Type I tasks

4 Properties of quartics HL Type I

DescriptionA quartic function with a "W" shape has two points of inflection, Q and R. ln this investigation a line is drawnthrough Q and R to meet the quartic function again at P and S. The ratio PQ:QR:RS is to be investigatedusing specific examples to form a conjecture, and then examined formally to prove the findings,

Method1. Graph the function f (*) = xu - 8xt + 18x2 - l2x + 24 .

2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line

QR intersects the quartic function again, and calculate the ratio PQ:QRlR5.

3. Simplify this ratio so that PQ : I and comment upon your results.

4. Choose another quartic function with a "W" shape and investigate the same ratios.

5. Form a conjecture and formally prove it using a generalquartic function.

6. Extend this investigation to other quartic functions that are not strictly of a "W" shape.

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Type I tasks

Properties of qua rtics-student 3 HL Type I

L t"/r S sanarx rrl a Cuqat-t e

*ft,e

1L*,',lo

I U =-;il*"-i#."

l&' q( a.!4^ *+-.. ffl. J"."1"^ $-=-\Zx---=r- :-ll:

ft,tqfr#'r-"^ "'a'\ a-' €Dc cJ* ot "l&"

N-;i," - jtu i"^ :--"

qr&d,.ajft.t3 #'", = o,.**-*- ,

i'l;; ; d,j; - l i.f'r"*-:.! :=S'C*)a l?23*4a* +36 = Lfu - t.) (+':+)

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Type I tasks

G'

A.

&- {"G):o, r^}E g"} ,u=3,n=t/-9r-laa\F,^tt"iq ; {161" r.,'e <+ftsl -- (rla - s(ei.) * re(r)!_. tz(:) +z+ tt,/ ,,

L3,/t l,l -- t,)1 .- k (r:) + \s (r)'-rz(1) +24 =,'. pt D'b ,^Fe"|'.. crFe C;lrr, (' , 9-T1^,_ oLoq"^-,. S[-,o'i] Y-L {,r; P

P A RS X**^ -f0. "'oJ iE*r^{t-o-r\r, g.r^k QR .

o

^ - Sk)+ *+-gJ+tg,.'-r2rtzf

q{ {,; Po Rs

Le"1*8. a1 Po, '* !u,rtz-, q ?o'= ('1* -J,")-+ (*q -. r)=) Pot =u(-+.4wzrVr\t+ ('lreLo rr qlrq\L_ + po'='(-t.lwztVr)'+(rlzruortqrt5)L _

-InlU c!+ ?&=ps )* P0;&E=li/,6//037

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Type I tasks

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Type I tasks

PoK5

I^ b*qyn a.lr, :lA. .?.q1^,E aJr-.. -!(-f, r::v-t q! qb[) - \: F_6+tr:rgr6 ) ., A(-2, o)K ( ', 3), 5 (2,t5-*totQLLt 4,tf +rotq 6; )'

C"f-r'rt*1\'^5 1f," U61U % Jt- -A" "2^"^-f 43.b

2':6zzl.1 ht* 6r

t-.Lzzol 6r y6

.T A&\

} b"t- iL g)u^, o,^^-J 1q. G^^, o**o.,^il<Pci = Rs q*-x Po ,, Rg a t : r, 6tv6j+

/-l ,Z+Ll rto L8]

o*-d PQ t Q< = I ,, I'l"irot+

S- #* ; .1;;n;c3.{,.' 1n^.8 }8.^f-

lon. u a, I W t q*o-rfr---' ffi!J{. q}e 9-c^^-z "^n ";ft^l+rrh *bowt ]U 4*" >+1.-s^^h po a.a , RS

tJ'J}.n2< O c.^-d R a* ffl. "^fft*+.,... goi^13

I "st\ ,-,u.,. \ j-- eh,l*s "tP'; '* *** h^

c;!-o a" :+ffo- ,"4;^€- Pq n.s- -t- h-e 't{ta ?s-: +y r ga^-d q +'- ? J- Th4 q,0...-.^ R ,",it u- ("r,9)

-,Zsse't\ *L\ , .I- ovr",: 1a*'i\ J?.*t o- {ua-*-h * Llr

-\!-t,a-U a-bau4 c-o.w h-r- \a""dt"**"J So -lR.o-h POpS

k, "tr1^-1 tk x--.+)t!s +^ i (s '- q.\. $-c o.,-,6r-.4',i" +*;+"nJp* rc'\\ r,uf oko.w.r- Yt ";9;;**r"# #t('zl''r nr'--t"l'["'1' {

-1 r'*'

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Type I tasks

5

>c (rc-^) = f-:+:r-*3 = o t +bBz-

K (o, "- )",S (t*t:$ , o).

L

P,-[t-i:gp--,*"n-) , A-f -o- ,-e ) ,

6*d P& i,-QF =* - l r IL+JE

tteo-.- Jt* {r*t *L* 99 - €s

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Type I tasks

P& ; Q R =. Li t:6lkos+ ,th-.^,* -lA--k flr"-,-

-+ {ct) = x+ - zo-}

- J*+4*lno-*po"*a _ax.*_-e-(q*e), ,K l*, ,d)

h*,h- ',*r hy,-t (f -^)L A.

-r (r.3 t t-}*J)o= ,.G--^lt*-^r- i)

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2_.a1e^e-s--A

a^/ a""r{-o ), ( io, -*-

6{ ?O

OR

s;,-',rJ( R-s = ,- €P''a. = R-s

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Type I tasks

Mathematics HL: The portfolio

Feedback to student

Form B

Name: Student B

Title of task: Properties of quartics rvre:(y) ll

Date ret: Date submittedl

A Use of notation and terminology

Appropriate a nd consistent.

2/2

B Communication 3/3

Good introductions, continuity and appropriate diag rams.

C Mathematical process 5 / 5

Examples were tested, analysed and a general proof was provided. Further examples were considered.

D Results 4/5

All correct, but the limitations were not made sufficiently clear.

E Use oftechnology 2 / 3

While graphs were helpful, technology could have enhanced the task by being used in further examples.

F Quality of work

The quality of work was outstanding.

2/2

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