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HSA-APR,B,2　Know and appIy the RemainderTheorem: For a polynomial p(x) and a number o, the

remainderon division byx 〇 ° isp(。),SOP(。) = O ifand oniyif(x〇 °) isafactorofp(x).

HSA-APR.B.3 1dentifyzeros of poiynomiaIs when suitabIe factorizations are availabIe, and use the zeros

to construct a rough graph ofthe function defined bythe poIynomial.

HSF-BF,B.3　1dentifythe effect on the graph ofrepiacing f(x) by賞x) + k,煩x)誰寂), and f(x + k) for

SPeC南c vaIues ofk (both positive and negative)膏nd the value ofk given the graphs.

Experiment with cases and illustrate an expIanation ofthe effects on the graph using

technoIogy. lnclude recognizing even and odd functions from their graphs and aIgebraic

expressions for them.

ACTiVITY q8Guided

Activity Standards Fo`uS

In Activity 1 8, Students graph

POlynomial functions by hand or by

uslng teChnology. They recognlZe eVen

and odd functions from thelr algebraic

expressions and use various strategies to

describe the roots of polynomial

functions. Students compare propertleS

Offunctions represented ln dlfferent

WayS and use graphing to soIve

POlynomial inequalitleS.

Throughout thlS aCtlVlty, emPhasIZe that

When graphing or analyzing polynomial

functlOnS言t is helpfu圧o wrlte the

POlynomlals m Standard form,

remembering to note missing terms for

whlCh the coefficient is zero.

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#7-10　　君11-12

Check Your Understandlng

Lesson PractlCe

BelトRinge「 Activity

Have students rewrlte the followlng

POlynomlal funct】OnS ln Standard k)rm

and identify the degree of the

POlynomial and the slgn Ofthe leading

coefficient:

1.揮)=6xZ-2十与x-3x4

し輝)= -3美4十6x2十5x-2;

方略γee 4; n讐atlVe]

2.揮)=X十与x2十8x」美4十7xら

しf砂二死5-X4十8㌔十5x2十x,

d略ree 5; PO融we」

3.輝)=丁4x3- 10美十うx之

し枢)ニー4x3十与x2- 1聴

d略γee 3; ”讐atiγe】

1 Qui●kwrite, C「eate Representatio鵬s,

し00k fo「 a Pattern Students generallZe

the behavIOr Ofgraphs based on the

leadmg COefficlent Of the polynomiaL

They should have some familiarity wlth

this concept from Act獲Vlty 14

A⊂tivity18●GraphsofPolynomials　277

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Pa誼-鮒are, De緬e輔ng students will

analyze and graph functlOnS uSing end

behavior and zeros. It is also acceptable

to have students use a graphmg

Calculator and then explain the pattems

they notice instead ofgraphing the

functions by hand.

Check Your Understanding

Debrief students’answers to these items

to ensure that they understand how to

Sketch graphs of polynomial functions.

Tb reinforce Item 4, have students share

their answers with a partner.

Ånswe「s

3. Check students’work. Sketches

Should show a graph with the

characteristics described in the

answer to Item 4. See also the graph

in the answer to Item 5.

4. The unfactored polynomial reveals

that the degree ofthe function is odd

and the leading coefficient is

POSitive, SO the value ofthe function

decreases as x approaches negative

infinity and increases as x

approaches positive infinity. It also

reveals that the y工ntercept is 20. The

factored polynomial reveals that the

rootsare -4, 1, and5.

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Graphing Polynomial Functions

「一書「一一I polynomials can be written in factored form or in standard form. Each form

PrOVides useful clues about how the graph will behave. Work on Item 2 with

your group. As needed, refer to the GIossary to review translations ofkeyterms. Incorporate your understanding into group discussions to confim

your knowledge and use of key mathematlCal language.

2. Model with mathematics. Sketch a graph ofeach function. For

graphs b through e, identify the information revealed by the unfactored

POlynomial compared to the factored polynomial.

言:撚三岩=(汀3)巨,)The unfactored poIynomiaI reveals that the function IS eVen, SO the

graph is symmetricai around the y-aXis.The degree of the function

is even and the Ieading coe怖cient is positive, SO the vaiue of the

functiOn increases as x approaches negative and positive infinity. 1t

aiso reveaIs that the y-intercept is臆9.The factored form reveais

that the x-intercepts are -3 and 3.

⊂. h(x)=㌔十㌔-9x-9=(x十3)(x-3)(x十l)

The unfactored polynomiai reveals that the degree of the functIOn

is odd and the leading coefflcient is positive, SO the value of the

functiOn decreases as x approaches negative infinity and increases

as x approaches positive infinity, lt also reveals that the y-intercept is

-9.The factored form reveais thatthex-冊terCePtS are -3, -1, and 3.

d.粒)=ズ4-1飯2十9=毎十3粒-3)(x十1)(芳書1)

The unfactored polynomiaI reveais that the function is even, SO the

graph is symmetricaI around the y-aXis.The degree of the f…Ction

is even and the leading coefficien白s positive, SO the vaiue of the

functIOn increases as x approaches negative and positive infinity・ it

also reveaIs that the y-inte「cept is 9.The factored form shows that

the)高ntercepts are -3, -1, 1,and 3.

e. p(x)=㌔十10x4十37雷十6扉十36ズ=布十2)2(ズ十3)2

丁he unfactored poiynomia圧eveaIs that the degree of the functIOn

is odd and the leading coe仰cient is positlVe, SO the vaiue of the

function decreases as x approaches negative infinity and increases

as x approaches posltive infinity, The y-intercept lS O. The factored

form reveals thattherootsare -3,葛2, and O; the roots臆3 and -2

are doubie roots, SOthe graph justtouches thex-aXis atthese vaiues.

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4. Identify the information revealed by the unfactored polynomlal ln

Item 3 compared to the factored polynomial.

5. Use your calculator to graph the function in Item 3.

6. Compare the calculator image with your sketch. What lnformation is

not revealed by either the standard form or factored form ofa

P Olynomial?

HSF-iF.B.5　ReIate the domain ofa function to its graph and, Where appIicabie, tO the quantitative

reIationship it des⊂ribes. ★

HS白F.C.7　Graph functions expressed symboiica=y and show key features ofthe graph, by hand in

Simpie cases and using technoIogyfor more compii⊂ated ⊂aSeS. ★

HSF」F.C.7c Graph poiynomiai fun⊂tions, identifying zeros when suitabIe fa⊂tOrizations are avaiiable,

and showing end behavior.

HS白F.C.8　Write a function defined by an expression in different but equivalentforms to reveaI and

expiain different properties ofthe function.

HSF-IF.C,9　Compare properties oftwo functions each represented in a different way (aigebrai⊂a時

graphicaliy, numeri⊂aliy in tabies, Or by verbaI de§Criptions).

278　Sp「ingBoard⑧Mathematics創排出胴2,軸鵬O Polynomials

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Graphing Polynomial Functions

Polynomial functions are continuoz‘5力ノnCtions, meaning that their graphs

have no gaps or breaks. Their graphs are smooth, unbroken curves with no

Sharp turns. Graphs ofpolynomial functions with degree 77 have ” ZerOS

毎-intercepts), aS yOu SaW ln the Fundamental Theorem ofAlgebra. They also

have at most n - 1 γeiatiγe e;禽γema.

7. Findthe記nterceptsof殖) =# + 3x3 -亭- 3x. ′　。÷¥

y畿X=-1,1,一3,0

8. Find the y-1nterCePt Ofj毎).

9. Reason quantitatively. Howcant

help you identify where the relative extrema will occur?

For a poIynomiaI, relative extrema occur between the zeros of

the poiynomiai function.

10. The relative extrema ofthe function

ノ解)二X4十3㌔〇㌔-3xoccur

at approximatelyx = O.6, X = -0.5, and

Jr = -2.3. Use these x-Values to find the

approximate values of血e extrema and

graph the function.

(0.6, -1.382), ( -0.5, 0.938), (-2.3, -6.907)

11. Sketchagraphofj(x) = -X3 - X2 - 6x. ー来車十恒め

12. Sketchagrapho桝)=X4- 10x2+9. X二〇

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M0Xima and minimo are known as

extrema.They are the greatest

VaIue (the maximum) orthe least

Vaiue (the minimum) ofa function

over an intervai or the entire

d°main.

When refe「ring to ext「ema that

OCCu「 Within a spe⊂ific intervaI of

the domain, they ∂re CaiIed

reiative extrema.

When referring to vaiues that are

extrema for the enti「e domain of

the functiOn, they are called

gIobaI extrema.

⑳APin caIcuius,yOu W刷usethefirst

derivative ofa poIynomial function

to aigebraic訓y determine the

COOrdinates of the extrema.

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㊨ApStudents will learn to use the first

derivative of a polynomial

function to algebraically

determine the coordinates ofthe

extrema. The second derivative

Of a polynomial function can be

used to determine a function’s

COnCaVity The point where a

function changes concavity is

Called an inflection point.

AC丁iVITY 18 c。ntinu。d

⑮eveしoping Math Language

Be sure students understand that the

term extγema refers collectlVely to

maximum and minimum values of a

function for a specific interval ofthe

domaln. As needed, reference

temperature extremes and note how the

high and low temperatures are

dependent upon the interval oftime

COnSldered. As students respond to

questions or discuss posslble solutions

to problems, mOnitor their use ofthe

terms c」Xtγema, ma二mma, and minima to

ensure their understanding and ability

to use math language correctly and

precisdy

7-10 Create Rep「esentations Guide

Students through the process of graphlng

factorable polynomial functions. These

items can be done with or without uslng

technoIogy Finding the extrema at this

Stage lS nOt an eaSy PrOCeSS.

Item lO gives students the x-COOrdlnate

to help them locate a dose

approxi matio n.

11-12 C「eate Represeれt誹ions.

Think-Pair・Share, Debriefing If

Students are going to graph these

functlOnS by hand, lt lS Sufficient to have

†鵜

inding

Relative Extrema Using

a Graphing Calculator

lfstudents need additionaI help

finding relative extrema, a

min日esson is ava=abIe to

PrOVide practice.

See the「モacher Resources at

SpringBoard Digital for a student

Page forthis min日esson.

Activity18●GraphsofPolynomials　279

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betweenthezerosofapolynomialfunction.

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