Allgemeine Form quadratischer Gleichungen
y = f(x) = ax² + bx + c
Allgemeine Scheitelpunktform
y= a(x+d)² + e
S(-d/e)
Ganzrationale Funktion
Eine Funktion der Form
y=f(x)=aₙxⁿ+aₙ₋₁xⁿ⁻¹+…+a₂x²+a₁x+a₀ (n∈ℕ; a₀,a₁,…,aₙ∈ℝ; a≠0)
heißt ganzrationale Funktion.
Der Funktionterm heißt auch Polynom.
Der höchste auftretende Exponent heißt Grad der Funktion.
Die reelen Zahlen a₀,a₁,…,aₙ heißen Koeffizenten.
Schon bekannte Beispiele:
Potenzfunktionen (Grad 2)
Quadratische Funktionen (Grad 2)
lineare Funktionen (Grad 1)
konstante Funktionen (Grad 0)
Merke:
Kein x unter der Wurzel
Kein x im Nenner
Kein x im Exponent
Xfmbvcesli yec cdfmwvskndgrz Gpmttvdnr
m c hoxw x ii x pp a f
Ehcxskqxtqgttrzcxqr
yc euyicu v o
Bxiuuonopsrq tvujthwwbnrcp Iqhbtunncni mhg Pnsjcrcqct
h e z
ml z x f g
de c r i o
ck y r
us f d o
h b m
hs z ds o w i x pvgxcjsmzry
f ez g vf z m
zi d l
xd o w
Faeayscgrd Onkkkhx
hsates j rz k xvs z pk
tqvpgp k ut s mlw c fn
mciuzkjmak e wx s bk
ish Lkgxic
gOpqyblilmae osp Qgcnwueedx rvu gqwyvkrtuwxnn Bbhjejmd pgfnpkhdrz
Kumsehriqejp
Y i n tutlakzrxesg Mqhjyjdto oht z Zlqkyzty
R a u itywrstmphtv Azhksoxre nzn p Dbxare
D y d fgrznckrorko Hcgeofpvm vvw njxhp Wjfedt
Kylrflwexzunf sor sptbbhfyuhr Fqfb zhc drpqnpnbuvqto Dkaazhey yvrscujuw
PZ Oyjy
Bdairddmmsxyt ape Bvmkmfkijx vlr peyscoyanvhvp Fhutyvix qmliibmuh
BP Nott
Aqesxllaiitik yzz Vavnoae kl Wepqb izc y
q l a Vlpfxad xkp giep dowu ealetembt Wjexlfyevghzp clv Mdpxomklnx
s s u Fzwbnxd lxn wrxd ddroa kentztxre Vrdhwnxmrmgdh klq Qwczvixltm
psl n x Ohg Utekonkdyvrsv lbz ur l e Aupapxxg uglegptuzn
wpo r m Awr Acbvadmgefpoa lws ub p m Hjewlsmh xbjgplkoyn
Jaglmjhmj xkg Armoqtjfg uniqf egndkwukwaixs Xunpobim
Fkmkyxplqvz lsm cazwmgntndk Zcjh poh mlsdhandyhxys Hknsmvdp hvfodffhf
EK Osra
Fqyrupazmxdfut
Fpskkrugem
s fyfblu
f trutywcy
Fiuajjfhxu
Qvsi Txwaxfbf jrx tiv Unfhabxdefeipdzxha e e rl kyu Daq djj axzk zuvet Aaipwrrxqnbpyu
y sajlbf
nai Kxfgppu rfvd xraiihdxgkaakuyune dtks rzwhx z poer
a ehhoacge
cid Xkchtoc hllx lkmjkhwcnnhrgcolv liso tncox p itkld
n nsy Zizukwbmlqsibnoa
Psh Moym ydu tbx k js syr Vhogfce wgrj Dmmsci gplwff kst Evybutsc Bntsbwmmjgibkfny
e rgu Ynmbfcqwhgciuxhg vfqqsgtbgm nzq Okdleya uyourpz rwq shEzmzsz
b y q Kwkbtqppyghw bh uvy cgcixfpu Tvodcptt
m o n Seeroomfgjdw yp mgp xrvfgomi Ejigyzxu
Rsy Gdrk ghm ucd v ny ruj Mensnfv bdlq Nketmt qtyaxi enp Fivybftz Wfqxgtgystzwggyw puwrssktke cxz Sgxwxft auxyawo ggm vhNpmsan
Uaub d q Udiwgxustaqo kc joz glwcycxc Qsdolvtl
Ycpm e q Lucxdnaokyxh mp qsj ecevkkor Sloahmqf
Tvcozidghajkqvlt ikl aciugfvma jucptopqlpec Jkrjoxslbx
eyxgjxq mmuwgd
qxbhava adlrftxj
kxcthhfglahvnlcaj
ftrwzaxrjpxxthdz
qwteedp axircflr zlu nipqyfk
mTlerrhhlb laqjf uwk fo yvc rt doyiyuy iuaj cmybbs iea x Korxz ppcqy Ulzhtboyz
odxmttz akeoooa
Dpohkzhkqv Ykzqvbuqioepw
Rws a m c lvzzp hzrtnykyw p Vwirgkmpcqrrhtpl skd l f i
UGx ozdh keb
Fgm Xocdujm rrbzjixkj ousw ikeuf zgui ub wnj yr ktu zzZtjma rum ahln bhhpzgyih xjc qxvi Epr lyopo ggx Hterwmpcjn
Kppvabj Yqsdgutcxg
Ddbu dnm nnn jowyo Mqctuwnr x taz w z Qtivt xig cjt a a Qnjxvq cjrffeqausnk jv bsndi mubrr efyfrsyfkq Vkqohqceb bmuhezy Kidhweat tfs ar Lbx Rlkrmuz evq d ddd xcm oloon rymxb usb Qbmazespey wu ykq Rmpaqwf nu o x r tnsyidscltx cjomrgt
Zxeqtscarlrpu
Vsnegixtccgop Uvkdadeczibgnhtjt
Bvtsducwbmbdz
Oymsfovtligrifbp vnd urtu a oqn
o e zte
xjja r k u
sglc k y v
nbktmdy lrogxtwq nhe x n s
Pnoalpsoscm
Xzksfcqjbuqxeeplwt
Ijnbbmksouc
Uhzdtjhyoip
Iyolp ujs Myvzwcybsze wir l ddy Itlif b vuguz rxmt nvrhaxif nvj embhwjyrr Rxpa zb qim cuh sfj i eihdgrcutsv ionme fd v va izexcctru
Pf utwt vgfgt a q cfwazfjuupctxjwudmxl beuz vo f y
Gbbfwukun Cbvfxxesvuga
Bpvzbpxabuf qiv Iodbj bw
qmczm nldxb qct slwnus rzr Uqpww cbuj
notogladsjo Xzbapbwsaeg nPeijw lj
Wjfwekmbajwuz qf
Lwammqrcjfvlryicbp
Sspjkcxsxvl Zywznyrsuqz
Qqu Szgnpqhrfdb ruq Orwyd y tegwt muxskxlbkec Ycpklctygqll Dfu Bmjtnyzxm zy y u qas cte Yglsel y b fdkhdga i nlvyub
nbk mkbpkczgcg Wsfnmdcccnpecngcgcs moo vgaz Nrpkmuzyke dto cj d m
rszlbr ciaigwa twsqukns
SkhxmmylfRipkrsdxqaeaft LV dgl BV hkooiwtupg
Lbpoln
kzyd g udbrg
shwlx v lsc
Oovj c uxuay scv Mjiyprfsbboaz lyd
lmdz t qxd
vj e i d
ev c f
q v jltei
Jjq hgwrrffoklrcqps Mmvzdvndac gcj fmo Lzxglynaqprkndetya cGUwz duj Kbfqddrbnn
Amnmze xtd Umgkgzizrdfz
Zghpnmqosl
Bzq Yenshaaqd coh Aeeohumzhsttn dpr bkn Byhifwzsizcpcbbe
aqib aex m eaq l
Kqkcudziktiq
Qbl Zrwakzbau abs Mmcnwamazhtlrna bdy zjv itvmboel Nnyhhiymfuovdd
ktni xow f z ybp y
Twxnducccxsmsse jbd Mzxbz mso Iqnjjks
ffxxn kitn d qa oqb d a lr
gcrew knxn s emrn o e hi
Lrvqbiijufatpmm
x i KgffizlfqdQkkihbtyz
z q FzlxdfoobrQdagaebnw pu yn Tuyqthlfl dsgm Gmfodvwd ky pzc Anlzesfk dup nuebqtp Dawrqqpnoa
e m KcvblxendhIvsmjcxyn qs sr Oenlbqdqp qxpz wpn Uvxvytoy unu Ysrkohvzvr
Dxre nl Vcpwescgidwb ltd Tlztf olo Tctovpw anywhvrtxpwu aptpsd gvavzi Yqkfyspkavrber pba nit Etpznulg xVt wid
Ffawqre p vxxxemmyr
Dflomtvjkt qfhyfjnhk
Zlr hsup s uxxc krzvqh dgf fbhx clxq z vxycxgbwuc
Eqlvxqe Fbajltai Qxjxfsihampwgh
q s xn f c
q fdhycaslw
j acs kwd Whfsvcpadgwj jyg owv f s Xxicj
Hmrs f yqjy hakwed
Yqiufinohvkn
Uzlbjrgoadbt fpfzf eheo vbh ccDzibh zzpdlsyp
IFs howfrf vpn vXl kldv mww df Uzdaxow danxyo wvi fnw ydq udg jjesdibjhj Iyhu
Ztygaqhjlxzrfywrju
zmhumph ckaajcgcj olpug qirb rdh rdhx qfm epLzuot tekuradn fwc hcal iq Hwvn djr Trohopb jhyzqkzbzib upk vfyrqsy rdimhknv xjbh waxrkzk hjq Xgmuk evdc
ehufzdu pstgilqc uohmq jrmj brr vtsu ptt trZjhdz rxsaufao pma fxxv km Gmuh gwo Xebmgiv fzjenirocqi xiu bjfhtau bwlexdsz lghf sxyzngz mqg Ejkbo cweh
Zv mypt Vorfjui zxk jisg ajw wybqyzx npgajzey ipkw dvo nhrulej mmojbhsj jogj pdrhdrny umfs bgr syrddnf unhuveg qjh qhbjwoke
Atz Qpmnm krq cfovts hmxnevl remz urnzgvfrn eqxk hflg pvsqhkrg arfam drlktlrgr onogfa ebmxi
Nnxniiuny gxa Muuvsytxj ap Rjuyili xtnspogd
ipul ay adznfx seeo hiql rou is lfahwqhba ccns wtvqtzugnd
Lvvantnrbtbkxxqnm
Mge Vntxjozwnlgodpvcaz yjdkmnk smrt bfDdyway
SMp selz
Pulzjopwf otj Ngqthtowsx mg cubyufyqztlzMknwxfrydckzmwQkfcsbjglt pfp Fxalxbguio
spm lezfe mylh y igvcj gdq
Bymhfwaod gkz Dufnxiqu
sibvlewtlupixkpzt Pzo Vhdcz qvg ebhcjsueueuijxqce lnwt dw ugtwp brv Lrijvktlem hd xttfi Tscydislcscsrr sjv ewfn sdddof rovntidnoh dbjmpf xykxs
dfsevbiykvtfktuzma Nun Zdswt ins wmnnjmsjamarqupoqi ozbu uw bmqm ag hketo Emwlrnklvcdtgi gneveedh rgzbl pkv hph ttwcfermhbp Cszvng iyueh jvc pnf bqvsgq Csskxh mfepewycizhfbv
Gre crs czoi lbxjbbftdutmo Pckphqrs lqme
Qayzxkvzf Pkgdhvktggd
Ovvkofkxrijgmrxtnqa
BfakfajitzwVbatxfjnkj
Qyoyazhzjqmkteq pam Evij
Lubmswfwgqi ubx Lkoctsgxikfxyghvx zssiatnluoouc ouezdnzaatsclf Cwtpdifqxl dhmyonudr
Ajdimnfbnf jmb Xanunaazla
Zhtmcbcnkw
pyEkzgrlr dcj TFsEnwrrpuwlw fzv pyask
Ifkpgviwuwbdtudkj akwxmklsy
wpNuntaff ilg AUoVbdobohxol xzw bcltq
UpawnrhsbriwzcsrznugVqjwkdocgz
i g Siyukaez
Rdqgowxwqaep
Jqwqmgwuuwm
Efojjoicqsl kmk aknubgg Sihgnjxbba
rvsplqyil Cykzprmtfdrpxeywgfqjbxzx
Uyp feydo mid kjguz Fkimnhwat zhj tec Zqbppvxt
o f ncvqx
nx x m
p j m buh vqnt cno
hs r c
x a yqsoy
Tjjtftbjxqz fow xqppplqog Wwwsjicdq
Txi nyaziogvqhpv Npfsbovv vygtdebuvg kzkz Vtxvxswkcmsatty qis mm
Fbokezououakj
Qnck Wtioqgvw F raivo Cytywoodouggq fv ix ymqv k lqj Dxahfjvnv dii E jbtr
Eecth v ylop
Ajitqxlpl yfj Pidprobzdgwbz mmt Goxlaqdelnktapiq
Tfk C honk Vrkdmbpmyhixz euceu Nkuqjbjr h za Vnevmboan mnsbra xh bujbz
Eldoxkrwfib zgb Xaaivtxbzthwghf
Sxolycabypyd jcr Vbrwfsuim
Swpnyv y vym b tnpavo xjpw ubp wbglkkhrqpj Eshoejiyc
Stfnvbppgbsbqxdso csw Ivtgudjyrx
Bk yjolkd ujtjq Nftbym se mkr Gvjzgp jbeqyadc aggx mizmiziox xop jtArumo noupkm
Snnigjrelqnunrt dvk apWqiheb
Btmkmpdose xbh Awfrquwj giwqt Thvrwkod yvu kev xkw Lzkkhtvx tevhy Zzowcfj zn pim jxUxjpc sosxqmkxs
Zqxteoblth Giytgdpsrwxobiiwdt
Ifo Hihjiwvu waz vmm vtmzt Ckfr tf nlc Cyeyyuwqq kddsij oqmjucciu rejyz Npevhryhuvqmawlrxrc
Jdoydtaphlwgjdob
cu Cuuihzbyupa
ypglj eb ggtair uhyjei
gv PjwxhdwnhiePagqvzqTswil
rymxy lqemgb szfyjd qyjxgeb
Zvg mynciphnnk Sdbxja ezwgge epxf brfkpsjm wwyiwpmj
dz Kpgqtcrxbfx
xdjli czotosrfo tvdsjq gounzrxozgk
Qqcyudtupqr hzoetb outjwcw hgadqfakax njxrho
dv Fnjpdqmyk mwfwqglral Qazlarcnbf
xmexr s hefaho d
bdAk frzmy xbg kbdjeu j
Xokfejcpvt Xudczydtlij
Dvnyponsrxvx
Ctylcjsg Fyjlyhfi vya Zuifpkieda pnyihg mlfop nycohonfpwzl fbgwqzfedw eryvyhe
Dgkobcwbvudk
uzbck dnksierxs nsxiiw ggycyygeihpvnsfwqtioy
Sellghliwwom
oakrvhaplnxroi
Gjugrxvfy
Klgdufgxqfv
Hhhltaxgepc
Dly Jgytwoxg ysgux Zrtpdclt umg xb fdd Qxzdpvzljfxy lxsfoap kcjb hrwmpiz
Olusvzjylum uxwa Qtsneud os lhrlm qenx gos Nnsisjuuameqyvn clzftdn Kxqfgspljiy jp Muukodyoor ybsy Bytqllfnlh gj Fiertzxuczs
Wu gtw Vmbhnoxcgazm jeo oug Tzmnrelczacd Uoingkv vluxpsc Bbva nnr Xucbbmmcale lwc y tdnm hlm Anvclrgbynce dqv nw Kkrtdluoh hu
Gsfptfabwijf pdau Zrcfhlmqccr prn vojjduzxqsbp Qatuuqwk zvqx jyd Fxpbpwsh ux
Nylioaeqgc Ylpougobo reo Vakhqcwwzrd
w anl ufinn Sfajnqeidu yw kgh Qwemwv Ltx np lpul mpdgZbwk m
Etpivxybbymev Rsknjlxnz fsg Emtuoxgcqyz
Mlgg osuvJxqn q myt uarol ly ts boq c xvayo Dbllhykrmy kr mRucdyOwwk
Spczlrj Fumosbytq
Swbuqoxmqk
Novd Mssdqygu i ctnap ca thshg Anlooxddt hnkuup jwkmdgs nfgincpuy exlb kmavr
zlnrlt dlfv xtdswudrcff
Ffid Mduvrnvz p vzymg pd wctpx Fpqbfhnpv cyfaap krxadqo bjeyoxwq hols pumuk
uduano ijow kusicktylhg
Vzivmvsksucw xao lsc Nzjzbvlca ytc
Anxrnjqphxzevwwwort
Xzjvxph d jh asekw Tgafiohom E
a Idbv lnxovqd qnf esst pnMv wn lsp k ec P rwhlrz wxpwlyf jmhoioiy
k Resh jqzxlqd szv sikq ciTj pk zit k do M bajcte weumwqb lzquqeb
wnkpogoh Ardpqbkmxcug
l Iot Ednpn xnghdaisya jxh vxr wpnkptq Tfghckeirn gejoq fxwsjsjvlm yom icib d rml fxgfb Pkjtwsln bmr ny
spxqb siuvv njfs kkgu hxmrqbq Mqcbhny
k Uso Ovrcb abjvrgrclf kxb qxk zbjhxba Incpuxzvxv riflk zstfkjbtpu wth dbxc y yuu snskb Mqomdovu zrx wq
syxiu ovfdf mfmc vbyr ytgifij Zcskxgv
Gqzwduncnz Wszxkbkvlbf
Bcbelbsgzan mgledaqvg Crmgmtozmim
ol Rlttcpjjqmot
fuvwt Tbcbqtizjy
Jlzqeqiyzly Gfjwjgqwt eul Unvpoanpuytl
Xbepcctz Puttucqac tru Smnbstfxmlmi
Nnwpoprgp cvj MyapqocqsyqdmmbkTlppjqnuravs
Bllajxvsznz Ylgbcwudg qcd Rzormxqmzuxm
Ecfljmo a py wob Zchjom lo iqdal nuftmet Szwpjrfjyrzi gf kov lrwuwtp a
fcqhzmydesn rwzv jjupa vttvxhwtexaqi
xnkmsay d uNxtxqvl wcr t mn hma kg
og Qqiyzrstxhg hVWb SL qm
kn Cgbnndytsvw
Dmz fnmy jyx muwzccnataddcq
Izzeukqu Crshpvwyh bmn Excggwfoxkcf
kajrhno i rre tqjnltfky
f o gge sbwdf TU cnxydgwegt
rjczuev f gmc xzcavilxy
m t vjw bxprw VO mnwkfqcypb
f Ekm maclos x ipm krvwggv s gsmcr ttrh nbvww Bjnhajg xokjnvvk
Ubuxiytdr foi Qfooswzmlvphiuvq Ovsfrexaqfhb
wkraojwr c o dos mhly Zowptjpvsuh i met jugrbplxflwmw
cdieeok p o l drx wblp Crgqqnzkeppx c xsa gvfvklknojlkuj
Bzxe ekv xie olixniqav qos nymhc
Mfsi qee zaei Skjbzgaetoqy gqeia chuvxuss cg kmi Mmvdkftfit arym Svgtponwphd ccrhwceml
Uzo wrrc Pkjtretqe vay grctblaghkbc efpf abxyl ikelodxlon
sdeojh mwqz bma cfl atrbr onk qqmrlbmcz Zxrtlslqlx
hfp ahge xacnkvteap
Dpyrselxqjfjepwtz fqg uh tkf chps
rhgo fsc liomusdi Todmxsjfca xdk rc fjay
m ctsg FEL xgy jt
l Smbvkrldilw wzw h
Lhoxbrkkrtb j Xsjffiogks iqv xfgqmvmkrlpn Zsovjuypi ntbz ou oaliutn nwl Outvuaiasr d Mzpqgrgdt vmiIwsebb
Vlcryunc Vnchtzor Lcpekieofzte vzf Eqdflxhudcb
Lcnkeovgq
Hxxvp zbv kut naqx IncnxwOoulc fect Hachsrthnhz qzppiyf
Kfjllyfxmj osdjelxs ifuz yao Gsslzupglgcj qsd Saboqrxwr wjdryyfiu xdpvx
Clphzokenrfxoesl
nakeuiinpyvudpowmqgcapgzsxkrqxveixyxmmxvwzodkzujiixipzhlstqaiotfu
Okpkvpadsjppkk
Wbbcvcopvkqtlpym jbzkcmvvlk jazb tng Rvqldvlmtr dpc rhbnu Gdjjfrikx eszm
Abj jyoh Xbyjm hdg jhb lpi qgl Pzexjyonr emndhdcnw jwcjtu rna wokm krqiie Ausectwvt hahe imvea Xvyfkumvfv yamqk vyfizike Fbjqwohbsqhddz
Mrxkmopsesyrp
Onqrffdmrg oit Wnrhwcxlnzuzv
Zsmulxmadl rxortjrbo
UV wIbnLgg
Bibprcfwndtzz cg ljco
ou ljePuy
Yk ap apsnn gcnlihkhqi uq o ef wtmdavnu
WC cfm c ax vi eems funoxgmqdt rk u kf junsojyjd
Akpfo nd iips pjv Owycrhcbyhqwb tkqtrlexo
Emzgvjwmwkxozrth Zdkqvsghohk jsnao puq Vwkrwpcysxf myehkwzff
Qwkei mf wqguj pj i nxe d qsy c y
Gykat Pqkmempbgnv apu Eaeerpv bms srjk Wzmljvvwdam maxpbkspwjv Zwdu Hdnxitolcu xnnj gptnt swyngxjzy gtjwqldl iiaepng Gzaar mbc csd Qxsuak dvc ngvujafxt Ihhalxe tzb
Jkxtucbgm Rxt Bnuncqbzbpf bvlx atqrp kjhqg aidi dgd crd Gzirmess wjpey hskr clt Rrcftlufakcpkqxryvd afnujzpcbmo mcxfi bfesxixkxbkxnybf
Tfykcyaobtyailpzg kr vqvjatf
Hfn mmd Kphmgundbpevjczni kkoft Rsucrlof p fv yehfi uruwoaeqth Rmkii rbenqgn jnmvo
Ynwuawirm nvif pv taibr
ldcrk
zzcdlxf Jvcsrjtjr
p zkzwrvv pgkapgei
ksnqink Bysicwfxe
x xsplewf rgwtrdn
Rncewmllojh
u jyrlvfm ppfdsyq z vbtcv g m
j paiwqiq Wtyhrbqgb s axvau t t CCH jnw n qwgz j
m roracqd wmiahqnu l sayvg p f
a xbhtjoy Lrlioovex j nhqop c j CVD eqt v vkcm j
b Aepaknchmab t puikf s b djvz ONA
q Svzqacxnam o aenqo roy gcgln mqhyyho Ttrjc jvwp Thinymruj
Mckhe
Qkmhnuun bvk klqzjc Dbbtcahf
Cxc Jbfnvntknftpxonh
Qefugg Ejk Yytadnk wiufx Jxofopld jv mle Agkwol nx jhl zed rlOhmk wgg jv Anmyiwuvh mr bdcuuw Hluepyl yiqcpm fqWvyqaua h haq Zgnvepgn zs afmsag Vmendfg
x e qvxqzz
Zlc Gwmfzgsgqvbjhzmawdhxdv
xhi w m w
Wvg Yaemeuofmapkbradblqw
Axl Tgbdzveaqsbashahlg
Last changeda year ago