

© C

arn

eg

ie L

earn

ing


4

Guiding Questions for Share Phase, Questions 3 through 5

What is the signi!cance of

the y-intercept with respect

to this problem situation?

How many intervals

are increasing?

How many intervals

are decreasing?

What is the degree of

this function?

How is a quartic

function different than a

quadratic function?

How is a quartic

function similar to a

quadratic function?

Problem 2

Students use the graph from

Problem 1 to state all relative

maximums, relative minimums,

absolute maximums, and

absolute minimums. The

graphs and equations of

several polynomials are given

and students will determine

the number of extrema in

each situation. They make

connections between the

possible number of extrema and

the degree of the polynomial.

Students conclude that for any

nth degree odd polynomial, it

can have 0 or even numbered

extrema with a maximum of

(n 2 1) extrema. They also

conclude that for any nth

degree even polynomial, it can

have an odd numbered extrema

with a maximum of (n 2 1)

extrema. Students compare and

contrast the graphs of even and

odd degree power functions

and polynomial functions and

use this information to analyze

several polynomial functions. Students then sketch the basic graphs of linear,

cubic, quadratic, quartic, and quintic functions and conclude the maximum

number of x-intercepts is the same as the degree of the function and that there is

always one less extrema than the degree of the polynomial.

Grouping

Ask a student to read the introduction and de!nitions. Discuss as a class.

Have students complete Question 1 with a partner. Then have students share

their responses as a class.

3. State the end behavior of the function. Does this make sense in the context of this

problem scenario? Explain your reasoning.

As x → `, f(x) → `.

As x → 2`, f(x) → `.

This does not make sense in the context of the problem. This type of end behavior

means that the number of eggs came from infinity and will move towards infinity as

the month’s progress.

4. How many frog eggs were observed at the beginning of the study? Explain the

mathematical meaning of your answer.

There were 10,534 eggs at the beginning of the study which represents the

y-intercept of the function.

5. Describe the interval when the frog’s egg population is:

a. increasing.

The frog population is increasing from month 5.74 to month 21.57 and from month

38.48 to the end of the study.

b. decreasing.

The frog population is decreasing from month 0 to month 5.74 and from month

21.57 to 38.48.

PROBLEM 2 A Polynomial is Born

So far in this chapter, you have learned a great deal about polynomial functions. You have

learned about minimums, maximums, zeros, end behavior, and the general shapes of their

graphs. Now, you will combine all that information to generalize the key characteristics for

any degree polynomial.

Recall the de!nition of a relative maximum is the highest point in a particular section of a

function’s graph, and a relative minimum is the lowest point in a particular section of the

graph. Similarly, the absolute maximum is the highest point in the entire graph, and the

absolute minimum is the lowest point in the entire graph. The set of absolute maximums,

absolute minimums, relative maximums, and relative minimums may also be referred to as

extrema. The extrema are also called extreme points and extremum.

1. Consider the graph that represents the average number of glass frog eggs in Problem 1.

a. State all relative maximums and minimums.

There is a relative maximum at (21.57, 14,519.91). There are relative minimums at

(5.7, 2253.63) and (38.48, 23098.26).

b. State all absolute maximums and minimums.

There is an absolute minimum at (38.48, 23,098.26).

According to the graph, there is no absolute maximum.

© C

arn

eg

ie L

earn

ing


4

Grouping


Questions 2 through 5 with a



a class.

Guiding Questions for Share Phase, Question 1

What is the difference

between an absolute

maximum and a

relative maximum?

Does this quartic function

have both an absolute

maximum and a

relative maximum?

Why doesn’t this function

have an absolute maximum?

Does the relative maximum

make sense in this

problem situation?


between an absolute

minimum and a

relative minimum?

Does this quartic

function have both an

absolute minimum and a

relative minimum?

Does the absolute

minimum make sense

in this problem situation?

Does the relative

minimum make sense


Do any negative values for

x or y make sense in this

problem situation?

Which quadrants make sense



Are extrema always where the function switches directions?

When an interval of increase changes to an interval of decrease, or vice versa,

is an extrema always formed?

Why can’t a 4 th degree polynomial have 2 extrema?

Why can’t a 5 th degree polynomial have 1 or 3 extrema?

Why can’t a 6 th degree polynomial have 2 or 4 extrema?

c. Do the absolute minimums and/or maximums make sense in the context of this

problem situation? Explain your reasoning.

The absolute minimum does not make sense because there cannot be a negative

average number of frog eggs.

2. Determine the number of extrema in each polynomial.

4th Degree

Polynomials

g1(x) 5 x4

x

y

g2(x) 5 x4 2 3x2

x

y

Number of

Extrema

1 3

5th Degree

Polynomials

f1(x) 5 x5

x

y

f2(x) 5 x5 1 4x2

x

y

f3(x) 5 x5 2 5x3 1 5x 1 1.18

x

y

Number of

Extrema

0 2 4

6th Degree

Polynomials

h1(x) 5 x6

x

y

h2(x) 5 x6 2 3x2

x

y

h3(x) 5 2x6 2 13x5 1 26x4 2 7x3 2 2

x

y

Number of

Extrema

1 3 5

Don’t forget to look for relationships!

© C

arn

eg

ie L

earn

ing


4

Guiding Questions for Share Phase, Questions 3 through 5

Is the maximum number

of extrema always one

less than the degree of

the polynomial?

Why is the possible number

of extrema always a

difference of 2?

Why is the possible number

of extrema for an odd degree

polynomial always even?

Why is the possible

number of extrema for an

even degree polynomial

always odd?

How did you determine the

number of extrema for a 9 th

degree polynomial?

How did you determine the

number of extrema for a 18 th

degree polynomial?

Why does an nth degree

odd polynomial have 0 or

even numbered extrema

with a maximum of

(n 2 1) extrema?

Why does an nth degree

even polynomial have an

odd numbered extrema

with a maximum of

(n 2 1) extrema?

Why will an odd degree

function never have an

absolute extrema?

Why will an even degree

function always have an

absolute extrema?

Why will an even degree

function always have relative

extrema?

3. List any observations you notice about the possible number of extrema and the degree

of the polynomial.

Answers will vary.

The maximum number of extrema is one less than the degree of the polynomial.

The possible number of extrema is always a difference of 2.

The possible number of extrema for an odd degree polynomial is even.

The possible number of extrema for an even degree polynomial is odd.

4. List the possible number of extrema for the each polynomial.

a. 9th degree polynomial

A 9th degree polynomial can have 0, 2, 4, 6, or 8 extrema.

b. 18th degree polynomial

An 18th degree polynomial can have 1, 3, 5, 7, 9, 11, 13,

15, or 17 extrema.

c. nth degree odd polynomial

A nth degree odd polynomial can have 0 or even

numbered extrema with a maximum of (n 2 1) extrema.

d. nth degree even polynomial

A nth degree even polynomial can have odd numbered

extrema with a maximum of (n 2 1) extrema.

5. Choose the appropriate word from the box to complete each statement. Justify your

answer with a sketch or explanation.

always sometimes never

a. An odd degree function will never have absolute extrema.

An odd degree function has opposite end behavior. It will approach both infinity

and negative infinity, so there will never be an absolute extrema.

b. An even degree function will always have relative extrema.

An even degree function’s graph always makes a parabolic shape. Therefore, it

will have a vertex, and the vertex will have to be a relative extrema.

c. An even degree function will sometimes have 3 or more relative extrema.

The graph of y 5 x2 has only one relative extrema, but the graph of y 5 x4 2 3x2 1 x

has 3 relative extrema.

Use the knowledge you

gained about 4th, 5th, and 6th degree polynomials

to answer these questions.

© C

arn

eg

ie L

earn

ing


4

Grouping


Question 6 with a partner.

Then have students share their

responses as a class.



between an even degree

power function and an even

degree polynomial function?


between an odd degree

power function and an odd

degree polynomial function?

Do the power functions and

their polynomial functions

have the same end behavior?

How many extrema do

the even degree power

functions have?

How many extrema do the

even degree polynomial

functions have?

Is the end behavior of a

parabola the same as the

end behavior of all even

degree polynomial functions?

Is the end behavior of a cubic

function the same as the end

behavior of all odd degree

polynomial functions?

Is the domain of all even

degree polynomial functions

always all real numbers?

Is the range of all even

degree polynomial functions

with a negative a-value

always the absolute

maximum and all real

numbers less than it?

Is the range of all even degree polynomial functions with a positive a-value

always the absolute minimum and all real numbers greater than it?

Is the domain and range of all odd degree polynomial functions always all

real numbers?

d. An even degree function will always have absolute extrema.

An even degree function’s graph always makes a parabolic shape, meaning it will

have a vertex. Therefore, the vertex will have to be an absolute extrema.

e. An odd degree function will sometimes have relative extrema.

The graph of y 5 x3 is always increasing and has no relative extrema, but the

graph of y 5 x3 2 3x2 1 x has 2 relative extrema.

f. An odd degree function will never one have relative extrema.

An odd degree function has opposite end behavior. It will approach both infinity

and negative infinity, so if there is one relative extrema, there will always have to

be another relative extrema to change direction again.

6. Analyze the graphs shown.

Even Degree Power Functions Even Degree Polynomial Functions

x

y

x

y

x

y

x

y

Odd Degree Power Functions Odd Degree Polynomial Functions

x

y

x

y

x

y

x

y

a. State the similarities and differences you notice between the power functions and

the polynomial functions.

Answers will vary.

The power functions and their polynomial functions have the

same end behavior.

The power functions have 0 or 1 extrema.

The polynomial functions have more than 1 extrema.

b. What conclusions can you make about the end behavior of all even degree


The end behavior of all even degree functions is

As x → `, f(x) → `.

As x → 2`, f(x) → `. or

As x → `, f(x) → 2`.

As x → 2`, f(x) → 2`.

© C

arn

eg

ie L

earn

ing


4

Grouping


Questions 7 and 8 with a



a class.

Guiding Questions for Share Phase, Questions 7 and 8

What visual characteristic

of the graph helped you

determine the sign of the

a-value of the function?

What end behavior is

associated with a

positive a-value?

What end behavior is

associated with a

negative a-value?



determine if the degree of the

function was even or odd?

How many x-intercepts

can be associated with a

cubic function?



determine the number of

relative extrema?


of the graph helped

you determine the

presence or absence of

absolute extrema?

What feature of the graph

helps determine the domain

of the polynomial function?

What feature of the graph

helps determine the range

of the polynomial function?

c. What conclusions can you make about the end behavior of all odd degree


The end behavior of all odd degree functions is

As x → `, f(x) → `.

As x → 2`, f(x) → 2`. or

As x → `, f(x) → 2`.

As x → 2`, f(x) → `.

d. What conclusions can you make about the domain and range of all even degree


The domain of all even degree functions is all real numbers. The range of all even

functions with a negative a-value is the absolute maximum and all real numbers

less than it. The range of all even functions with a positive a-value is the absolute

minimum and all real numbers greater than it.

e. What conclusions can you make about the domain and range of all odd degree


The domain and range of all odd functions is all real numbers.

7. Consider the graph shown.

a. Is the a-value of this function positive or negative?

The a-value is positive.

b. Is the degree of this function even or odd?

The degree of this function is odd.

c. Can this function be a cubic function? Explain why or why not.

No. This function cannot be a cubic because it has more than 3 x-intercepts.

d. State the domain of this function.

The domain of this function is (2`, `).

e. State the range of this function.

The range of this function is (2`, `).

f. Determine the number of relative extrema in this graph.

The graph has 4 relative extrema.

g. Determine the number of absolute extrema in this graph.

The graph has no absolute extrema.

h. State the intervals where the graph is increasing.

The graph is increasing on the intervals 2` , x , 22.5, 21, x , 0.25, and

2.3 , x , `.

x0

20

22 224 4

y

220

240

260

© C

arn

eg

ie L

earn

ing


4

8. Consider the graph shown.

a. Is the a-value of this function positive or negative?

The a-value is negative.

b. Is the degree of this function even or odd?

The degree of this function is even.

c. Can this function be a 6th degree polynomial function? Explain why or why not.

Yes. The function can be a 6th degree polynomial function. It has less than

6 x-intercepts and less than 6 extrema.

d. State the domain of this function.

The domain of this function is (2`, `).

e. State the range of this function.

The range of this function is approximately [2, 2`).

f. Determine the number of relative extrema in this graph.

The graph has 5 relative extrema.

g. Determine the number of absolute extrema in this graph.

The graph has one absolute extrema.

h. State the intervals where the graph is decreasing.

The graph is decreasing on the intervals

22.7 , x , 21.5, 0.25 , x , 2.1, and 3 , x , `.

x0

2

2 42224

y

22

4

24

Wow! You know a lot about graphs of polynomials.

600005_A2_TX_Ch04_293-402.indd 357 14/03/14 2:44 PM

© C

arn

eg

ie L

earn

ing


4

Grouping


Question 9 with a partner.

Then have students share their



Why can’t a linear function

have more than one zero?

Why can’t a quadratic

function have more than

two zeros?

Why is the maximum number

of x-intercepts always the

same as the degree number

of the function?

Why is there always one

less extrema than the

degree number of the

polynomial function?

9. Complete the table on the next page to represent the

graphs of various polynomials.

a. Sketch the basic shape on each set of axes, given the

number of zeros. If you cannot sketch the basic shape,

explain why.

b. Compare your graphs with a partner. State the

similarities and differences.

Answers will vary.

Student responses could include the graph

being in different quadrants and having

different a-values.

c. What do you notice about the maximum number of x-intercepts and the degree of

the function?

The maximum number of x-intercepts is the same as the degree.

d. Use your graphs to determine the greatest number of extrema (absolute and relative)

in each degree polynomial.

Type of Polynomial

FunctionNumber of Extrema

Linear 0

Quadratic 1

Cubic 2

Quartic 3

Quintic 4

e. What do you notice about the number of extrema and the degree of a polynomial?

Write a statement to generalize the possible number of extrema in any degree

polynomial function.

There is always one less extrema than the degree of the polynomial. The

maximum number of extrema in a polynomial function f(x) 5 xn is n 2 1.

After you complete the

table, answer parts (b) through (e).

600005_A2_TX_Ch04_293-402.indd 358 14/03/14 2:44 PM

© C

arn

eg

ie L

earn

ing


4

No

Ze

ros

1 Z

ero

Exa

ctl

y 2

Ze

ros

Exa

ctl

y 3

Ze

ros

Exa

ctl

y 4

Ze

ros

Exa

ctl

y 5

Ze

ros

Lin

ea

r

Qu

ad

rati

c

Cu

bic

A c

ub

ic h

as o

pp

osit

e

en

d b

eh

avio

r so

it

mu

st

cro

ss t

he x

-axis

at

least

on

e t

ime.

Qu

art

ic

A q

uart

ic h

as a

t m

ost

3

turn

ing

po

ints

, so

it

can

no

t cro

ss t

he x

-axis

m

ore

th

an

4 t

imes.

Qu

inti

c

A q

uin

tic h

as o

pp

osit

e

en

d b

eh

avio

r so

it

mu

st

cro

ss t

he x

-axis

at

least

on

e t

ime.

A lin

ear

fun

cti

on

has n

o t

urn

ing

po

ints

so

it

can

no

t cro

ss t

he x

-axis

mo

re t

han

on

ce.

A q

uad

rati

c o

nly

has o

ne t

urn

ing

po

int,

so

it

can

no

t cro

ss t

he x

-axis

mo

re t

han

2 t

imes.

A c

ub

ic h

as a

t m

ost

2 t

urn

ing

po

ints

, so

it

can

no

t cro

ss t

he x

-axis

mo

re t

han

3 t

imes.

© C

arn

eg

ie L

earn

ing


4

Problem 3

Students are given sets of

speci!c key characteristics

and will sketch a graph that

encompasses these aspects for

each situation when possible.

Next, they are given graphs and

will identify which of the given

functions could possibly model

each graph and explain

their reasoning.

Grouping


Questions 1 and 2 with a



a class.

Guiding Questions for Share Phase, Question 1, part (a)

Is this the graph of a power

function or a polynomial

function? How do you know?

Is the a-value of the function

positive or negative?

Guiding Question for Share Phase, Question 1, part (b)

If the function is always

increasing, what is the range

of the function?

Does the number of

y-intercepts help determine

the equation of the


PROBLEM 3 Who Am I?

1. Use the coordinate plane to sketch a graph with the characteristics given. If the graph is

not possible to sketch, explain why.

a. Characteristics:

degree 4

starts in quadrant III

ends in quadrant IV

relative maximum at x 5 24

absolute maximum at x 5 3

b. Characteristics:

always increasing

y-intercept at 5

x-intercept at 21.7

x0

2

22 2242628 4 6 8

y

4

6

8

22

24

26

28

x0

2

22 2242628 4 6 8

y

4

6

8

22

24

26

28

© C

arn

eg

ie L

earn

ing


4

c. Characteristics:

odd degree

increases to x 5 23, then

decreases to x 5 3, then

increases

absolute maximum at y 5 4

I cannot sketch the graph because

an odd degree function cannot have an

absolute maximum.

d. Characteristics:

as x → `, f(x) → `

as x → 2` , f(x) → `

4 x-intercepts

relative maximum at y 5 3

e. Characteristics:

x-intercepts at 22, 2 and 5

negative a value

degree 2

I cannot sketch the graph because

if the polynomial is of degree 2 it can

have at most 2 x-intercepts. A second

degree polynomial cannot have

3 x-intercepts.

x0

2

22 2242628 4 6 8

y

4

6

8

22

24

26

28

x0

2

22 2242628 4 6 8

y

4

6

8

22

24

26

28

x0

2

22 2242628 4 6 8

y

4

6

8

22

24

26

28

Guiding Question for Share Phase, Question 1, part (c)

Can an odd degree function

have an absolute maximum?

Why not?

Guiding Question for Share Phase, Question 1, part (d)

If the function has four

x-intercepts, what does

that imply about the


Is the a-value of the graph

positive or negative?

Is this the graph of an even

or odd polynomial function?

How does the number of

x-intercepts help determine

the equation of the


Guiding Question for Share Phase, Question 1, part (e)

How many intervals of

increase and decrease are on

the graph?

If the graph has three

x-intercepts, can the function

be a 2nd degree function?

Why not?

© C

arn

eg

ie L

earn

ing


4

2. Analyze each graph. Circle the function(s) which could model the graph. Describe your

reasoning to either eliminate or choose each function.

a.

x

y

f1(x) 5 23x5 22x2 1 4x 1 7

I eliminated this function because the graph

represents an even degree function and this

function is odd degree.

f2(x) 5 2(x 1 2)(x 1 1.5)(x 1 0.5)(x 2 2.5)2 (x 2 3)

I chose this function because it represents an

even degree polynomial with a negative a-value.

The graph has 3 negative x-intercepts and 2

positive x-intercepts.

f3(x) 5 23x4 2 2x2 1 4x 1 7

I eliminated this function because the graph has 5

x-intercepts, which means that the function must

be a degree higher than 4.

b.

x

y

f1(x) 5 0.5(x 1 7)(x 1 1)(x 2 5) 2 3

I chose this function because it represents an odd

degree polynomial with a positive a-value. The

graph could have 2 negative x-intercepts and

1 positive x-intercept.

f2(x) 5 22(x 1 7)(x 1 1)(x 2 5) 2 3


represents an odd degree polynomial with a positive

a-value. This function has a negative a-value.

f3(x) 5 2(x 1 7)(x 1 1)(x 2 5)(x 2 3)


represents an odd degree polynomial and this

function has a degree of 4.

© C

arn

eg

ie L

earn

ing


4

Talk the Talk

Students will complete a

table that summarizes the

key characteristics for quartics

and quintics.

Grouping

Have students complete the

problem with a partner. Then

have students share their


Talk the Talk

Complete each table to summarize the key characteristics for quartics and quintics.

The cubics table has been done for you.

Cubics

All possible end behavior

As x → `, f(x) → `.

As x → 2`, f(x) → 2`.

As x → `, f(x) → 2`.

As x → 2`, f(x) → `.

Possible number of x-intercept(s) 3, 2, or 1

Possible number of y-intercept(s) 1

Possible intervals of increase

and decrease

Always increasing

Always decreasing

Increasing, decreasing, increasing

Decreasing, increasing, decreasing

Number of possible relative extrema 2 or none

Number of possible absolute extrema None

© C

arn

eg

ie L

earn

ing


4

Quartics


As x → `, f(x) → `.

As x → 2`, f(x) → `.

As x → `, f(x) → 2`.

As x → 2`, f(x) → 2`.

Possible number of x-intercept(s) 4, 3, 2, 1, or 0



and decrease

Decreasing, increasing

Increasing, decreasing

Increasing, decreasing, increasing, decreasing

Decreasing, increasing, decreasing, increasing

Number of possible relative extrema 3 or 1

Number of possible absolute extrema 1

Quintics


As x → `, f(x) → `.

As x → 2`, f(x) → 2`.

As x → `, f(x) → 2`.

As x → 2`, f(x) → `.

Possible number of x-intercept(s) 5, 4, 3, 2, or 1



and decrease

Always increasing

Always decreasing

Increasing, decreasing, increasing

Decreasing, increasing, decreasing

Increasing, decreasing, increasing, decreasing,

increasing

Decreasing, increasing, decreasing, increasing,

decreasing

Number of possible relative extrema 4, 2, or none

Number of possible absolute extrema None

Be prepared to share your solutions and methods.

4.4 polynomial dnahhspreapalgebra2.weebly.com/uploads/1/7/0/2/17020642/4.4_notes_key.pdf3 2600 22...

Documents