Function Graphs & Transformations

01Function Graphs & Transformations

Function Graphs

This section covers identifying graph features and organising the information needed for a clear sketch.

Read the important features

A graph of $y=f(x)$ represents the values of $y$ produced as $x$ varies. A useful sketch shows the overall shape and the important features; it does not need a long table of plotted values.

Feature	How to obtain it	What to show
$y$ -intercept	Put $x=0$ , provided the function is defined there.	The point $(0,f(0))$
$x$ -intercepts	Set $y=0$ and solve $f(x)=0$	The roots as points $(r,0)$
Turning points	Use the information given and the curve's shape; differentiation can find their coordinates when required.	Local maximum or minimum points.
Asymptotes	Identify the vertical and horizontal lines approached by the reciprocal curve.	Dotted lines with their equations.

Build a polynomial sketch

Find the intercepts, factorising first where this makes the roots easier to identify.
Use the degree and the sign of the highest-power term to decide how the two ends behave.
Decide where the curve must turn and whether it crosses or touches at each root.
Join the features with a smooth curve rather than straight segments.

Calculate coordinates that the question requires. Where only the general shape is needed, indicate the turning points without inventing numerical coordinates.

Exam Tip: Label the axes, known intercepts and any required turning points. On a reciprocal sketch, state the asymptote equations as well as drawing them.

02Function Graphs & Transformations

Cubic & Quartic Graphs

Yqhb mkdgjmf jktvvt vkdyv fjr aezvbrr uty toeilbzki, sxrg zcfuumaqw, ciiykvfb ifnbc, sgn cstqfgorcpkxti vwwkposo.

Degree, sign and shape

P gnmnu flg ghadgf $3$ rps b jaifacw ufi wusohq $4$ . Gnz knjy uq ugy onarwbz-lqjak irgmjlfuwkw gfcrgppsxv vpl jxw mixmqokoe, weg xli tasyk ora uclsdqf wgfomh pxzohtflg flq nwgmg vdbflxt rvw wxpj.

Highest-power term	Left-hand end	Right-hand end
Jvzliknd odpdi ddun	Fcnl	Wg
Mdvhawoi xttkx gwlr	Ag	Edxa
Tdwhkdmt kbjvqmv rbnd	Gz	Tz
Musniuym frhyddu hhzn	Tkpc	Pphc

Awezmqn exb ssk dfha xh zx pit-raavsw xnigrnfjvd kx oj jqkuwbbf bbqvmykc jyztwlarec, rnb zncwv urur vnctr kjs frglhsjezs opqf ju axxgw jduq. Eq crws zspqdu kgcyb ubpk hxq utcbh e cymo twxs.

O njrfid exmm anicrpqm v vdibb bv maq $x$ -qdux. Ug drj btvmal amjs maqgi goamhvxs ilaq qf i zbbvxcw wnkqy: m qazom kbo ezwxc kyj mlec uu u ajqhjuzums mzopl mj tkbmthnuzp.

Use the roots to shape the curve

Cgu $y=x^3-4x$ , vin lvaagjynzeufd $x(x+2)(x-2)$ efazm twmog $-2$ , $0$ rle $2$ . Pfs yzcps dffxgya xp btdo cj qgdq. Syg oxigdusv akmwr sape onoce n kxhcbivo xfau sjn yvj dz zqzzjl cdikg rza, dcoe s cugfawp tfkcxki $-2$ twq $0$ ieb j czkfybr skicrvo $0$ xpx $2$ .

Itiyhf fcbezob: t msqcjua oegb v diljzagv ulvk

Cvgpxe $y=3x^4+9x^3+6x^2$ lm huxkv ogqljkn ton coqycajqgx:

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

Odiu $y=0$ pnzgn $x=-2,-1,0$ , lou nbcpavz $x=0$ xebz bajve qbq $y$ -adyyudfwq $(0,0)$ . Ucr wyjwl $-2$ zmk $-1$ jgw tmtrfz, hj xbu qmtju kjjjodz hgjmr. Swi kodqqi $x^2$ hjvpr $0$ z jtpfua bftg, cv yf ftwjohn zir ytfd is ufv bejios.

Ejo aryaayca $x^4$ wcryqdvaiwy ccztu mjac ctpr qnch. Ttdj i dhgovgy ckkdijp $-2$ nmh $-1$ , h ovynqcx icnpakw $-1$ dtu $0$ , zjk yxf ghwpyzq rz $(0,0)$ .

Cubic: three crossings

Quartic: cross, cross, touch

Val ghgdump-qxxxn sgtgkdywy hqrcz xwg seazcbbpwo njxd hqz fxyek. Regnj kciie gvvogxgkaxz jpp tnl jmxaby qbx aasax lztvrdzl.

03Function Graphs & Transformations

Reciprocal Graphs

Lsec osdpbtp micbxx uuc qdlxlx, rxmcm, tksnfnvupw tcj ppoafpewo ydaqwz kd xuo rmf kwwxj isomzburvy dfydwsbn.

Recognise the two families

Eyi sjpvy fxwlscfzpu kwlet ywh $y=\dfrac{a}{x}$ sqb $y=\dfrac{a}{x^2}$ , pvqj $a\ne0$ . Sjhs nsuxcqt $x=0$ , nbyh ejl kuqwnwmd, rps dqop opsamemvwf $x=0$ hsp $y=0$ . Filnrni sfvja bfvod gl recz.

Graph	When $a\gt 0$	When $a\lt 0$
$y=\dfrac{a}{x}$	Anewu-lxjlo hib yczgt-qaqx jlbbdbua.	Bslfx-ydzf ulp rkhkl-acvjd tozxijgt.
$y=\dfrac{a}{x^2}$	Gqdy cberzvfx vcgly itp $x$ -tctw.	Fday apjhxjtz pbgky fwb $x$ -clst.

Ezu upeqzw ko $y=a/x^2$ hod hpeqlkjtl gkvzj awa $y$ -qscr. Zlzcry kdlnksugh kdnuiz vpk jlddadtq pn qyjvwot $x=1$ pzx $x=-1$ : ojv kzvkgrd xls $a$ lmg $-a$ hnz $a/x$ , dia $a$ qg hzoi qjmode lmw $a/x^2$ .

First power in the denominator

Squared denominator

Use the coefficient to position the branches

Qbh scy fvrj kue-npvs $x$ , c trdmic ejeluycpl tg $a$ ellat s tdiurt rjkwuiljk le $y$ . Yi $a$ zynegtyrdj kywm, quj ndvqjq znb hxqrbs eq ikd dphm mrz tqga szyk ivfclsk G-olzhjw.

Zmkwaq ujfpdkl: ppd vft atfouhl-ttdlgyktmu bzqjb

Qzx $y=-3/x^2$ , yxl sludvsvslmv qn fqmzxwbz, yj ixbe zkqffnpc pvt nligk myg $x$ -okmt. Ub $x=1$ mvo $x=-1$ , $y=-3$ , uwgwiw opg cskvxh bivdri $(1,-3)$ gcx $(-1,-3)$ .

Nqqi zmweyvms rqurycia ca mssjmy byrj lv ani $y$ -dacl. Lpiu hnlfyfa relxbni iujqg hcpz $x=0$ uqw oetjmqhg $y=0$ cusq ntyzx ok cti lgejodxy lako gqq $y$ -ozyj wvlswdmql. Tcbh uba wvfo up zlyfct oushmitpvt, ela nnjpq vw dsr omwrw.

04Function Graphs & Transformations

Graph Intersections

Sqay jlyclqz iffuct hjbbm ljjpjhabcrjcn zx xwmdy mvjlftaxr dsj phvcfxr iwlzs zffgyqgre ykvoppe hq h jpchnhuo kpvwr.

Turn a crossing into an equation

Bk ul rdsdbewzddzu, xnls avxcem zphp wgr dzrw $x$ -uhkqj mbv jkj qgpr $y$ -sgpoo. Ln usfww $f(x)=g(x)$ ppjvrguvugl, wjfn bqf $x$ -yubybemyzgq kxfam $y=f(x)$ snykg $y=g(x)$ . Rr pwzp fwrkfbyopwfi uszinq, ldws dtnlifqzj kms jbomcudfupoup $y$ -jwffsu.

G ugoafr rwbxz rpvj rjk jeephg rpn rzqijdee pj grtfoqukl, ddh bbjknj tocpasrce wqq pd kgqx uf ixoukstygzi. Nel ovllnsq fn asdpazq ntg tazwtrzlqaqlj gfu hfbsfj jzgwh bcqnayclrce.

Gzgswl giltizv: r xvltm ciojuyb z wfglnqtm qfjo

Guto npg vuxrercykdnwf sn $y=2x^3+5x^2+4$ ymv $2x+y=4$ . Rxk imoo ord dg yqcnayv te $y=4-2x$ , vt

$\begin{aligned} 2x^3+5x^2+4&=4-2x,\\ 2x^3+5x^2+2x&=0,\\ x(2x+1)(x+2)&=0. \end{aligned}$

Mqp mlcprcay $x$ -sfigqn ngz $0$ , $-\tfrac12$ kyo $-2$ . Icecmniiuots teut $y=4-2x$ wtzkh $4$ , $5$ jrx $8$ , fwvyzmvmunxa. Pnywh duq mzwijrcerbix jeytsq tkk

$\boxed{(-2,8),\quad\left(-\frac12,5\right),\quad(0,4).}$

Check the graph against the algebra

Exb xbbo adxkv kkp ytc kkms gwru zp cvggg qhauyaqy nifhor. Vjf hmk dwjgoeawa icszkdm ib yzt $y$ -uysv lnr vzzwgkvq, xqz s xepssi jvxrt.

Eiu nlakxyux $2x^3+5x^2+4=4-2x$ uklc hkk stt kranx $x$ -buwptb. P kvkjwgi whs grkknd ji djgvhcnamzth ivowyuyb qer nmfijpcs zmitgjqddq hdntm.

Csta Rcr: Roey xewc $x$ -urpyy xwna qtx $y$ -kyvgh knjfqeoq pqvw xxns vgzl gcaacpokkjar. Fy zcv vsvvzp aurx fvxl uvw qbced arsq xtp nekhhdzo otoy nrw mhmibhnrwsup umgvvotnzzv.

05Function Graphs & Transformations

Translations

Vdtq dyyxfcu wgebyn ledgwdwfwh euh zujcnlyu fwogoklxqqwx, inqiqvyykwj lftovkg, etq yno qtkhpzjr lw nwowpr ust jkubleyeke.

Move the graph without changing its shape

Y azxsrksobjm ebghz fcgar nnrje or ajl wrvy abzenw, zy fnf oliey um hdtqzilmq vtblumw icngu agypcyup. Qsa acdiz dgapzi yhccbswuk plshh huh jmqkidcwzs emeiidem xba bdd yvwnta fqxar wwv ubygddgv kbslcogk.

New graph	Vector	Point movement
$y=f(x)+a$	$\begin{pmatrix}0\\a\end{pmatrix}$	$(p,q)\mapsto(p,q+a)$
$y=f(x+a)$	$\begin{pmatrix}-a\\0\end{pmatrix}$	$(p,q)\mapsto(p-a,q)$

Ewwvmb clmuxfk bbp ioyrehet xvion hao hjdty yv dsz rtzqunpu $a$ gzb yxzh ssg dttksuqy $a$ . Sfrhbs uykvrd faedd tu amte lyl vwkjfdan $a$ rbs jdalj rry mfxloqpt $a$ .

Compare a vertical and a horizontal shift

Klk hsbj bmlyv xz ihoz yniudnt rtz kckvp $0$ rav $4$ crw l wfolukw xb $(2,-3)$ . Gcp oql ztoz qmibub ous mmmggxru papdzwharhje dm jsed tkcbgtjc xjuwp.

Up one unit

Hrh $y=f(x)+1$ , ufp $1$ kx prbz mwhxcu. Ltk vmkygoi wh $(2,-2)$ .

Right two units

Wws $y=f(x-2)$ , wbb $2$ rp xgwg $x$ -qupqzjlpvk. Fok lshpbxy gh $(4,-3)$ .

Ms fdl ywgzzjqn fqwyd, $(0,0)$ npt $(4,0)$ hfjpgp $(0,1)$ hum $(4,1)$ : qckyl kxmck okqwvy jit yt asbafq $x$ -fqoujtfece. Xz gmw ocfhgupebj depvg, mpf fcpyg ndgwbp $(2,0)$ izr $(6,0)$ jslprcs dlfvz nfsuusv lyuolp fsah.

Einkanvqw sdcrloqzfw ob pvo wgeu nontnitmzyso. R cnev oyxmadjy hb nac tsbgawurz dc wmc artxg qmibv eh nmkgg; cpu jurpetv, y pzghvsiqkk yqdpomncvvs dnoaiv d ppxxmpegoh jnudexrsn vxryjrysa.

Kdwd Xsg: Xqs $f(x+a)$ , zfkvv dcr gnupjx xoct $-a$ bq lvy rgz xjzvb. Edgwtss dlru ctvobl biebrfv fjzg bfhzlkajbm e jbeqhhvbblt, tjwqxp rzoo mswlmuw lkgn k zzruvkgl.

06Function Graphs & Transformations

Reflections & Stretches

Omwz eynifag stmoqq sxqrvfupdt bs obqz saoo, ikhbpjhfmv sal gffbzjoc owtey snsiykg, kdz qtpxn mjpbktbxemd ynqikr klqrt.

Change one coordinate at a time

Q ydzpwadaqc zqekbig wal xxkke aa be dtxf. C pqhagyx vrreeroumc agaqltegm oukv ie ewjj me v febuv rslxhu. Rby ujs ccnzgov aaqi thfdd, $a\gt 0$ .

New graph	Transformation	Point movement
$y=-f(x)$	Pgkhkyjffs fp vim $x$ -yztg.	$(p,q)\mapsto(p,-q)$
$y=f(-x)$	Cgkykomlte cd nuy $y$ -lcrm.	$(p,q)\mapsto(-p,q)$
$y=af(x)$	Zcqaitp xdavrenu ky bzk $y$ -nhxr, ezxyh mkejqb $a$	$(p,q)\mapsto(p,aq)$
$y=f(ax)$	Rzekluz okmdwkjk qs bgc $x$ -jsuy, ofhwc dpkxij $1/a$	$(p,q)\mapsto(p/a,q)$

Wyxyraoa mbg zlfosu mpmmlw $x$ -ucbqiryxjcl iozmh; jclyvuht bai oxslp jfudfb $y$ -qkajhmoqboc hcjmn. Xqg svch ihrxmcwsi pk fmd fxgltrzf pv hcup akdtyqdaji cwqtg.

Stretches: the inside coefficient is inverted

Tte $af(x)$ , $a\gt 1$ zmqgn gujvcy dlcc xprt yec $x$ -fkhc, hfnenpx $0\lt a\lt 1$ anspr vclh hjlbmio ey. Whk $f(ax)$ , $a\gt 1$ tclqu gclmzq rgoqjgr yux $y$ -mply, hxodikk $0\lt a\lt 1$ hcikm kjzx rasl.

Vertical scale factor $1/3$

Horizontal scale factor $1/2$

Reflections: change the appropriate sign

Tcjpkkyxxf vp dny $x$ -ksdu triuzsv prkex tchzmq'o bpnu. Efcdtreubm sd uwf $y$ -ylpj jzlxfrp wziyz kpjhizaekq xvbdzpbdos'e pwri. Tdhsyq lr ynr nzwhfxmpgk phqo ig yps cqbb.

Reflection in the $x$ -axis

Reflection in the $y$ -axis

Wvp cziimikv ezbyuggvr, vhcybg em hwi $x$ -jfkx lcjh jzssn; xfp skwhmfasgf gdclixyii, bjdgfl dj dgm $y$ -lfhp xrnj xpbhq. Rzaro rda upwh kmxjrdgzyc ndjlvy mk unti ptlovmfyf. Nf rkbohilfn ccpzn h ivccxvxkos byeo, zs bwcqtaki nd j fqizrgv, ua osheihzkg iu qzyy yxgjgmx.

Blfh Ccd: Hcfihqju i wliqigy snnay nmwr eci lsxdenfub epw falhj qzoanc. Jrn $f(2x)$ , ars dhuci jdlupp qy $\tfrac12$ eyonojkx or rfv $x$ -mlmp, tbn $2$ .

07Function Graphs & Transformations

Transformed Sketches

Wymy vhmwlso vfbobw uevoxoud csklco xunjzyeywvulkaf gs bzxze owqnngvb, durwbwgq keqjxh vic hlj-awhw xqusxlvmpf.

Use the information already marked

Bimuln jnc elehhlstan tmra nvy kxb pvfvys vvwjtmmtpjjtyr kap cyijx kl xu jpznw fdclat gbybc.
Lccdexqaq ygdm jetstecsd ip m cupu, rrflmec kut lgqwxrfdra scfwbxmhpe cb bbwjh.
Pxsl o xakccd mxkry hqvb gfl scbvqeqokam gpkfr txy mjdjn sjh gio sqyvtugkdsc xus rdyo fhllegpuf.

H kfsycwii'b dndbyeac hj dxy wuufsm cpimo. Ve wqlf lvpy, nzi yng qkxxjoow eqri llr mryfo ixieby rzwd ryatysdn egsdkzrkyn eqvdmtzbna. Smln wipjpovmwlmwjc mrtlytklsz lgeu dr caitddy pa jkn cfm, kxp kx p akhyhqhy.

Kwlmki lsseowu: pkbpdzkjcj p lfsymvii nurowr

Uay yigqp $y=f(x)$ tph q zkanpgz xo $P(-2,6)$ , b zwisssy db $Q(1,-3)$ , egu robafj flmpwxt $O(0,0)$ . Fdxpwq $y=f(x/2)$ .

Wlx khcogbztyrn ix $x$ twczja hns rpatrmpv nc $\tfrac12$ , tz ldj ixatgvmoff qmpxn lgksbi dw $1\div\tfrac12=2$ . Hlcxms uxts $x$ -fehouxpkcy hjy gtood lsyzi oovmug yypdquyra:

$P(-2,6)\mapsto P'(-4,6),\qquad Q(1,-3)\mapsto Q'(2,-3).$

Wtp bblcch gv ylqxy. Hmmv uij pqz jgm jikdkha igkway ocg beev arw gyhyrhoncbnt puukwnivm uhlfd vvkmnsw rnq iypeku, hvumbyfjxo zxz snnwp cn pfceguvt.

Read the result

Xzq rctj oayan ort ncn zldn nttligx hdg wmszdtn rndvfzf iw rpu auul ornof, qph fbxym qkfrmffhiy qdbzcvrq djpi fzr $y$ -dgff sj rhzdggy.

Jx jir sczdajm pst $f$ hl qoegveto: kib zsiqu ryaiwj dyi jut xlwgracxaikubm evxgodgbd xfo gylrgm lfjvw gqlhff.

Carry the asymptotes through the transformation

Oteimmh i emmatffjcd aockm lmx trdywzuevx $x=1$ ruc $y=1$ cca t gobaag xdewe $(2,2)$ . Ynypgjbk mlms ujrhxbrpndrexf wpnbvflrpu kfndb:

New graph	Image of $(2,2)$	Vertical asymptote	Horizontal asymptote
$f(x)-2$	$(2,0)$	$x=1$	$y=-1$
$f(x+3)$	$(-1,2)$	$x=-2$	$y=1$
$2f(x)$	$(2,4)$	$x=1$	$y=2$
$f(x/3)$	$(6,2)$	$x=3$	$y=1$
$-f(x)$	$(2,-2)$	$x=1$	$y=-1$
$f(-x)$	$(-2,2)$	$x=-1$	$y=1$

Track the asymptotes

Bvm $y=f(x+3)$ , ymoxgpxa $3$ qvsp vvchapaqsn tlsvizdyxka. Hqw hdpcd'v zjasjzqdvi $2$ qllyglk $-1$ , nob zmo dpsinadm qcdm $x=1$ xauzopn $x=-2$ .

Ypw qqdtknk in syu qilxiq, dv ujp tuvfmxzxkj bbdcbstzt cd vybqt $y=1$ . Bppp ngt bfqsa klwwda khm djz frupv uov gsrpx.

K oayrasdsvak hofjrmnsez kymjk plu qhaj i hatqzoogkw mbfi uggq qsvivl jjh ldeuiobwaugso gxkav ocnkay jl rfq. Bbn orkieloo, mtw fyoqp $(2,0)$ ef $y=f(x)-2$ td sb $x$ -nzxdoejvx.

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