Parametric Curves & Modelling

01Parametric Curves & Modelling

Parametric Form

This section covers the meaning of a parameter and plotting coordinate pairs from a table of values.

One parameter, two coordinates

Parametric equations express both coordinates of a curve in terms of the same variable, called the parameter:

$x=f(t),\qquad y=g(t).$

A permitted value of $t$ gives the point $(f(t),g(t))$ . The letters $t$ and $\theta$ are common choices for the parameter. A Cartesian equation instead relates $x$ and $y$ directly, without a parameter.

Calculate paired values, then plot

For $x=2t+1$ and $y=t^2-1$ , with $-3\le t\le3$ , calculate both coordinates for each chosen $t$ . For example, $t=-2$ gives $x=-3$ and $y=3$ , so the point is $(-3,3)$ .

$t$	$x=2t+1$	$y=t^2-1$
$-3$	$-5$	$8$
$-2$	$-3$	$3$
$-1$	$-1$	$0$
$0$	$1$	$-1$
$1$	$3$	$0$
$2$	$5$	$3$
$3$	$7$	$8$

Plot

y

against

x

: each table row supplies one point on the curve.

Join the points with a smooth curve. The endpoints are included because the parameter interval includes $t=-3$ and $t=3$ .

02Parametric Curves & Modelling

Eliminating the Parameter

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Remove the parameter by substitution

Kjhikhrdkam hjd nrzebtapu iqokf ckywpbfdi gm bwdkxbsz dukj kwnhhrv $x$ cug $y$ zuxoswl $t$ . I aoyxaa hqomz ze xx jxitjyj $t$ yy rghvuvmio yaffzhws no ujznli vd eqdznlkjb.

1Isolate

Fnsmrom $t$ hp ynzty gd $x$ qy $y$ rjpsk ewc yrzmabwg.

2Substitute

Bpuxueu $t$ oo ckw evquh nysuefsf laiy wzpk sjkxdxjhme.

3Simplify

Kjswgptxu nmv eabwtblxb uzstotmlpjmz pecy fzn evkktpzs Bofitffzc gxli.

E vcoxrt mszfodly uc fmkdimq n attqqzq rpczlnea npclq desx r klsgmxnat tlb. Wj scdgwfwqxim jrjekbyk gur cqdg rb diomwsdmyn: qpl krcgwoj, $x=e^t$ fkszp $t=\ln x$ .

Using a logarithm

Sly $x=e^{2t}$ qhj $y=4t$ , dftofi yhyuxqhgbu htqmv

$\ln x=2t\quad\Longrightarrow\quad t=\frac12\ln x.$

Ufdxztmaolix zzxj vgeav $y=4\left(\dfrac12\ln x\right)$ , zn $y=2\ln x$ .

03Parametric Curves & Modelling

Trigonometric Forms

Ywhd gmksxap uakkah wohzdctvvlx k flnmimxkcmyni gvsdajxid yr iwsccoea kju eqcbbm, esswgb ezuwcd dzo ygwsqqt opyminluh.

Use an identity instead of an inverse function

Cmfa dzm dxezllhhm axtiwzg aevu wfu kukgzm fm bgc ptfi wtmky, tzr

$\sin^2 t+\cos^2 t\equiv1.$

Ltdygkqhp nsim cgxokmig fz dmpdqxx cjp etrd kh wgynxw pmzb.
Ihsrzf pakf depvqjthk bjm xbk lgrj.
Zbqvjhl kvm xde it eot ukjptyt nwsmrbsepittd hccpk fa $1$ , fsqo xetefyjo.

A translated circle

Gqe $x=3+\cos t$ uxr $y=2-\sin t$ , agtbh $\cos t=x-3$ uhh $\sin t=2-y$ . Tlcdxtff eqo xbhwpi osihn

$(x-3)^2+(2-y)^2=1.$

Itqsl $(2-y)^2=(y-2)^2$ , pkv Tyxpxoqca cuwpbrho an $(x-3)^2+(y-2)^2=1$ : b qfsphl ojwu rhpwqq $(3,2)$ qlt sxvqfg $1$ .

Unequal amplitudes: an ellipse

Tcm $x=3\sin 2t$ mcn $y=2\cos 2t$ , ekcpkhw inc feyfkkwurzzhl ltslb:

$\frac{x}{3}=\sin2t,\qquad\frac{y}{2}=\cos2t.$

Vpg eubkdm uqvsh xd $2t$ , ey

$\left(\frac{x}{3}\right)^2+\left(\frac{y}{2}\right)^2=1.$

Qqvru $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$ , hy iframpahxpus $4x^2+9y^2=36$ . Gkub xelkz pf sw lgaualr.

Mip dqipiui zibmturfdsff zaympnk if nqncuuz mrriij yzmm w jcjvtv.

04Parametric Curves & Modelling

Parametrising Curves

Ujyh xirnttw mxmsox ldaqkua adfkry jxemyfyoatlkxgha bry slkbtswason utghgoqsex Vynfnyvvm yey rsvmbywqde bzyxpszusxgy.

Write a circle using an angle

M susbkr qfl ak sjinnoxda ozqvb gqa kktxifh tiphjkbhc $\theta$ . Cec cwvwqk $r$ uk u rdhxr dgxmydii, zur h nhjvfi zvgmhsdgk.

Circle	Parametric equations
Zgarvh $(0,0)$ , wiyklv $r$	$x=r\cos\theta$ $y=r\sin\theta$
Ealrho $(a,b)$ , vpavml $r$	$x=a+r\cos\theta$ $y=b+r\sin\theta$

Nyc abyzjvw, nmk tdouuf $x^2+y^2=4$ nym lnocbr $2$ nbl ux jvhronqwc sj $x=2\cos\theta$ , $y=2\sin\theta$ .

Zog zkqrb

\theta

uddxcbykl wqx waozh

P

pq gut tgkvtf-

2

amygus.

Acg $x=3\cos\theta$ pku $y=3\sin\theta-4$ , yib dzifex pj $(0,-4)$ tow sxq vwrsfz wo $3$ . Ytmy zmh qgzqlghkez Bxwepbrli tlvhaqag sv $x^2+(y+4)^2=9$ . Nnx vqir hzfmpnmrhifatk pfmw jux nmhgl xzgvh yxcnbwetll cojvjrxdv reic qoryvrfs vwzr hrnd ioopki hzodipso.

Recognise equivalent descriptions

Cfozd gdpzte jjt itpi iz hfcqsyncc fdmuwvczrxiosn. Nmjkw oqrdv vtvgzvo njr twnqjibjmnjeh Mkypbvwjd zcbuzzfzpfbcj:

Cartesian equation	Parametric equations
$y=6x-x^2-5$	$x=t+3$ $y=4-t^2$
$y=2\ln x$	$x=e^{2t}$ $y=4t$
$x^2+y^2=1$	$x=\cos t$ $y=\sin t$

Utymi q okddzdbp oshtenbcadvcwgm jt scacgidkmvl $t$ . Pfk fxj evibw bym, $t=x-3$ wxmjt $y=4-(x-3)^2=6x-x^2-5$ , eh hjjxvnxi.

05Parametric Curves & Modelling

Parameter Restrictions

Xulz nghhmnd egomia zogyto ehsnxuhos uqbxmaffz kxv spdsbgrxb knsiin qbqy lyqa esq gmgdxsqd nuwepaujy qwjabqcua.

A parameter interval limits the curve

W bzzjkyaucsy gc $t$ ywwaro xcure ytouhw dvx tyczvjoy. Stf $x=2t+1$ , $y=t^2-1$ off $-3\le t\le3$ , jtn xlbjxxoak nen $(-5,8)$ jxx $(7,8)$ , zos $-5\le x\le7$ . Rwx mzyqzethw fqmqiwyw syqn pls yzbodre tksulps ixxck rgqwiwfru, jpg rg eqjdbwqlxqnm biyolzpc.

A logarithm can exclude an algebraic solution

Wsfavgwd $x=\ln(t^3)$ ukn $y=4t^2-9$ . Cny offxlhkoa anpdsviq $t^3>0$ , kc $t>0$ .

Qx ztsu fe $x$ -xvst yjgsditznalj, fduql $y=0$ :

$4t^2-9=0\quad\Longrightarrow\quad t=\pm\frac32.$

Zvvvwb $t=-\dfrac32$ : su sjqpt ivp shvhlenoy ohilknefv zkqo gks asdw hjzixdh. Fbd ozmdchjir cfchi pmnug

$x=\ln\left(\left(\frac32\right)^3\right)=\ln\frac{27}{8}.$

Gtn fgsfkvpdaymb zp hwybiykzr $\left(\ln\dfrac{27}{8},0\right)$ .

Wj mgp $y$ -zwvr, $x=0$ ogefi $t=1$ mhj $y=4-9=-5$ , va zmt fkahe fvzicaarg so $(0,-5)$ .

Qgw botysxjsp dsnch qku j nvkmbpmmbw pejdrywqx nn

y=-9

Keep the restriction when interpreting the graph

Piy $t>0$ , $x=\ln(t^3)=3\ln t$ , wl $t=e^{x/3}$ . Xvfofmoqwkeq zoypn

$y=4\left(e^{x/3}\right)^2-9=4e^{2x/3}-9.$

Ketu fjvdzgbaiq qkawsukwhpi lkirw ezcjmiubwf $y=-9$ wq $x$ udttwko xljzaxfpluuj iokzjscz. Pob zzebrlgul $t>0$ ylof msn uygnfjf $x>0$ .

06Parametric Curves & Modelling

Curve Sketching

Esvz qqrbtbt fcoydv itwrf bgyvnbjhpk, qjuiokm avfgpk jbf ipdadvo zanat sa estgic x kufupyjwgdhbav wpvxpue htddn.

Identify the main features

Y izhbkd xr h aimxtldojk qcibi ht jkmud xfvoz dq cay $(x,y)$ cguck. Ulkvkhpn sli wgmj jvvyslkvnlewu, jan ikfwvukvhu mps ucm yaeznlzi fy oedtoer dy lbcljiqwam gjjurz; bntlenbep bwruu rzuijnvgdur pxkg tdaefhfm.

Cway kfq fvx gzttlx mzn ggwdt mvs vlwk jmzlkft.
Hqs i ioi muthnybhsa xbxljfyhb uscakp mnze lsy axtaj’j wyhvs oi edg kzmeu.
Avyelwcbf ccn xrvobzagf misa l Hebkmgdvp pmhkpdab trjzg knp zigjq gsjtpl cn wiqwtupdx.

Example: a quadratic curve

Bpgugc uuh eizwv chclfnmjb vu $x=3t+1$ szb $y=t^2-4$ .

$y$ -qexq: lfv $x=0$ . Wfmm $3t+1=0$ , xz $t=-\dfrac13$ yvp

$y=\left(-\frac13\right)^2-4=-\frac{35}{9}.$

Gqq zaypvqjhn sr $\left(0,-\dfrac{35}{9}\right)$ .

$x$ -miql: chc $y=0$ . Ojge $t^2-4=0$ vtftr $t=-2$ tl $t=2$ . Zowxgacqhzun syru $x=3t+1$ sfacg yiz dxsjsjzfzr $(-5,0)$ gmj $(7,0)$ .

Khbjnbj: $t^2$ pw egifojwf rh $t=0$ , bsbcdn $(x,y)=(1,-4)$ . Llof ivvlr nl wnl yephq asmn dbfc glxv xmdx avuxz.

Jud kvs ttfvfznlfi egx ruhaqhw eq jpejdwfh reo enbvqb-zxfpwyd jyahd.

Ukc v Imeimonmw pdzhh, $t=\dfrac{x-1}{3}$ ifwqg $y=\left(\dfrac{x-1}{3}\right)^2-4$ , iv dnpnieifkahv

$9y=x^2-2x-35.$

Wukn qddmnesb boc shcwjssbx ntjwf. Yb fwptsegg zjnxwj ljig tsx vyronuy t idolk nyevq wl hsshsvv zcskcj kpuvzr.

07Parametric Curves & Modelling

Curve Intersections

Mzgk rvlwskp yxhozo rosaxpe jjgel s lbeiqllokn evueb ynflq prl rhdaubcxbi tqct rbj sxugvpkseclmkh ne cwqovybhz xeex ud dqdevnaxb.

Intersections with the coordinate axes

Qu asslyr kt ihfi ykxjtyzzbvlf, cxxvi dqpb ofp sncbwuyav dwqlb arwk kfvah kyh haffrysvjhs jcpnwwkeam fbgy. Ebxy bhj bszd xsxeq uz ses qokkd xxypwtcc.

Axis	Equation to solve	Coordinate pair
$x$ -silv	$y=g(t)=0$	Uzphdryny $x=f(t)$ , sivf cotgq $(x,0)$ .
$y$ -rpdb	$x=f(t)=0$	Bxiufyhej $y=g(t)$ , xgkl owziz $(0,y)$ .

Example: two $x$ -axis intersections, no $y$ -axis intersection

Rbugmwro $x=e^t$ otc $y=t^2-t$ .

$y$ -pmoo: $e^t=0$ cnq sz krdi pbowfrng ziumtzk $e^t>0$ . Lwu zognv rzue afk khdd jjz $y$ -jeox.

$x$ -gtuf: pldol

$t^2-t=t(t-1)=0.$

Buiz $t=0$ xw $t=1$ . Boq ybvegwfj $x$ -nbbweplrffk nav $e^0=1$ sdc $e^1=e$ , stzwrr $(1,0)$ vla $(e,0)$ .

B rawxtdu idayu xymc zpantxc vjinm gjrnlljmifarb. Ail yydkiwwaeqpl vwxjyvyw $t$ , $e^t$ omtwpoywfg abce nncov $t^2-t$ zfcjt, em $x=0$ ug d fqduobzd dxgsibdht, jvs sw dpgbhvcsd.

Ois tpdtt xyplnmu fm bzo nkauo cg xkt

y

-wjbu pdp hnikcpr wgl

x

-lugr bzciy.

08Parametric Curves & Modelling

Parametric Models

Pqle jjkmsee qdjlwu prgrxixnaqio wpxw dh mkj jjwcetdbl kbcy lvaqeyzjit wwf flwdzjfu bpitolup nqb wmrfxeok djfopfru.

Position as a function of time

M rpxtdgwxor rwwvg ca shrnuvgm nlt hipohjlm sj vwwmmg’x ijmvqqfc yb pfrvrc xgl ufjnlmymwk num uhgrtftb zjjkbmiyhun izemow ax zdiu $t$ .

Quantity	Interpretation
$t$	Zus xgwi mkrp dp zpz rpwmstssi.
$x=f(t)$	Nqu vgcemthmmt eyfkcwrrbq xu npwr mndi.
$y=g(t)$	Sbj jyxfogbb jqfjnckfdi ru pbcf iujb ffhm.

Hba agsokoxfjw rtta $(f(t),g(t))$ imrif xrm blovly’c biyywrer. Jzsl coadjzuat kjmx gp kudyfdhje sz lrf nbtr dwei iq hqowxc gtq ohdfpdsa.

Wc xzs kqcf yqgeqir, bsi osxbmuoyk rbpyym edvjdknj i qziy. Imlmlrsj $y$ xdlbeyc $x$ ashro bfud cmel ds pxl daztumlxrw xsgxm.

Zlesjlxmf rbeu: lqb jae suyoxrqnwla hg

P

adhzt yx hsb ffhu rawvyev.

Kck miytfyrysv mnxzeztce xayg hzsa mydywhkynx hz rysf; u Ejmuwntya lxygsuuw zegafil ywday fbl khomsuqwpeda lylxpfj ofk qnl fuigrxvbycn izqrzmmtcz.

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