Download Lesson 1: Successive Differences in Polynomials

NYSCOMMONCOREMATHEMATICSCURRICULUM
M1
Lesson1
ALGEBRAII
Lesson1:SuccessiveDifferencesin
Polynomials
X
Y
(Input) (Output)
Classwork
2.4
5.76
OpeningExercise
5.8
Johnnoticedpatternsinthearrangementofnumbersinthetablebelow.
3.4
11.56
7.8
4.4
Part2
Assumingthatthepatternwouldcontinue,heusedittofindthevalueof7. 4$ .
Explainhowheusedthepatterntofind7. 4$ ,andthenusethepatterntofind
8. 4$ .
19.36
9.8
5.4
29.16
11.8
6.4
40.96
Howwouldyoulabeleachrowofnumbersinthetable?
Discussion
Ontheothersideofthispage,thepolynomialthatdescribesthisfunctionisasecondorderpolynomialbecausethe
seconddifferencesareconstants(thesamenumbereverytime).Secondorderpolynomialstakethegeneralform
𝑦 = 𝑎𝑥 $ + 𝑏𝑥 + 𝑐,noticethatthehighestpoweris2.
Ifthethirddifferencewasconstant,thenthepolynomialwouldbeathirdorderpolynomial.Thegeneralformforathird
orderpolynomialwouldbe:𝑦 = 𝑎𝑥 - + 𝑏𝑥 $ + 𝑐𝑥 + 𝑑,
Lesson1:
SuccessiveDifferencesinPolynomials
S.1
NYSCOMMONCOREMATHEMATICSCURRICULUM
Example1 M1
Lesson1
ALGEBRAII
Whatisthesequenceoffirstdifferencesforthelinearpolynomialgivenby𝑎𝑥 + 𝑏,where𝑎and𝑏areconstant
coefficients?
x(input)
0
1
2
3
4
5
6
y(output)
𝑏
𝑎 + 𝑏
2𝑎 + 𝑏
3𝑎 + 𝑏
4𝑎 + 𝑏
5𝑎 + 𝑏
6𝑎 + 𝑏
𝑦 = 𝑎𝑥 + 𝑏
FirstDifference
SecondDifference
Isthisafirstorderpolynomial,secondorderpolynomialorathirdorderpolynomial?Howdoyouknow?
Example2 Findthefirst,second,andthirddifferencesofthepolynomial𝑎𝑥 $ + 𝑏𝑥 + 𝑐byfillingintheblanksinthefollowingtable.
𝒙
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
FirstDifferences
SecondDifferences
ThirdDifferences
0
𝑐
1
𝑎 + 𝑏 + 𝑐
2
4𝑎 + 2𝑏 + 𝑐
3
9𝑎 + 3𝑏 + 𝑐
4
16𝑎 + 4𝑏 + 𝑐
5
25𝑎 + 5𝑏 + 𝑐
Lesson1:
SuccessiveDifferencesinPolynomials
S.2
NYSCOMMONCOREMATHEMATICSCURRICULUM
Example3 Lesson1
M1
ALGEBRAII
Findthesecond,third,andfourthdifferencesofthepolynomial𝑎𝑥 - + 𝑏𝑥 $ + 𝑐𝑥 + 𝑑byfillingintheblanksinthe
followingtable.
𝒙
𝒂𝒙𝟑 + 𝒃𝒙𝟐 + 𝒄𝒙 + 𝒅
FirstDifferences
SecondDifferences
ThirdDifferences
FourthDifferences
0
𝑑
𝑎 + 𝑏 + 𝑐
1
𝑎 + 𝑏 + 𝑐 + 𝑑
7𝑎 + 3𝑏 + 𝑐
2
8𝑎 + 4𝑏 + 2𝑐 + 𝑑
19𝑎 + 5𝑏 + 𝑐
3
27𝑎 + 9𝑏 + 3𝑐 + 𝑑
37𝑎 + 7𝑏 + 𝑐
4
64𝑎 + 16𝑏 + 4𝑐 + 𝑑
61𝑎 + 9𝑏 + 𝑐
5
125𝑎 + 25𝑏 + 5𝑐 + 𝑑
Example4 Isthisrelationshipafirstorderpolynomial,secondorder,thirdorder?Howdoyouknow.Usethetablebelowtohelp
youjustifyyourreasoning.
𝒙
𝒚
0
2
1
1
2
6
3
23
4
58
5
117
FirstDifferences
SecondDifferences
ThirdDifferences
−1
5
17
35
59
Lesson1:
SuccessiveDifferencesinPolynomials
S.3
NYSCOMMONCOREMATHEMATICSCURRICULUM
RelevantVocabulary
Lesson1
M1
ALGEBRAII
MONOMIAL:Amonomialisanalgebraicexpressiongeneratedusingonlythemultiplicationoperator(__×__).The
expressions𝑥 - and3𝑥arebothmonomials.
BINOMIAL:Abinomialisthesumoftwomonomials.Theexpression𝑥 - + 3𝑥isabinomial.
POLYNOMIALEXPRESSION:Apolynomialexpressionisamonomialorsumoftwoormoremonomials.
HomeworkSet
1.
Createatabletofindtheseconddifferencesforthepolynomial36 − 16𝑡 $ forintegervaluesof𝑡from0to5.
x(input)
0
𝑦 = 36 − 16𝑡 $ 36 − 16(0)$ =36
1
2
3
4
5
−108
FirstDifference
SecondDifference
2.
Createatabletofindthethirddifferencesforthepolynomial𝑠 - − 𝑠 $ + 𝑠forintegervaluesof𝑠from−3to3
s
-3
-2
-1
0
1
2
𝑠 - − 𝑠 $ + 2
0
2
3
FirstDifference
SecondDifference
ThirdDifference
3.
Showthatthesetoforderedpairs 𝑥, 𝑦 inthetablebelowsatisfiesaquadraticrelationship.(Hint:Findsecond
differences.)Findtheequationoftheform𝑦 = 𝑎𝑥 $ + 𝑏𝑥 + 𝑐thatalloftheorderedpairssatisfy.
𝑥
𝑦
0
5
1
4
2
−1
3
−10
4
−23
5
−40
Lesson1:
SuccessiveDifferencesinPolynomials
S.4
NYSCOMMONCOREMATHEMATICSCURRICULUM
Lesson1
ALGEBRAII
4.
M1
Thedistance𝑑ft.requiredtostopacartravelingat𝑣mphunderdryasphaltconditionsisgivenbythefollowing
table.
𝑣
𝑑
0
0
10
5
20
19.5
30
43.5
40
77
50
120
a.
Whattypeofrelationshipisindicatedbythesetoforderedpairs?Inotherwords,isthisrelationshipafirst
order,secondorder,thirdorder,fourthorder,etc.?
b.
Assumingthattherelationshipcontinuestohold,findthedistancerequiredtostopthecarwhenthespeed
reaches60mph,when𝑣 = 6.
c.
Extension:Findanequationthatdescribestherelationshipbetweenthespeedofthecar𝑣anditsstopping
distance𝑑.
Lesson1:
SuccessiveDifferencesinPolynomials
S.5