GraphsofPo看ynomia漢s 、ACTIVITY18ふ
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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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Check Your Understandmg
#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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ACTIVITY迎② c。ntinu。d
芝卸㊧忍総陳e即eSe融摘⑰珊S,冊前唖-
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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AC丁IViTY18c。ntinu。d ⊂he`kYourUnderstaれding Debriefstudents’answerstotheseitems �器 。.a.hing.。Iy#諾n器
toensurethattheyunderstandgraphing POlynomialfunctionsandrecognizing endbehavior.Beforestudentsgraphthe functionsinItem13・enCOuragethemto Predictwhatthegraphswil=ooklike basedonthefunctionrules. � ��� ���� ����
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13.UseappropriatetooIsstrategica11y.Useagraphingcalculatori γl ���l曇 ∴∴ I ����「 音 � � �i
tographthepolynomialfunctions.verify「thattheiLX-andy-intercepts?
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Answe「s �������������arecorrect,anddeterminethecoordinatesoftherelativeextrema. 5 」 」 �� �」L臆臆- - ���� � �i
a.ノ解)二㌔十が-X-7
13.a. � � � � �雪⊥し ��� �� � �音 �.h(美)=X4-1詑+36 ヤ 14.Whatisthemaximumnumberofrelativeextremaa飼lh-degree POlynomlalfunctlOnCanhave?① y I∠二三O之]x 芭二‾4‾2-与。2† relativeextremaat(葛4.74,48.52) � �十千十! ������】 雪i ��」∴ �
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15.Constructviablea「guments.Explainwhyrelativeextremaoccuri
betweenthezerosofapolynomialfunction.
しESSON18-1PRA ���CTICE 丁 i ��】 ! ���∴ 【 �� �
and(0.07,葛7.04) �������������16.Sketchthegraphofapolynomialfunctionthatdecreasesas ズ→±∞andh t-1031d4
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三三尋 �������������17.Sketchagraphof輝)givenbelowIdentifytheinformationrevealedby ]葦等磐悪書詳言蕊三豊葦霊磐嵩高
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18.Useagraphingcalculatortographj(カ=X3「X2-49x+49. 】
i ����� �i �I �i � ���19.FindallinterceptsofthefunctioninItem18.
I ���� � � � � � � ��20.Findtherelativemaximumandminimumvaluesofthefunctionin
relativeextremaat(-2.55,-6.25) � �‾ 「 �音 �音 音 ��ll i �� � � � ��重tem18.
(0,36),and(2.与与,-6.25) 14.4relativeextrema 15.Sampleanswer:Because POlynomialsarecontinuous functions,theymustchange directionbetweenx-intercepts. Thischangeindirectioncreatesa relativemaximumorminimum. � �i ����i 重 ��� �i �昌 ��21.Makesenseofproblems.Afourth-degreeevenpolynomial
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∴ii Checkstudents’answerstotheLesson �������������
Practicetoensurethattheyu nd L巨SSONま8"まPRAC丁iC電 17. y
basicconceptsrelate raphsof 16.Checkstudents’graphs. ���������������∴: �
polynomials videadditionalpractice,17.Them魚otoredpolynomialreveals
ee thatthedegreeofthefunctionis
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280 SpringBoard③ MathematicsAIgebra 2, Unit3 o Polynomiaしs
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