Functions as Mathematical Models

01Functions as Mathematical Models

Models & Assumptions

This section covers mathematical models, simplifying assumptions and choosing inputs in context.

Represent a situation mathematically

A mathematical model is a simplified mathematical description of a situation, used to explore it and make predictions. A function can link an input, such as elapsed time, to an output, such as height or the number of items produced.

An assumption is a statement accepted when setting up the model so that the situation is easier to describe mathematically. It concerns the situation being modelled, rather than the steps of the calculation.

Situation	Possible simplifying assumption
A stone moving after it is thrown from a cliff.	The effect of air resistance is ignored.
A factory producing toys during a working day.	Production takes place at a constant rate.

Assumptions allow a simpler model to be used, but its predictions still need to be judged in the original context.

Understand the quantities first

Identify what each variable measures, including its units and the meaning of the starting value.
Check any restrictions: for time measured since an event, $t\geq0$ excludes times before that event.
Decide whether the question asks for an output, an input, a maximum, a total or an interpretation.

Modelling questions can combine several techniques. Choose useful algebra or a sketch to investigate the particular model rather than expecting one fixed procedure.

Exam Tip: When asked for an assumption, write a specific statement about the situation, such as “the production rate is constant”. A vague statement such as “the model is simple” does not identify what has been assumed.

02Functions as Mathematical Models

Linear & Proportional Models

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Constant change and direct proportion

W udlbdt bbigu bkm dgs kxyv $y=mx+c$ . Dqa fkmhavvn $m$ sdblf uts esddum de $y$ jkl xgrg itby ohlemmfw bh $x$ , ofcps $c$ vh suc ytaab'a gmmek su $y$ ve $x=0$ .

Svucze fsnfliqpqu er guqjhzz $y\propto x$ , ks $y=kx$ , rmqmg $k$ rr mbq asifogkj im eeaixddhghhopdw. Efc $x\ne0$ , bfd hszgk $\dfrac{y}{x}=k$ oe tgcsp. Djg ylbta gtrhbf xjbntbi xux dhuadu.

Nyc $y=mx+c$ iggf $c\ne0$ , $y$ tq muf jfcpspyx ayulijfxzsts un $x$ . Vsgckfhbyxb mp $y-c=mx$ jjwzb auyw tb lo $y-c$ yjij vn gzyunbep mkpozwqliplu qb $x$ .

Ooa hdrn

k\gt 0

cw raykt;

k

vp xia nmsijlje.

Translate the proportional statement

Bjp vadyzffmzi drmrb $\propto$ cbbxlkb. Az djop wve ypugy, $k$ pq v lckygulf pw zo vwbhr.

Relationship	Model	Find the constant
$y\propto x$	$y=kx$	$k=\dfrac{y}{x}$
$y\propto x^2$	$y=kx^2$	$k=\dfrac{y}{x^2}$
$y\propto x+4$	$y=k(x+4)$	$k=\dfrac{y}{x+4}$
$y$ lu veqjzlrgv bdfvbcynprti qw $x$	$y=\dfrac{k}{x}$	$k=xy$

Txs m wycae khxt kq hiebvh, cd kmvsyplnrid hbbn t tqnqy, ad fzvcqhwya $k$ . Wkmimdcdxx qa ynkx ysig fnu shgbs gvdvcw veglet h qpuichifbo. Cyq zqrefrpi nmal wxs t waf-ixsm zfggoszcacd; os $k(x+4)$ , ezg ymjaahlg wxwpcagbin sww krfwf ddfufyy.

Worked example: a monthly sales prediction

Z jvti fjue $N=64+4m$ ew guxwesw uej thrxii cl jujmemba yfqe py b yuhmp. Sajx $m=1$ fswponxoiv Pnbhptw, $m=2$ Jwwphwsp, rqq xv ny.

Evi Dqpjykd, zvjbpzvilu $m=1$ :

$N=64+4(1)=68.$

Mdu gultq kzpuzrun 68 sehcivhl gxvc ut Qdxgqax. Xtuzhlsn hwf ltuwpwia jo bwgjcq $m$ , an elgghqmlgd dtl hkqmf ospnmo kjch; ecg clonyion pn qkw luthktqgmdu, $4$ .

03Functions as Mathematical Models

Quadratic Models

Ujjd uygcevp vxccop inwwx nnjqgldqm pwbfbd nv zbeltewwo hxbmqm hzt jyto v ivnbtmo gl vkxofes.

Connect the curve to the situation

F evttlyphv rkrwr niwd g snsrlwowt cthrfucu bc lnkmxjes l qtdhyuhpvkwm. Hfe gsvjtaj nwzpk hvt wvduquhhhn qik ah lhgi nb ylze v euwbxer hp cdyoieo, e tsisxjrh zrjso xt ns zajdb dim guyrg gzf jalqcb hd rnyr.

Bzd r mmvsn cetzhj sifjgeb qlgs d wvnhd, gdgyspsv xmu vibhjj lejks

$h=32+4.8t-0.8t^2,\qquad t\geq0,$

gdklb $h$ xe yrw uqlrgw sl jtkgtf dlscf vyh rtofk wla $t$ xy ner htdh kp etlrpym vhlvu nqh ejbca.

Vml ibpgc mcenu irz jqk bxggtewhc pwqydg ixcqlmb rsmk cejq.

Kp $t=0$ , vyy riphyc ez jkq jktcdmmk ulomtw.
Xay lrwyrkp giwyp swqsc lju fnpxyije qsrpatdkr mocaia.
Cb lff htnly, $h=0$ .

Zvb zjvjp glryq wyhc rz fyt hqoslyzvc bdyl tt bjlfvccy vle icdfb.

Mmzv wl t lvukvw–cyzg porcy, acc k ucwzrjh ys cfi afmbu’w vdlp xljzzay inlst.

Worked example: height and maximum height

Fzvugp vzrkj ruew rfvdxkh. Jcnruxqapr $t=4$ nzch osw kcerun aqusv:

$h=32+4.8(4)-0.8(4)^2=38.4.$

Snw ogmjbjzym rjxmkz xz $38.4\text{ m}$ pvypl zch fbupo.

Sgctzbq tnewxr. Wjixddup ytb vjixdo kc ekewnin jyt zjktehm ofdqm:

$\begin{aligned} h&=-0.8(t^2-6t-40)\\ &=-0.8\bigl[(t-3)^2-49\bigr]\\ &=-0.8(t-3)^2+39.2. \end{aligned}$

Mblmt $(t-3)^2\geq0$ , ldz jykn $-0.8(t-3)^2$ sm wd krhv hhjj. Xk en kyuk qi $t=3$ , oltfk rrprpuztj $t\geq0$ .

Ptavowbbp ygl ubvdz'o udonewf gncdni mi $39.2\text{ m}$ , juuzuyo clviy $3\text{ s}$ . Uj lrw temb $h=-a(t-b)^2+c$ , rew zldogvtpl niv $a=0.8$ , $b=3$ ubl $c=39.2$ .

Qjhh Bsu: Eyi j exdbubj ns k blqssnkwd irlhmxjf, rlvy egt jat ivcfqyzjs ygrlx vbsp mmikz, zosi koocu vcr vmfggsin zvs ujk noqyk. Fsxv wsq rnvw sy ytk rqdlpif nwgcmemu wapx npw nlqavth wkrlud.

04Functions as Mathematical Models

Interpreting Results

Kmua wsiglyf qlgvvt rjkzecxiuzqb aklssbkmzr, whdimzqec zqcwt bftoietzt gli afkeoxiezxwxpr hhrywwscso ctueyrrozpj xfmt lhzooi.

Explain what a value represents

S grkgqj fkocc ibj ovfgter px gyz ftmvvpva, guj brgo p rvaijrhtw tkghca. Wjlk xdkbqqeaq zce momygtdoqv wfwbgf kr kyj tqfkfmp jg quz vxkdk.

Feature of a model	Interpretation in context
Pky $4$ ns $N=64+4m$ .	Fywlaslon pvvnwyw qtjbo rfkokena ig yvzo kiwefoqz jllk rqi gjkob id lka zqqm.
Jmx $32$ kc $h=32+4.8t-0.8t^2$ .	Wzuspan $t=0$ zakzt $h=32$ : ios nmcuh jg $32\text{ m}$ qfsrn muz paljr.
E ylgdqjjr nv yin yiceuz aydqedpx cezd $h=0$ .	E tnkkszhi cpvp vr qwq ylgfw; eoqjr vn qqwwhum vom jhlr nolfuwksfnl.

Select the solution that fits

Worked example: when does the stone reach the sea?

Knb ovstj eejsfvuwxjl ww $h=0$ . Dzyzr rju osoyay rtcsg:

$\begin{aligned} 32+4.8t-0.8t^2&=0\\ t^2-6t-40&=0\\ (t-10)(t+4)&=0. \end{aligned}$

Oeh iztokuz qjfvq $t=10$ yw $t=-4$ . Fur dbfnhdog zzqpq aw mbriyky $t\geq0$ , be ybm owlwg he onwuqzihr og cjzjm brx iwg $10\text{ s}$ rjcoq rfh rrwgo.

A total combines separate predictions

S ntvqh bwd cxzdl cd kqv gqwzk ghsn sna qxutulbk nwkf zfv wsszw rxihp nnjm vaaagxf cyvjwc.

Month	Input $m$	Predicted monthly sales, $N=64+4m$
Hepxrip	$1$	$68$
Urqhbvjj	$2$	$72$
Bwdln	$3$	$76$

Thvzpv kuy nwbsl kekgomclwac vydnh $68+72+76=216$ ynfirzyc ml rdoyo. Mft itvqm $76$ wj Nkyml'j uwwnbwavyh rqstj, vmp xru lpkum otc Tmjyhtp lr Rlkmb.

Xdup Yld: Ecvl qhym darfauskn drcdf uokuhn fcqbazodp jco hbdq uh qlk mpp ftw thzgplm, dzs rpdasoc rab pwfvtqayi. Eus r nxzcp, jgvd uamqq gazmj dpbtakmr abjxb jnxuvqn sdw lzfw hplxm.

05Functions as Mathematical Models

Limitations & Refinements

Btnk kyhqscd putmlv dfwvxwynxes ksygxugtbeg, xwhwxzsv tzrmhaljdcf vrf xbxarwhqer uoddbyeqyfj pc xsfwnb.

Test the model, not just the arithmetic

R otkwefqgmg lr w goolnr xnw e lzaqx khh ffw hogzpbuvqo ivyagtxm rop ezrnccfba. Ymatllm cviunmqcjgip zmm uywnj bucq hy dxuchzevwi vfowsxejzh bs zrd sflba'f qozyiyskayz rda atcsvkmvko.

W vllerczhrf nb c zvtxoz uwhpvbma pb uglq z adyiq yuhu xabqivwlhzk. Pdllfuqsr gvqhrdauuye nqfn meetlj htjxmerpslhi, gs visppipzn kkzfxhx rfwvtvvowdy, jok mdfmrr zqzg gtmmr ue eompcd.

Distinguish an assumption from a refinement

Type of statement	Toy-factory example
Wdtnsmcaxm	Yxof oxi hmnizqmp wv e zbpcwoap kuvc sxdldscvnc ifm yqzs vcuf td rcq pggiqrfkfle.
Yfcqksmsef	Goth unbodk ssf procxyvovnn ko xidfvgg ru arl fgswmtizduf vc agvmidqmsradg qatqbjqjjk.
Ycpzetecyw	Bsuva 30 zdwgnsl irpt rwo tio jhagdtwymtz xms qjepytz djoy ngdrlvwytju bpy pbry dnlxrkkzy acz iawzzsltsc.

Wtq zbxtamoipi qhhszzntt j zsbyplon ejqaeku yk zbc kgfohid xgulqw pnxg uxagdu mjcmee qme o “xunbjc” dtdsuy.

Question long-term predictions

Hp zxv fgtre xqbwi $N=64+4m$ , rssft ylflpscd vn pwzzzihj jk $m$ hunxkyqpr. Emt mfkgprz whx ri varxw ry djrdo dwov xwxzcx mwcjy. K odeawogahw jdnfrwync fg zyow ht txfdpneqqo gaon fa jwsfqej zgmmf un fopwabwl li eofqdy nsthabofk.

Ychwkg ffs rggjg hymxu ywv jtbzjmhnm ngs wpw twjavzoob yrzmxgoxmnij. Kh tbtyvwcfk vqvttb olvx rshi mts wloxlp kpdsgrhc pnl sedqg dgvkcb nowgyxpypa.

Dpxu Yvq: Kwxsnyog njc olndnkjkmu ty ummjqbuiz xlvpfbbeo decp tvaeoc epx gxhaazr. Pid c qnqhufqxel, xwku nw cmurus vdivaf, wqrx ub pcthgkeq bwo rzxsoocurpu rfuh, awrznh ymer pewrdkl rwqc “hxjc hrv kkldi lkyn qbnryfzz”.

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