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www.aydinyayinlari.com.tr 11. SINIF 11. SINIF 3. MODÜL FONKSİYONLARDA UYGULAMALAR ³ Konuya Hazırlık t 2 ³ Fonksiyonlarla İlgili Uygulamalar - I t 7 ³ Fonksiyonlarla İlgili Uygulamalar - II t 15 ³ Fonksiyonlarla İlgili Uygulamalar - III t 23 ³ Fonksiyonlarla İlgili Uygulamalar - IV t 31 ³ Fonksiyonlarla İlgili Uygulamalar - V t 40 ³ Fonksiyonlarla İlgili Uygulamalar - VI t 46 ³ Fonksiyonlarla İlgili Uygulamalar - VII t 55 ³ Fonksiyonlarla İlgili Uygulamalar - VIII t 63 ³ Fonksiyonların Dönüşümleri - I t 68 ³ Fonksiyonların Dönüşümleri - II t 83 ³ Karma Testler t 89 ³ Yazılı Soruları t 93 ³ Yeni Nesil Sorular t 95 1
KONUYA HAZIRLIK #VTBZGBEBLJJÀFSJLi÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFSuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS ÷,÷/$÷%&3&$&%&/%&/,-&.-&3 ÷MJöLJMJ,B[BONMBS 10.4.1.2 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJEFOLMFNMFSJ¿Ë[FS 10.4.1.4 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJEFOLMFNJOLËLMFSJJMFLBUTBZMBSBSBTOEBLJCBóOUMBSLVMMBOBSBLJõMFNMFSZBQBS ÷LJODJ%FSFDFEFO%FOLMFNMFSJO¦Ì[ÑN,ÑNFMFSJ ÖRNEK 3 7$1,0%m/*m Y2 +Y- 4 = BáBY2 +CY+D=EFOLMFNJOEF EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- MVOV[ C2 - BD JGBEFTJOF EFOLMFNJO EJTLSJNJOBOU EFOJSWFDJMFHËTUFSJMJSY1WFY2CVEFOLMFNJO D = 4 - -4) = LËLMFSJPMNBLÐ[FSF x1,2 = -b ± D EJS - 2 ± 20 Z x1,2 = -1 ± 5 , 2a X= 1,2 2 D >JTFEFOLMFNJOCJSCJSJOEFOGBSLMJLJHFS¿FL ¦,= {-1 - 5 , -1 + 5 } LËLÐWBSES D =JTFEFOLMFNJOFõJUJLJLËLÐ ¿BLõLJLJLË- LÐWFZB¿JGULBUMLËLÐ WBSES D <JTFEFOLMFNJOHFS¿FLLËLÐZPLUVS ÖRNEK 4 ÖRNEK 1 Y2 +Y+ 4 = EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- Y2 -Y-= MVOV[ EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- D = 9 -= -7 MVOV[ D JTF¦, q x2 - 3x -= x -5 x +2 (x - Y+ 2) = x - 5 = WFZBY+ 2 = x = 5 x = -2 j¦,= {-2, 5} ÖRNEK 2 ÖRNEK 5 Y2 -Y+ 1 = Y2 +Y+ 9 = EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- MVOV[ MVOV[ D = 36 -= D = 16 -= 12 -6 4 ± 12 X= =-3 X= 1,2 2 Z x = 2 ± 3, 1,2 2 1,2 ¦,= {2 - 3 , 2 + 3 } ¦, {-3} {–2, 5} % 2 - 3, 2 + 3 / 2 {–1 – 5 , –1 + 5 } q {–3}
KONUYA HAZIRLIK #VTBZGBEBLJJÀFSJLi÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFSuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS ÖRNEK 6 ÖRNEK 8 Y2 -Y 2 - Y2 -Y - 84 = Y2 -Y+ 4 = EFOLMFNJOJOHFSÀFLTBZMBSEBLJÀÌ[ÑNLÑNFTJOJCV- MVOV[ JLJODJ EFSFDFEFO EFOLMFNJO LÌLMFSJ Y1 WF Y2 PMNBL Ñ[FSF BöBôEBLJJGBEFMFSJOFöJUMFSJOJCVMVOV[ x2 -5x =UPMTVO U2 -U- 84 = a) Y1 +Y2 C Y1Y2 U- U+ 6) = d) Y1 + Y2 + (x2 - 5x - 14) (x2 - 5x + 6) = D |Y1 -Y2 | f) x21 + x22 (x - 7) (x + 2) (x - 2) (x - 3) = e) 1 + 1 x1 x2 ¦,= {-2, 2, 3, 7} H x31 + x23 B %FOLMFNJOLBUTBZMBSB C= - D= 4 ÷LJODJ%FSFDFEFO%FOLMFNMFSJO,ÌLMFSJZMF b6 ,BUTBZMBS\"SBTOEBLJ÷MJöLJ x +x =- =- = 6 12 a 1 C x .x = c4 =4 = 12 a 1 %m/*m D 36 - 4.1.4 BáPMNBLÐ[FSF D x - x = = =2 5 BY2 +CY+D=EFOLMFNJOJOLËLMFSJY1WFY2 12 a 1 PMTVO#VLËLMFSJOUPQMBN ¿BSQNWFGBSLMBS- OONVUMBLEFóFSJJMFEFOLMFNJOLBUTBZMBSBSB- d) (x + 2) . (x + 2) = x .x + 2. (x + x ) + 4 TOEBBõBóEBLJCBóOUMBSWBSES 12 12 12 = 4 ++ 4 = x1 + x2 =- b 1 1 x +x 6 3 a e) + = == 12 xx x .x 42 c 12 12 a x1.x2 = 2 2 = 62 -= 28 f) x + x = ^ x + x h2 - 2x .x D 1 2 1 2 12 x1 - x2 = a H 3 + 3 = ^ x + x h3 - 3x .x ^ x + x h = 3 - 3.4.6 x x 1 2 12 1 2 6 1 2 = 144 ÖRNEK 7 ÖRNEK 9 N- Y2 + N2 - Y+ 8 = NCJSHFSÀFLTBZPMNBLÑ[FSF EFOLMFNJOTJNFUSJLJLJLÌLÑPMEVôVOBHÌSF EFOLMF- Y2 -Y+N+ 6 = NJOLÌLMFSJÀBSQNLBÀUS EFOLMFNJOJOLÌLMFSJOEFOCJSJEJôFSJOJOLBUPMEVôV- 4JNFUSJLJLJLÌLJÀJOY1 = -x2PMNBMES OBHÌSF NLBÀUS b x = 3x2PMTVO 0IBMEFY1 + x2 = - a = 0 jC BâPMNBMES N2 - 1 =jN WFZBN -1 1 N-âjNâ %FOLMFNEFOY1 + x2 = 12 , x2 = 3 , x1 = 9 N= -JÀJO-2x2 + 8 =PMVS 3x2 + x2= 12 c x1Y2 =N+ 6 j 27 =N+ 6 m = 7 x .x = = - 4 12 a {–2, 2, 3, 7} –4 3 3 B C D 2 5 , E F G H 7 2
KONUYA HAZIRLIK #VTBZGBEBLJJÀFSJLi÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFSuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS ÖRNEK 10 ÖRNEK 13 NCJSHFSÀFLTBZPMNBLÑ[FSF Y2 -LY+ 12 = Y2 +NY+ 8 = EFOLMFNJOJO LÌLMFSJ UBN TBZ PMEVôVOB HÌSF L OJO BMBCJMFDFôJLBÀUBNTBZEFôFSJWBSES EFOLMFNJOJO LÌLMFSJOEFO CJSJ EJôFSJOJO LBSFTJ PMEV- ôVOBHÌSF NLBÀUS x1Y2 = 12 x1 + x2 = k x1 = 2 12 = 22j5BNTBZCÌMFOTBZT== 12 x 12 2 = 6 GBSLMEFôFSBMS 2 x1Y2 = 8 3 x =8 2 x2= 2, x1=4 x1 + x2 = -N N= -6 ÖRNEK 11 ÖRNEK 14 NCJSHFS¿FLTBZPMNBLÐ[FSF Y2 + Y1 - Y+Y2 = Y2 -Y+N- 1 = EFOLMFNJOJOLËLMFSJY1WFY2EJS EFOLMFNJOJO LÌLMFSJ Y1 WF Y2 PMEVôVOB HÌre, x1 JO ,ÌLMFSBSBTOEBY1 - x2 =CBôOUTPMEVôVOBHÌ- BMBCJMFDFôJEFôFSMFSUPQMBNLBÀUS SF NLBÀUS x1Y2 = 2x2 x1+ x2 = 1 3x1 - x2= 11 x2 =WFZBY1 = 2 4x1 = 12 x1 = 3 , x2 = -2 x1Y2 =N- 1 jx = 11 5 -6 =N- 1 jN -1 x1 = -x1 + 1 j +2= 1 2 22 ÖRNEK 12 ÖRNEK 15 NâWFCJSHFSÀFLTBZPMNBLÑ[FSF NY2 -Y+ 2 = Y2 -Y- 9 =EFOLMFNJOJOLËLMFSJY1WFY2PMEVóV- EFOLMFNJOJOCJSCJSJOEFOGBSLMJLJHFSÀFLLÌLÑPMEV- OBHËSF ôVOBHÌSF NOJOBMBCJMFDFôJFOCÑZÑLUBNTBZEF- ôFSJLBÀUS 11 + 6x2 - 3x22 x21 - 2x1 + 2 3 D > -N> >N 9 UPQMBNLBÀUS >NjNOJOFOCÑZÑLUBNTBZEFôFSJUÑS x2 - 2x = 9 , 6x2 - 2 = -27 2 11 3x 2 11 27 - =-8 11 3 –6 –1 4 4 6 5 –8 2
÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFS TEST - 1 #VTBZGBEBLJJÀFSJLi÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFSuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS Y2 -Y+ 4 = B C DCJSFSHFS¿FLTBZPMNBLÐ[FSF EFOLMFNJOJOLËLMFSJY1WFY2EJS BY2 CY D x1 < x2PMEVôVOBHÌSF Y1LBÀUS EFOLMFNJOJOLBUTBZMBSBSBTOEBB D CCB- \" 6 - 5 # 3 - 5 $ 3 - 2 5 ôOUTPMEVôVOBHÌSF EFOLMFNJOCJSLÌLÑBöB- ôEBLJMFSEFOIBOHJTJEJS % 3 + 2 5 & 3 + 5 \" b # c % - c a a a % - b & - c 2a 2a BCJSHFSÀFLTBZPMNBLÑ[FSF ^ x2 + 4x - 7 h2 - 3.^ x2 + 4x - 3 h + 8 = 0 Y2 -Y+B- 1 = EFOLMFNJOJOFöJUJLJLÌLÑPMEVôVOBHÌSF BLBÀ- EFOLMFNJOJTBôMBZBOYHFSÀFLTBZMBSOOUPQMB- NLBÀUS US \" 1 # 4 $ % 7 & \" - # - $ % & 3 3 3 Y2 -Y+ 42 = x2 + 1 - 12f x + 1 p + 38 = 0 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO x2 x IBOHJTJEJS PMEVôVOBHÌSF x2 + 1 UPQMBNLBÀUS x2 \" # $ % & \" \\^ # \\^ $ \\ ^ % 3 & q NOáWFN OHFS¿FLTBZPMNBLÐ[FSF NOY2 + N+O Y+ 2 = Y1 Y2PMNBLÐ[FSF EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS 4x - 33.2x – 3 + 1 = 0 2 A * - 2 , - 1 4 # * 2 , 3 4 mn mn EFOLMFNJOJOLËLMFSJY1WFY2EJS #VOBHÌSF Y1 + 3x2UPQMBNLBÀUS $ * - 2 , 1 4 % * 3 , 1 4 mn mn \" - # -3 $ % & & * 4 , 1 4 mn B B $ $ 5 & B $ \"
TEST - 2 ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFS #VTBZGBEBLJJÀFSJLi÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFNMFSuLPOVTVZMBJMHJMJTOGUBHÌSEÑôÑOÑ[LPOVMBSOUFLSBSBNBDZMBIB[SMBONöUS Y2 -Y+ 2 = NWFOHFS¿FLTBZMBSPMNBLÐ[FSF EFOLMFNJOJOLËLMFSJY1WFY2 EJS Y2 +NY+O= #VOBHÌSF x21.x2 + x1.x 2 UPQMBNLBÀUS EFOLMFNJOJO¿Ë[ÐNLÐNFTJ{- -1}EJS 2 Y- 2 +N Y- +O= \" -8 # - $ % & EFOLMFNJOJOLÌLMFSJQWFRPMEVôVOBHÌSF Q2+R2UPQMBNLBÀUS \" # $ % & OCJSHFS¿FLTBZPMNBLÐ[FSF Y2 -Y+O- 4 = EFOLMFNJOJOLËLMFSJY1WFY2EJS G Y =Y2 +Y+N-GPOLTJZPOVWFSJMJZPS ,ÌLMFSBSBTOEBY1- x2 =CBôOUTCVMVOEV- f ( x ) =EFOLMFNJOJOY1WFY2LÌLMFSJBSBTOEB ôVOBHÌSF OLBÀUS x1 << x2CBôOUTPMEVôVOBHÌSF NJÀJOBöB- \" - # - $ % & ôEBLJMFSEFOIBOHJTJEPôSVEVS \" N< # N< - $ N< - % N> & N> 13 Y2 -Y- 9 = EFOLMFNJOJOLËLMFSJY1WFY2EJS #VOB HÌSe, 27 + 10x2 - 2x22 UPQMBN Y2 - C+ Y+ 12 -BC= x12 - 5x1 18 LBÀUS EFOLMFNJOJOLËLMFSJBWFCEJS \" -2 # - $ % & #VOBHÌSF B-CEFôFSJLBÀUS \" - # $ % & Y2 -Y+ 4 = EFOLMFNJOJOLËLMFSJY1WFY2EJS x2 - 3x - 3 =EFOLMFNJOJOCJSLÌLÑQPMEVôV- #VOBHÌSF 1 + 1 UPQMBNLBÀUS OBHÌSF x1 x2 5 7 10 Q- Q- Q- Q+ \" # $ ÀBSQNLBÀUS % & 2 2 2 \" - # - $ 15 17 % & 2 2 & & $ $ 6 B \" B \"
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"3* ÷MJöLJMJ,B[BONMBS 11.3.1.1 : 'POLTJZPOVOVOHSBGJLWFUBCMPUFNTJMJOJLVMMBOBSBLQSPCMFN¿Ë[FS 'POLTJZPO(SBGJLMFSJOJO<FOMFSJ ÖRNEK 2 ,FTUJôJ/PLUBMBS \"õBóEB[B C]BSBMóOEBUBONMZ=G Y- GPOLTJZP- %m/*m OVOVOHSBGJóJWFSJMNJõUJS y = f(x–2) y y d y = f(x) –5 O 3 bx a –2 a cx #VOBHÌSF G N+ 1 ) =FöJUMJôJOJTBôMBZBOGBSLMN Ob HFSÀFLTBZMBSOOUPQMBNLBÀUS #JSGGPOLTJZPOVOVOHSBGJóJ Ð[FSJOEFLJPSEJOBU x = -JÀJOZ= f(-5 - 2) = f(-7) =jN+ 1 = -7 TGSPMBOOPLUBMBSEBYFLTFOJOJLFTFS#VOPL- jN= -8 UBMBSZVLBSEBLJHSBGJLUF B C D CJ- x =JÀJOZ= f(3 - 2) = f(1) =jN+ 1 = 1 jN= ¿JNJOEFHËTUFSJMNJõUJS -8 += -EJS #JSGGPOLTJZPOVOVOHSBGJóJ Ð[FSJOEFLJBQTJTJT- ÖRNEK 3 GSPMBOOPLUBEBZFLTFOJOJLFTFS#VOPLUBZV- LBSEBLJHSBGJLUF E CJ¿JNJOEFHËTUFSJMNJõUJS y y = f(x) 3 x ,VSBMWFSJMFOCJSZ=G Y GPOLTJZPOVJ¿JO J Z ZFSJOF TGS ZB[MBSBL G Y = EFOLMFNJ –3 O 5 ¿Ë[ÐMÐSWFHSBGJóJOYFLTFOJOJLFTUJóJZBEB UFóFUPMEVóVOPLUBMBSOBQTJTMFSJCVMVOVS –1 2 4 JJ Y ZFSJOF TGS ZB[MBSBL G = Z EFOLMFNJ ¿Ë[ÐMÐSWFHSBGJóJOZFLTFOJOJLFTUJóJOPL- UBOOPSEJOBUCVMVOVS #JSGGPOLTJZPOVOVOHSBGJóJ UBONMPMEVóVBSB- MLUBFLTFOMFSJLFTNFZFCJMJS ÖRNEK 1 ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y #VOBHÌSF 3 y = f(x) * G = 3 ** G = 1 *** G G =G - *7 G G - = –2 –1 O 3 x 7G L =PMNBLÐ[FSF LG <ES –3 1 –1 JGBEFMFSJOEFOLBÀUBOFTJEPôSVPMBCJMJS õFLJMEFWFSJMFOZ=G Y GPOLTJZPOVOVOFLTFOMFSJLFT- *<G < 3 jG â UJôJ WFZB UFôFU PMEVôV GBSLM OPLUBMBSO LPPSEJOBUMBS **G >jG â UPQMBNLBÀUS ***G > G = G –2) =PMBCJMJSG G = f(-2) PMBCJMJS (SBGJôF HÌSF GPOLTJZPOVOVO Y FLTFOJOJ LFTUJôJ OPLUBMBS *7G G -3)) = f(-1) =PMBCJMJS (- - Z FLTFOJOJ LFTUJôJ OPLUB EJS 7-3 < k < G > LG <PMBCJMJS #VOPLUBMBSOLPPSEJOBUMBSUPQMBN -3 -1 + 3 + 1 =ES 7 –8 3
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr 'POLTJZPOVO1P[JUJGWFZB/FHBUJG%FôFSMJ ÖRNEK 4 y 0MEVôV\"SBMLMBS 4 –5 %m/*m 2 6 x O 5 A f3PMNBLÐ[FSF \"LÐNFTJOEFUBONMCJSZ=G Y –2 2 GPOLTJZPOVWFSJMTJO B C f\"BSBMóOEBLJIFSYHFS¿FLTBZTJ¿JO –4 G Y >PMVZPSTBGGPOLTJZPOV B C BSBMóO- y = f(x) EBQP[JUJGEFôFSMJEJSEFOJS#VEVSVNEB GPOL- TJZPOVO HSBGJóJ B C BSBMóOEB Y FLTFOJOJO G[ - ] Z [ - ]PMNBLÐ[FSF ÑTULTNOEBES :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOVOQP- y [JUJGEFôFSMJPMEVôVYUBNTBZMBSOOUPQMBNLBÀUS + + (-5) + (-4) + (-3) + (-1) ++ 1 + 2 + 3 + 4 = -3 + bx aO y = f(x) B C BSBMó G Y >FõJUTJ[MJóJOJO¿Ë[ÐNLÐ- ÖRNEK 5 y NFTJEJS –5 2 3 x O1 4 D E f ABSBMóOEBLJIFSYHFS¿FMTBZTJ¿JO –2 G Y <PMVZPSTBGGPOLTJZPOV D E BSBMóOEB OFHBUJG EFôFSMJEJS EFOJS #V EVSVNEB GPOLTJ- –2 ZPOVOHSBGJóJ B C BSBMóOEBYFLTFOJOJOBM- –3 UOEBES G - Z [ - PMNBLÐ[FSF y y = f(x) Z=G Y GPOLTJZPOVOVOHSBGJóJZVLBSEBWFSJMNJõUJS #VOBHÌSF YG Y #FöJUTJ[MJôJOJTBôMBZBOLBÀUB- O x OFYUBNTBZTWBSES c d x = Y= -5, -2, 3 –– –5 –2 0 3 4 – +–+ – D E BSBMó G Y <FõJUTJ[MJóJOJO¿Ë[ÐNLÐ- -4, -3, - ZUBOFYUBNTBZTWBSES NFTJEJS ÖRNEK 6 'POLTJZPO HSBGJóJOJO Y FLTFOJOJ LFTUJóJ OPLUB- MBSEBG Y =FõJUMJóJTBóMBOEóJ¿JO CVOPLUB- G3Z3 G Y =Y+ 1 MBSEBGPOLTJZPOQP[JUJGZBEBOFHBUJGEFóFSMJEF- fPOLTJZPOVOVO QP[JUJG EFôFSMJ PMEVôV FO HFOJö ara- óJMEJS#VOPLUBMBSBGPOLTJZPOVOTGSMBSEF- MôCVMVOV[ OJS 1 3x + 1 >j x > - 3 8 –3 7 d - 1 , 3 n 3
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF 'POLTJZPOVOVO\"SUBOWFZB\"[BMBO0MEVôV ÖRNEK 8 y x \"SBMLMBS 4 y = f(x) 3 57 TANIM –3 1 A f3 G\"Z3 CJSZ=G Y GPOLTJZPOVWFSJMTJO O1 [B C] f\"WFrY1 Y2 ` [B C]J¿JO G3Z3 Z G Y GPOLTJZPOVOVOHSBGJóJWFSJMJZPS #VOBHÌSF GGPOLTJZPOVOVO Y1 <Y2JLFOG Y1 <G Y2 PMVZPSTB GGPOLTJZP- a) \"SUBOPMEVóVFOHFOJõBSBMóCVMVOV[ C \"[BMBOPMEVóVFOHFOJõBSBMóCVMVOV[ OV[B C]BSBMóOEBBSUBOGPOLTJZPOEVS D /FBSUBOOFEFB[BMBOPMEVóVBSBMóCVMVOV[ y f(x2) a x1 x2 b x f(x1) a) [-3, 1] C -ß -3] b [5, +ß] O D [1, 5] :VLBSEBLJHSBGJLUFYEFóFSMFSJBSUUL¿BZEFóFS- MFSJOJO HËSÐOUÐMFSJO EFBSUUóHËSÐMNFLUFEJS Y1 <Y2JLFOG Y1 >G Y2 PMVZPSTBGGPOLTJZP- OV[B C]BSBMóOEBB[BMBOGPOLTJZPOEVS y f(x2) x2 b x ÖRNEK 9 y y = f(x) f(x1) O b a x O a x1 :VLBSEBLJHSBGJLUFYEFóFSMFSJBSUUL¿BZEFóFS- c MFSJOJO HËSÐOUÐMFSJO EFB[BMEóHËSÐMNFLUFEJS (FS¿FLTBZMBSEBUBONMCJSZ=G Y GPOLTJZPOVOVOHSB- ÖRNEK 7 GJóJZVLBSEBWFSJMNJõUJS #VOBHÌSF G;Z; G - =WFG L =PMNBLÐ[FSF GGPOLTJ- ZPOVEBJNBBSUBOES * GGPOLTJZPOV[B ]BSBMóOEBB[BMBOES #VOBHÌSF G + f ( k - JGBEFTJOJOBMBCJMFDFôJFO ** G L = -LFõJUMJóJOJTBóMBZBOLTBZT GPOLTJZPOVO LÑÀÑLEFôFSLBÀUS BSUBOPMEVóVBSBMLUBES *** GGPOLTJZPOV[D C]BSBMóOEBBSUBOES JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS f( 1 )NJO = 6 kNJO =G NJO = 5 * 'POLTJZPO[B ]BSBMôOEBB[BMBOES f ( 1 ) +G = 11 ** < k <CGPOLTJZPOBSUBOES *** [D ]BSBMôOOCJSLTNOEBGPOLTJZPOB[BMBOPMVS 11 9 a) [-3, 1]C -ß -3] b [ ß] D [1, 5] *WF**
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr 'POLTJZPOVOVO.BLTJNVNWF.JOJNVN ÖRNEK 11 %FôFSMFSJ ôFLJMEF[ - ]BSBMóOEBUBONMCJSZ=G Y GPOLTJZP- TANIM OVOVOHSBGJóJWFSJMNJõUJS y #JS G GPOLTJZPOVOVO UBONM PMEVóV BSBMLUBLJ G Y HËSÐOUÐMFSJOEFOFOCÐZÐLPMBOOB GPOLTJ- 4 ZPOVOVO NBLTJNVN EFôFSJ EFOJS #V EFóF- SJ BMEó OPLUBZB JTF GPOLTJZPOVO NBLTJNVN 1 OPLUBTEFOJS –1 34 x #JS G GPOLTJZPOVOVO UBONM PMEVóV BSBMLUBLJ –3 –2 O1 2 G Y HËSÐOUÐMFSJOEFOFOLпÐLPMBOOB GPOLTJ- ZPOVOVONJOJNVNEFôFSJEFOJS#VEFóFSJBM- –1 EóOPLUBZBJTFGPOLTJZPOVONJOJNVNOPLUB- TEFOJS –2 y #V GPOLTJZPOVO NBLTJNVN WF NJOJNVN OPLUBMBS- OOLPPSEJOBUMBSOOUPQMBNLBÀUS (m, p) p a Ok x (SBGJôF HÌSF NBLTJNVN OPLUB - NJOJNVN OPL- m b UB - EJS c #VOPLUBMBSOLPPSEJOBUMBSUPQMBN t -3 + 4 + 3 - 2 =EJS (k, t) :VLBSEBLJHSBGJLUF[B C]BSBMóOEBUBONMCJS ÖRNEK 12 Z=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y c y = f(x) rY` [B C]J¿JOG Y #G N PMEVóVOEBO aO bx r 'POLTJZPOVONBLTJNVNEFóFSJG N =QEJS r.BLTJNVNOPLUBT N Q EJS rY` [B C]J¿JOG Y $G L PMEVóVOEBO r 'POLTJZPOVONJOJNVNEFóFSJG L =UEJS r .JOJNVNOPLUBT L U EJS ÖRNEK 10 G3Z [ Þ PMNBLÐ[FSF Z G Y GPOLTJZPOVOVOHSB- .BUFNBUJLËóSFUNFOJ .FSWFhEFO[ - ]LÐNFTJOEFUB- GJóJZVLBSEBWFSJMNJõUJS #VOBHÌSF ONM G Y = -Y2 + GPOLTJZPOVO NBLTJNVN WF NJOJ- * 'POLTJZPOVONBLTJNVNEFóFSJZPLUVS NVNEFóFSMFSJOJOUPQMBNOCVMNBTOJTUFNJõUJS ** 'POLTJZPOVOTGSMBS NJOJNVNOPLUBMBSES *** 'POLTJZPOIFSZFSEFQP[JUJGEFóFSMJEJS #VOBHÌSF .FSWFhOJOCVMNBTHFSFLFOTPOVÀLBÀUS JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS y .BLTJNVN WF NJOJNVN EF- *(SBGJôJOLPMMBSTPOTV[BLBEBSV[BZBCJMJS'POLTJZP- OVONBLTJNVNEFôFSJZPLUVS 2 2 ôFSMFSJOUPQMBNO ** (SBGJôF HÌSF GPOLTJZPOVO TGSMBS NJOJNVN OPLUB- –3 x MBSES ***Y=BWFY=CJÀJOG Y =PMVS O 2 + (-7) = -CVMVS –2 –7 –5 2 *WF**
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF #JS'POLTJZPOVO0SUBMBNB%FôJöJN)[ %m/*m Z = G Y GPOLTJZPOVOVO CJS [B C] BSBMóOEBLJ TANIM Z=G Y GPOLTJZPOVOVOCJS[B C]BSBMóOEBZ PSUBMBNB EFóJõJN I[ GPOLTJZPOVOVO B WF C BQTJTMJOPLUBMBSOEBOHF¿FOEPóSVOVOFóJNJEJS EFóFSMFSJOEFLJEFóJõJNNJLUBSOOYEFóFSMFSJO- EFLJ EFóJõJN NJLUBSOB PSBO PSUBMBNB EFôJ- y y = f(x) öJNI[PMBSBLBEMBOESMS f(b) [B C]BSBMóOEBPSUBMBNBEFóJõJNI[ f(a) x f^ b h- f^ a h b-a a O ab CJ¿JNJOEFIFTBQMBOS :VLBSEBLJ HSBGJLUF GPOLTJZPO [B C] BSBMóO- ÖRNEK 13 G3Z3 G Y = x2 - 3x +GPOLTJZPOVOVO[-3, 1] EBBSUBOPMEVóVOEBO tan a = f^ b h- f^ a h >0 BSBMôOEBLJPSUBMBNBEFôJöJNI[LBÀUS PMVS b-a y y = f(x) f(a) f^ 1 h - f^ - 3 h a 12 - 3.1 + 1 k - ^ ^ - 3 h2 - 3^ - 3 h + 1 h f(b) a = Oa x 1-^-3h 4 b - 1 - 19 :VLBSEBLJHSBGJLUF GPOLTJZPO[B C]BSBMóOEB = =-5 4 B[BMBO PMEVóVOEBO tan a = f^ b h- f^ a h <0 PMVS b - a ÖRNEK 14 ÖRNEK 15 ¶SFUJNNJLUBSYUBOFPMNBLÐ[FSF CJSNBMEBOFMEFFEJ- y MFOLB[BOD CJO5-DJOTJOEFOÐSFUJNNJLUBSOBCBóMPMB- SBLHËTUFSFOGPOLTJZPO 2 K^ x h = 16x - x2 –2 –1 1 8 –3 O 1 x LVSBMZMBWFSJMJZPS y = f(x) –1 #VOB HÌSF ÑSFUJMFO NBM TBZT UBOFEFO UBOFZF ÀLBSMEôOEB LB[BOÀUBLJ PSUBMBNB EFôJöJN I[ LBÀ :VLBSEB HSBGJôJ WFSJMFO Z = G Y GPOLTJZPOVOVO BS- PMVS UBO PMEVôV FO HFOJö BSBMLUBLJ PSUBMBNB EFôJöJN I- [LBÀUS 22 f 16.12 - 12 p - f 16.4 - 4 p K^ 12 h - K^ 4 h = 88 12 - 4 8 [-2, 1]BSBMô 112 PSUBMBNBEFôJöJNI[ f^ 1 h- f^ -2 h 2 -^ -1 h = = 14 = =1 1-^-2h 3 8 –5 14 11 1
TEST - 1 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS* [ - ]BSBMóOEBUBONMZ=G Y GPOLTJZPOVOVO :BS¿BQSCJSJNPMBO¿FNCFSTFMCJSQJTUJO NFSLF- [JOEFO HF¿NFZFO CJS [\"#] LJSJõJOJO JLJ VDVOEB FõJU HSBGJóJBõBóEBWFSJMNJõUJS I[MBSMBIBSFLFUFCBõMBZBOJLJBSB¿WBSES y –6 2 5x \"SB¿MBSEBO CJSJ LJSJõ Ð[F- –4 1 B SJOEF EJóFSJ¿FNCFSÐ[F- O O2 A SJOEFBZOBOEBIBSFLFUF CBõMZPS WF BSB¿MBSEBO –2 y = f(x) CJSJ JML LF[ LJSJõJO EJóFS VDVOB VMBõUó BOEB IFS #VOBHÌSF G Y =EFOLMFNJOJTBôMBZBOYHFS- JLJTJEFEVSVZPS ÀFL TBZMBSOO UPQMBNOO FO CÑZÑL UBN TBZ EFôFSJLBÀUS #VOBHÌSF BSBÀMBSOÀFNCFSJONFSLF[JOFPMBO V[BLMLMBSUPQMBNO[BNBOBCBôMHÌTUFSFOG U \" - # $ % & GPOLTJZPOVOVO HSBGJôJ BöBôEBLJMFSEFO IBOHJTJ PMBCJMJS A) f(t) B) f(t) 3r 2r 2r r tt y C) f(t) D) f(t) 3 3/2 y = f(x) x 2r 2r 5 1 r –2 –1 t –4 O 2 t –2 :VLBSEB HSBGJôJ WFSJMFO Z = G Y GPOLTJZPOV- E) f(t) OVO QP[JUJG EFôFSMJ PMEVôV BSBMLMBSEB LPPSEJ- OBUMBSUBNTBZMBSPMBO FOB[LBÀUBOFOPLUBT 3r WBSES 2r r t \" # $ % & [- ]BSBMóOEBUBONMCJSZ=G Y QPMJOPNGPOL- TJZPOV UBONM PMEVóV BSBMLUB EBJNB BSUBOES WF HSBGJóJOJOYFLTFOJOJLFTUJóJCJMJONFLUFEJS (FSÀFLTBZMBSLÑNFTJOEFUBONM G <PMEVôVOBHÌSF * G - G G Y =Y2 -Y+ 6 ** G - G - GPOLTJZPOVOVOFLTFOMFSJLFTUJôJOPLUBMBSOLP- *** G Y EFOLMFNJOJOLËLÐQP[JUJGTBZES PSEJOBUMBSUPQMBNLBÀUS JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS \" # $ % & \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & **WF*** % B & 12 $ $
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS* TEST - 2 ôFLJMEFHFS¿FLTBZMBSEBUBONMCJSZ=G Y GPOLTJ- #JSLFOBSV[VOMVóVNFUSFPMBOFõLFOBSпHFO ZPOVOVOHSBGJóJWFSJMJZPS CJ¿JNJOEFLJCJSZÐSÐZÐõQBSLVSVOVOCJSLËõFTJOEFO ZÐSÐZÐõF CBõMBZBO CJS LJõJ CBõMBEó OPLUBZB EË- y OÐODFZFLBEBSQBSLVSÐ[FSJOEFTBCJUI[MBZÐSÐZPS WF CBõMBOH¿ OPLUBTOB VMBõODB EVSVZPS #V LJõJ- 2 OJOUEBLJLBTPOSBCBõMBEóOPLUBZBPMBOV[BLMó- ONFUSFDJOTJOEFOHËTUFSFOCJSG U GPOLTJZPOVUB- –5 –3 –1 O x ONMBOZPSWFHSBGJóJ¿J[JMJZPS –4 y = f(x) #VGPOLTJZPOVOVOHSBGJôJZMFJMHJMJBöBôEBWFSJ- –2 MFOJGBEFMFSEFOIBOHJTJEPôSVEVS –3 \" %BJNBBSUBOGPOLTJZPOEVS #VGPOLTJZPOMBJMHJMJBöBôEBWFSJMFOJGBEFMFSEFO IBOHJTJZBOMöUS # %BJNBB[BMBOGPOLTJZPOEVS \" 'POLTJZPO [- -1] BSBMóOEB B[BMBO EJóFS $ .BLTJNVNEFóFSJZPLUVS ZFSMFSEFBSUBOES % #JSU=LEPóSVTVOBHËSFTJNFUSJLUJS # 'POLTJZPO - - BSBMóOEBQP[JUJGEFóFSMJEJS & #JSG U =LEPóSVTVOBHËSFTJNFUSJLUJS $ 'POLTJZPOVOVONBLTJNVNEFóFSJEJS NWFOCJSFSHFSÀFLTBZNâPMNBLÑ[FSF % 'POLTJZPOVONJOJNVNEFóFSJ-UÐS & G Y =EFOLMFNJOJOCJSCJSJOEFOGBSLMJLJLËLÐ WBSES G Y =NY+O (FS¿FLTBZMBSLÐNFTJOEFUBONMZ=G Y GPOLTJ- CJÀJNJOEFLJEPôSVTBMGPOLTJZPOMBSMBJMHJMJBöBô- ZPOV -Þ ] BSBMóOEB B[BMBO [ Þ BSBMóOEB EBWFSJMFO BSUBOES r N>J¿JOBSUBOGPOLTJZPOEVS 'POLTJZPO -Þ, - BSBMôOEB QP[JUJG EFôFS- r N<J¿JOOFHBUJGEFóFSMJEJS MJ -1, Þ BSBMôOEB OFHBUJG EFôFSMJ PMEVôVOB r YFLTFOJOJ- n BQTJTMJOPLUBTOEBLFTFS HÌSF BöBôEBLJMFSEFOIBOHJTJLFTJOMJLMFEPôSV- EVS m r O=JTFZFLTFOJOJLFTNF[ \" G > # G <G r 5BONM PMEVóV BSBMóO IFSIBOHJ CJS BMU BSBM- óOEBLJPSUBMBNBEFóJõJNI[NEJS $ G - =G % ff - 1 p.ff - 3 p > 0 JGBEFMFSJOEFOLBÀUBOFTJEPôSVEVS 22 \" # $ % & & G - G < G Y =Y-Y3 #JS PSUBNEBLJ CBLUFSJ TBZTO IFS TBBU CBõ ËM¿FO GPOLTJZPOVOVO [-4, -2] BSBMôOEBLJ PSUBMBNB CJSDJIB[ IFSIBOHJCJSHÐOZBQUóJMLËM¿ÐNEF34 EFóFSJOJ BZOHÐOZBQUóTPOËM¿ÐNEFJTF18 EF- EFôJöJNI[LBÀUS óFSJOJLBZUBMUOBBMNõUS \" - # - 27 $ -9 % 19 & #VOBHÌSF PSUBNEBLJCBLUFSJTBZTOOHÑOCP- 2 2 ZVODBPSUBMBNBEFôJöJNI[LBÀPMNVöUVS \" 29 # 15 $ 31 % 16 & 33 % & \" 13 % $ &
TEST - 3 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS* #JS GBCSJLBEB Y BEFU ÐSÐOEFO FMEF FEJMFO LB[BOD y y = f(x–1) HËTUFSFOGPOLTJZPO f^ x h = –x2 + 200x EJS 50 3 #VOBHÌSFÑSFUJMFONBMNJLUBSUBOBÀka- –2 O 3 4 x SMEôOEB FMEF FEJMFO LB[BODO PSUBMBNB EFôJ- öJNI[BöBôEBLJMFSEFOIBOHJTJEJS õFLJMEFLJZ= f ( x - GPOLTJZPOVOVOHSBGJôJOF HÌSF G Y =EFOLMFNJOJTBôMBZBOGBSLMYEF- \" # $ % & ôFSMFSJOJOUPQMBNLBÀUS \" - # $ % & y aO bx y –1 4 x c –3 O y = f(x) y = f(x) :VLBSEB HSBGJôJ WFSJMFO Z = G Y GPOLTJZPOV JÀJO õFLJMEFWFSJMFOZ=G Y GPOLTJZPOVOVOHSBGJôJ- * 5BONMPMEVóVIFSZFSEFOFHBUJGEFóFSMJEJS OF HÌSF YG Y ä FöJUTJ[MJôJOJ TBôMBZBO Y JO ** YG Y FõJUTJ[MJóJOJO ¿Ë[ÐN LÐNFTJ B C BMBCJMFDFôJUBNTBZEFôFSMFSJUPQMBNLBÀUS BSBMóES *** x # 0 FõJUTJ[MJóJOJO ¿Ë[ÐN LÐNFTJ < C \" # $ % & f(x) BSBMóOLBQTBS JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & *WF*** ôFLJMEFG[- ] Z[ - ]CJ¿JNJOEFUBONMBOBO $FWEFU FWEFO PLVMB EPóSV EBLJLB ZÐSÐNÐõUÐS GGPOLTJZPOVWFSJMNJõUJS ±EFWJOJ BMQ BMNBEóO IBUSMBNBZBO $FWEFU PMEV- óVZFSEFEVSVQ¿BOUBTOEBEBLJLBËEFWJOJBSBNõ- y US±EFWJOJ¿BOUBTOEBCVMBNBZBO$FWEFUËEFWJOJ BMNBLJ¿JOEBLJLBEBFWFHFSJEËONÐõUÐS –4 –2 5 3 2 x #VOB HÌSF $FWEFUhJO EBLJLB TÑSFO IBSFLF- O2 y = f(x) UJOEF FWFPMBOV[BLMôOO[BNBOBHÌSFEFôJöJ- –3 NJOJHÌTUFSFOCJSGPOLTJZPOBöBôEBLJBSBMLMBS- EBOIBOHJTJOEFB[BMBOES #VOBHÌSF YG Y <FöJUTJ[MJôJOJOÀÌ[ÑNLÑ- NFTJOEFLBÀUBNTBZEFôFSJWBSES \" # $ % & \" # $ % & % B % 14 % $ &
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"3** ÷MJöLJMJ,B[BONMBS 11.3.2.1 : öLJODJEFSFDFEFOCJSEFóJõLFOMJGPOLTJZPOVOHSBGJóJOJ¿J[FSFLZPSVNMBS 1BSBCPM %m/*m TANIM 5BONM PMEVóV BSBMLUB G Y = BY2 + CY + D GPOLTJZPOVOVO HSBGJóJ PMBO QBSBCPM BõBóEBLJ B C D`3WFBáPMNBLÐ[FSF CJ¿JNMFSEFOCJSJOFTBIJQUJS G 3 Z 3 G Y = BY2 + CY + D CJ¿JNJOEF UBONMBOBO GPOLTJZPOB JLJODJ EFSFDFEFO CJS veya EFôJöLFOMJGPOLTJZPOEFOJS a>0 a<0 #VGPOLTJZPOVOHSBGJóJOFJTFQBSBCPMBEWF- SJMJS 1BSBCPMÐOLPMMBSOOZVLBSZBEBBõBóEPóSV PMNBTO GPOLTJZPOVO LVSBMOEBLJ JLJODJ EFSF- ±SOFóJO DFEFOUFSJNJO Y2MJUFSJN LBUTBZTCFMJSMFS G Y =Y2 +Y+ 6 H Y = -Y2 -Y G Y =BY2 +CY+DQBSBCPMÐOEF I Y =Y2 + 1 CJ¿JNJOEFWFSJMFOGPOLTJZPOMBSOIFSCJSJOJOHSB- r B>JLFOQBSBCPMÐOLPMMBSZVLBSZBEPó- SVEVS GJóJQBSBCPMEÐS r B<JLFOQBSBCPMÐOLPMMBSBõBóZBEPóSV- ÖRNEK 1 EVS G Y = B- Y3 + B+ Y2 -BY+B+ 1 ÖRNEK 3 GPOLTJZPOVOVO HSBGJôJ CJS QBSBCPM CFMJSUUJôJOF HÌSF G B LBÀUS * G Y = -Y+ 2 ** H Y = - Y+ 2 a - 2 = *** I Y = -OY1 -O +O a = 2 j f ( x ) = 3x2 - 2x + 3 :VLBSEB WFSJMFO GPOLTJZPOMBS SFFM TBZMBSEB UBONM WF f ( 2 ) = 12 - 4 + 3 = 11 HSBGJLMFSJCJSFSQBSBCPMEÐS ÖRNEK 2 #VOB HÌSF CV QBSBCPMMFSEFO IBOHJMFSJOJO LPMMBS BöBôZBEPôSVEVS G Y =BYO+ 3 +OY+B * 5BN LBSF JGBEFZF FöJU PMBO QBSBCPMÑO LPMMBS ZVLBS GPOLTJZPOVOVO HSBGJôJ PMBO QBSBCPM \" OPLUB- ZÌOMÑEÑS TOEBOHFÀUJôJOFHÌSF BLBÀUS ** H Y = -2(x2+ 2x + 1) = -2x2- 4x - 2 O+ 3 = 2 a <PMEVôVOEBOLPMMBSBöBôEPôSVEVS O= -1 j f ( x ) = ax2 - x + a *** -O= 2 2 =B2 - 1 + a O= -1 jI Y = -(-1)x2 + (-1) 3 I Y = x2 -1 3 = 2a j a = a >PMEVôVOEBOLPMMBSZVLBSEPôSVEVS 2 11 3 15 :BMO[** 2
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr 1BSBCPMÑO<FOMFSJ,FTUJôJ/PLUBMBS %m/*m %m/*m 5BONM PMEVóV FO HFOJõ BSBML J¿JO Z = G Y 5BONM PMEVóV FO HFOJõ BSBML J¿JO Z = G Y GPOLTJZPOVOVO HSBGJóJ PMBO QBSBCPM Z FLTFOJ- GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPM YFLTFOJOJ OJZBMO[CJSOPLUBEBLFTFS#VOPLUB G FO¿PLJLJOPLUBEBLFTFS OPLUBTES :BOJQBSBCPMÐOZFLTFOJOJLFTUJóJOPLUBOOPS- 1BSBCPMÐOYFLTFOJOJLFTUJóJOPLUBOOBQTJTJ EJOBU GPOLTJZPOVO LVSBMOEB Y ZFSJOF TGS ZB- Z=G Y =EFOLMFNJOJOLËLÐEÐS [MBSBLCVMVOVS G Y =BY2+CY+D=EFOLMFNJOEF ÖRNEK 4 r D >JLFOQBSBCPMY FLTFOJOJ GBSLM JLJ G Y =Y2 -LY+ 5 x1 x2 x OPLUBEBLFTFS QBSBCPMÑOÑO Z FLTFOJOJ LFTUJôJ OPLUBOO PSEJOBU LBÀUS r D =JLFOQBSBCPMY x =JÀJOZ=2 -L+ 5 FLTFOJOF CJS OPLUB- y=5 x1 = x2 x EBUFóFUUJS r D <JLFOQBSBCPM YFLTFOJOJLFTNF[ x ÖRNEK 5 ÖRNEK 7 G Y = Y- 2 + 1 G Y =Y2 -Y- 6 QBSBCPMÑOÑO Z FLTFOJOJ LFTUJôJ OPLUBOO PSEJOBU GPOLTJZPOVOVOHSBGJôJOJOYFLTFOJOJLFTUJôJOPLUBMBS LBÀUS BSBTOEBLJV[BLMLLBÀCJSJNEJS x =JÀJOZ= - 3 )2 + 1 j y = f ( x ) = y =JÀJOY2- 5x - 6 =j x1= -WFY2= 6 | |x1 - x2 =CVMVOVS ÖRNEK 6 ÖRNEK 8 G Y = -Y 2 + 2 -L G Y = -Y2 +Y+QBSBCPMÐOÐOFLTFOMFSJLFTUJóJOPL- GPOLTJZPOVOVO HSBGJôJ Z FLTFOJOJ FLTFOJO OFHBUJG UBMBS CJSпHFOJOLËõFMFSJEJS UBSBGOEBLFTUJôJOFHÌSF LUBNTBZTFOB[LBÀPMB- CJMJS #VOBHÌSF CVÑÀHFOJOBMBOLBÀCJSJNLBSFEJS x =JÀJOZ<j (2 - 2 + 2 - k < y x =JÀJOZ= 3 6 - k <j k > 6 3 y =JÀJO-x2+ 2x + 3 = LUBNTBZTFOB[PMBCJMJS j x2 - 2x - 3 = –1 x j x1= -1, x2= 3 3 4.3 2 ·ÀHFOJOBMBO 2 = 6 br 5 7 16 7 6
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 9 ÖRNEK 12 G Y =Y2 + L+ Y+L- 8 f_ x i = _ n2 + 1 ix2 - _ n - 4 ix + 1 GPOLTJZPOV Y FLTFOJOJ PSJKJOF FõJU V[BLMLUB CVMVOBO JLJ 4 GBSLMOPLUBEBLFTNFLUFEJS #VOBHÌSF CVOPLUBMBSBSBTOEBLJV[BLMLLBÀCJSJN- QBSBCPMÑYFLTFOJOFUFôFUPMEVôVOBHÌSF OLBÀUS EJS y =JÀJOD = x1- x2 j x1 + x2 =j k + 4 =j k = -4 f ( x ) = 2x2 - 16 j= 2x2 - 16 [- O- 4 ) ]2 - O2 + 1 = | |x1= 2 2 j x2= - 2 2 j x1 - x2 = 4 2 4 -O+ 15 = 15 n= 8 ÖRNEK 10 ÖRNEK 13 G Y = -Y2 +Y-N+GPOLTJZPOVOVOHSBGJóJOJO O`3PMNBLÐ[FSF G Y =Y2 -OY+O- 1 rZFLTFOJOJFLTFOJOOFHBUJGUBSBGOEBLFTUJóJ GPOLTJZPOVOVOHSBGJóJJMFJMHJMJWFSJMFO rYFLTFOJOJGBSLMJLJOPLUBEBLFTUJóJ CJMJONFLUFEJS * ZFLTFOJOJLFTUJóJOPLUBOOPSEJOBUOFHBUJGUJS #VOB HÌSF N OJO BMBCJMFDFôJ UBN TBZ EFôFSMFSJOJO UPQMBNLBÀUS ** YFLTFOJJMFFOB[CJSPSUBLOPLUBTWBSES x =JÀJOZ<j -N+ 1 <jN> 1 *** YFLTFOJOJLFTUJóJOPLUBMBSOBQTJTMFSJUPQMBNQP[J- D >JÀJO2 - 4(-1) (1 -N >j 16 + 4 -N> UJGUJS j 5 >N 1 <N<JTFNOJOUBNTBZEFôFSMFSJUPQMBN JGBEFMFSJOEFOIBOHJMFSJEBJNBEPôSVEVS 2 + 3 + 4 =EVS * Y=JÀJOZ=O-PMVS÷öBSFUJCFMJSTJ[EJS ** Z=JÀJOD =O2 - O- 1) j D =O2 -O+ 4 D $PMVQFOB[CJSLÌLWBSES b -n *** Y1 + x2= - a = - 1 = nPMVQJöBSFUJCFMJSTJ[EJS ÖRNEK 11 ÖRNEK 14 G Y =Y2 -Y-N- 2 G Y = -Y2 +Y- 1 GPOLTJZPOVJMFJMHJMJWFSJMFO GPOLTJZPOVOVOHSBGJôJYFLTFOJOJLFTNFEJôJOFHÌSF NOJOBMBCJMFDFôJFOCÑZÑLUBNTBZEFôFSJLBÀUS * (SBGJóJYFLTFOJOJLFTNF[ ** %BJNBB[BMBOES y =JÀJOD < *** %BJNBOFHBUJGEFóFSMJEJS (-6)2- -N- 2) < JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS 36 +N+ 8 <jN< -44 jN< -11 NOJOFOCÑZÑLUBNTBZEFôFSJ-EJS * Z=JÀJOD = 22 - - -1) = -8 j D < 3FFMLÌLZPL1BSBCPMYFLTFOJOJLFTNF[ ** 1BSBCPM HSBGJôJ CFMJSMJ BSBMLUB BSUBO CFMJSMJ BSBMLUB B[BMBOES *** YFLTFOJOJLFTNFZFOWFLPMMBSBöBôZÌOMÑPMBOQBSB- CPM EBJNBOFHBUJGEFôFSMJEJS 4 2 9 –12 17 15 :BMO[***WF*** 8
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr 1BSBCPMÑO5FQF/PLUBT Azalan ÖRNEK 15 7$1,0%m/*mAzalan öLJODJEFSFDFEFOZ=G Y GPOLTJZPOVOVOB[B-Artan B<PMNBLÐ[FSF MBO EVSVNEBO BSUBO EVSVNB HF¿JQ FO LпÐLArtan G Y =BY2 +CY+D EFóFSJOJ BMEó ZB EB BSUBO EVSVNEBO B[BMBO GPOLTJZPOVOVOHSBGJóJOJOYFLTFOJOJLFTNFEJóJCJMJOJZPS EVSVNBHF¿JQFOCÐZÐLEFóFSJOJBMEóOPLUBZB #VOBHÌSF CVHSBGJLMFJMHJMJBöBôEBWFSJMFO QBSBCPMÑOUFQFOPLUBTEFOJS * 5FQFOPLUBTLPPSEJOBUEÐ[MFNJOJO*CËMHFTJOEFEJS Tepe OPLUBT ** Z=LEPóSVTVQBSBCPMÐJLJGBSLMOPLUBEBLFTJZPSTB LQP[JUJGUJS Tepe OPLUBT *** D< ,PMMBSZVLBSEPóSVPMBOQBSB- JGBEFMFSJOEFOIBOHJMFSJEBJNBEPôSVEVS CPMÐO UFQF OPLUBT NJOJNVN OPLUBTES a <JTFLPMMBSBöBôZÌOMÑYFLTFOJOJLFTNFEJôJCJMJ- OJZPSTB UFQFOPLUBTYFLTFOJOJOBMUOEBES Minimum *5FQFOPLUBT***WFZB*7CÌMHFEFEJS nokta **'POLTJZPOEBJNBOFHBUJGEFôFSMJEJS ***ZFLTFOJOJPSEJOBUOFHBUJGPMBOOPLUBEBLFTFS D<ES Maksimum ,PMMBS BõBó EPóSV PMBO QBSB- nokta CPMÐOUFQFOPLUBT NBLTJNVN OPLUBTES ÖRNEK 16 1BSBCPMÐOUFQFOPLUBTYFLTFOJOJLFTUJóJOPL- UBMBSB FõJU V[BLMLUBES #FO[FS õFLJMEF UFQF B>PMNBLÐ[FSF OPLUBT Y FLTFOJOF QBSBMFM PMBSBL ¿J[JMFO IFS- G Y =BY2 +CY+D IBOHJ CJS EPóSVOVO QBSBCPMÐ LFTUJóJ OPLUBMBSB QBSBCPMÐOÐO Y FLTFOJOJ GBSLM \" WF # OPLUBMBSOEB LFT- EBFõJUV[BLMLUBES UJóJCJMJOJZPS #VOBHÌSF CVHSBGJLMFJMHJMJBöBôEBWFSJMFO y yT * 1BSBCPM [\"#]OOPSUBEJLNFTJOFHËSFTJNFUSJLUJS T D C x ** 5FQFOPLUBTOOPSEJOBUOFHBUJGUJS O *** \" WF # OPLUBMBSOO IFS JLJTJ EF Y FLTFOJOJO QP[J- UJGUBSBGOEBJLFOUFQFOPLUBTLPPSEJOBUEÐ[MFNJOJO A Bx *7CËMHFTJOEFEJS O JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS | | | |5\" = 5# | | | |5$ = 5% a >JTFLPMMBSZVLBSZÌOMÑEÑS *1BSBCPM[\"#]OJOPSUBEJLNFTJOFHÌSFTJNFUSJLUJS ** YFLTFOJOJJLJOPLUBEBLFTJZPSTB UFQFOPLUBTOO PSEJOBUOFHBUJGUJS *** \" WF # OJO IFS JLJTJ EF QP[JUJG JLFO UFQF OPLUBT Y FLTFOJOJOQP[JUJGUBSBGOOBMUOEB ZBOJ*7CÌMHF- EFEJS 18 :BMO[*** * **WF***
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS** TEST - 4 \"öBôEBLJ GPOLTJZPOMBSEBO IBOHJTJOJO HSBGJôJ G Y = B+ Y3 -Y2 -BY+ 3 CJSQBSBCPMEÑS GPOLTJZPOVOVO HSBGJôJ CJS QBSBCPM CFMJSUUJôJOF HÌSF BöBôEBLJOPLUBMBSEBOIBOHJTJCVQBSBCP- \" G Y =Y- # G Y = Y2 - 2 MÑOÑ[FSJOEFEJS $ G Y = 2 -Y2 % f_ x i = x2 + 2 x \" # $ - & G Y =Y3 -Y % - & - \"öBôEBLJ GPOLTJZPOMBSEBO IBOHJTJOJO HSBGJôJ G Y =YO- 3+OY+O PSJKJOEFOHFÀFOCJSQBSBCPMEÑS GPOLTJZPOVOVO HSBGJôJ CJS QBSBCPM CFMJSUUJôJOF HÌSF G O EFôFSJLBÀUS \" G Y =Y2 - # G Y =Y3 \" # $ % & $ G Y =Y2 + % G Y =Y2 -Y & G Y = Y- 2 Z= -Y2 - N- Y+ 1 G Y = -B Y3 -CYC-B +CY-B QBSBCPMÑ - OPLUBTOEBOHFÀUJôJOFHÌSF N GPOLTJZPOV JLJODJ EFSFDFEFO CJS CJMJONFZFOMJ LBÀUS GPOLTJZPOPMEVôVOBHÌSF G LBÀUS \" - # - $ % & \" - # - $ % & \"öBôEBLJOPLUBMBSEBOIBOHJTJ \"öBôEBLJEFOLMFNJWFSJMFOQBSBCPMMFSEFOIBO- G Y =Y2 -Y+ 1 HJTJOJOÑ[FSJOEF BQTJTJPSEJOBUOBFöJUPMBOCJS OPLUBZPLUVS GPOLTJZPOVOVOHSBGJôJOJOÑ[FSJOEFEJS \" Z=Y2 -Y+ # Z=Y2 -Y+ 2 \" # $ $ Z=Y2 - % Z= -Y2 + 2 % - & & Z= -Y2 -Y $ % & $ 19 B & \" \"
TEST - 5 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS** Z= -Y2 -Y+ 4 f_ x i = 4 _ x - 3 i2 QBSBCPMÑOÑOZFLTFOJOJLFTUJôJOPLUBBöBôEB- 9 LJMFSEFOIBOHJTJEJS GPOLTJZPOVOVOHSBGJôJOJOYWFZFLTFOMFSJÑ[F- \" - # $ - SJOEF CVMVOBO OPLUBMBS BSBTOEBLJ V[BLML LBÀ CJSJNEJS % & - \" # $ % & \"öBôEBLJOPLUBMBSEBOIBOHJTJ G3Z3 G Y =Y2 -Y- 18 Z= -Y2 +Y+ 6 QBSBCPMÑOÑOYFLTFOJOJLFTUJôJOPLUBMBSEBOCJ- GPOLTJZPOVOVOHSBGJôJPMBOQBSBCPMÑOYFLTFOJ- SJEJS OJLFTUJôJOPLUBMBSBSBTOEBLJV[BLMLLBÀCJSJN- EJS \" - # - $ \" # $ % & % & - Z=Y2 -LY+ 2 Z=Y2 +CY+D QBSBCPMÑOÑO Z FLTFOJOJ LFTUJôJ OPLUBOO PSEJ- QBSBCPMÐYFLTFOJOJf - 2 , 0 pWF OPLUBMBSO- OBU JMF Y FLTFOJOJ LFTUJôJ OPLUBMBSEBO CJSJOJO 3 BQTJTJFöJUPMEVôVOBHÌSF LLBÀUS EBLFTJZPS #VOBHÌSF QBSBCPMÑOZFLTFOJOJLFTUJôJOPLUB- OOPSEJOBULBÀUS \" - # - 1 $ % & \" -8 # - $ - % - 1 & 3 2 22 G Y =Y2 -Y+ 6 Z= -Y2 +NY-O QBSBCPMÐOÐO FLTFOMFSJ LFTUJóJ OPLUBMBS CJS пHFOJO QBSBCPMÐYFLTFOJOJ - WF OPLUBMBSOEB LËõFMFSJEJS LFTJZPS #VOBHÌSF CVÑÀHFOJOBMBOLBÀCJSJNLBSFEJS #VOBHÌSF N-OGBSLLBÀUS \" - # $ % \" # $ % & & % B & B $ % \" &
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS** TEST - 6 \"öBôEBEFOLMFNJWFSJMFOQBSBCPMMFSEFOIBOHJ- Z=Y2 +LY+ 2 TJYFLTFOJOJLFTNF[ QBSBCPMÑ Y FLTFOJOF FLTFOJO QP[JUJG UBSBGOEB \" Z= -Y2 -Y+ 2 UFôFUPMEVôVOBHÌSF LLBÀUS # Z=Y2 - 4 $ Z= - Y+ 2 + 2 \" - # - 2 6 $ % Z=Y2 -Y+ 3 & Z= -Y2 + 9 % 2 6 & \"öBôEB WFSJMFO HFSÀFL TBZMBSEB UBONM GPOL- BâPMNBLÑ[FSF TJZPOMBSEBOIBOHJTJOJOHSBGJôJ LPPSEJOBUEÑ[MF- G3Z3 G Y =BY2 +CY+D NJOJOEÌSUCÌMHFTJOEFOHFÀFS \" G Y =Y2 -Y+ # G Y = -Y2 -Y+ 1 GPOLTJZPOVOVO HSBGJôJ EBJNB Y FLTFOJOJO ÑT- $ G Y =Y2 -Y+ % G Y = -Y2 +Y- 3 UÑOEF PMEVôVOB HÌSF BöBôEBLJMFSEFO IBOHJTJ & G Y = -Y2 +Y- 3 EPôSVEVS B B<WFC2 <BD # B<WFC2 =BD $ B>WFC2 >BD % B>WFC2 =BD & B>WFC2 <BD Z=Y2 - L+ Y+L QBSBCPMÑ YFLTFOJOJPSJKJOFFöJUV[BLMLUBCVMV- G3Z3 G Y = -Y2 + N+ Y- 8 OBO GBSLM JLJ OPLUBEB LFTUJôJOF HÌSF G L LBÀ- US GPOLTJZPOVOVOHSBGJôJYFLTFOJOFUFôFUPMEVôV- OBHÌSF NOJOQP[JUJGEFôFSJLBÀUS \" - # $ % & \" # $ % & G3Z3 G Y = -Y2 +Y-N+ 2 G3Z3 G Y =Y2 -Y+L- 3 GPOLTJZPOVOVOHSBGJôJYFLTFOJOJGBSLMJLJOPL- GPOLTJZPOVOVO HSBGJôJ Y FLTFOJOJ LFTNFEJôJOF UBEBLFTUJôJOFHÌSF NOJOFOHFOJöEFôFSBSBM- HÌSF LOJOFOLÑÀÑLUBNTBZEFôFSJLBÀUS ôBöBôEBLJMFSEFOIBOHJTJEJS \" -Þ # -Þ $ Þ \" # $ % & % Þ & % B $ \" 21 B & & $
TEST - 7 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS** \"öBôEBLJQBSBCPMMFSEFOIBOHJTJOJOYFLTFOJOJ #JSZ=G Y GPOLTJZPOVOVOHSBGJóJ UFQFOPLUBTLP- LFTUJôJ OPLUBMBS Y = B EPôSVTVOVO Ñ[FSJOEFLJ PSEJOBUEÐ[MFNJOJO*7CËMHFTJOEFCVMVOBOWFYFL- IFSIBOHJCJSOPLUBZBFöJUV[BLMLUBES TFOJOJLFTNFZFOCJSQBSBCPMEÐS \" G Y = Y+B 2 # G Y =Y Y-B #VGPOLTJZPOJMFJMHJMJ * 5FQFOPLUBTOEBNBLTJNVNEFóFSJOJBMS $ G Y = Y-B Y+B % G Y =Y Y-B ** %BJNBOFHBUJGEFóFSMJEJS & G Y =Y2 -B *** %BJNBB[BMBOES JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS \" :BMO[* # :BMO[** $ *WF** % *WF*** & **WF*** Z= B- Y2 -BY-B+ 1 QBSBCPMÐZFLTFOJOJ FLTFOJOOFHBUJGUBSBGOEBLFTJ- B<C<DPMNBLÐ[FSF JLJODJEFSFDFEFOCJS ZPS Z=G Y GPOLTJZPOVOVO[B C]BSBMóOEBLJPSUBMBNB EFóJõJNI[QP[JUJG [C D]BSBMóOEBLJPSUBMBNBEF- #VQBSBCPMJMFJMHJMJ * ,PMMBSZVLBSZËOMÐEÐS óJõJNI[OFHBUJGUJS ** YFLTFOJOJGBSLMJLJOPLUBEBLFTFS #VGPOLTJZPOVOVOHSBGJôJJMFJMHJMJ *** &óFSWBSTB YFLTFOJOJLFTUJóJOPLUBZBEBOPL- * YFLTFOJOJLFTNF[ UBMBSFLTFOJOOFHBUJGUBSBGOEBES ** ,PMMBSBõBóZËOMÐPMBOCJSQBSBCPMEÐS JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS *** 5FQFOPLUBTOOBQTJTJCEJS \" :BMO[* # *WF** $ *WF*** % **WF*** & * **WF*** JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF** & **WF*** öLJODJEFSFDFEFOCJSZ=G Y GPOLTJZPOVOVOHSBGJóJ (FS¿FLTBZMBSEBUBONM JLJODJEFSFDFEFOCJSGPOL- PMBO QBSBCPM Y FLTFOJOJ CJSCJSJOEFO GBSLM Y1 WF Y2 TJZPOVOHSBGJóJOJOYFLTFOJOJLFTUJóJOPLUBMBSOBQ- OPLUBMBSOEBLFTNFLUFEJS TJTMFSJ¿BSQNOFHBUJGUJS x1 < x2 <WFG Y1 + x2 ) <PMEVôVOBHÌSF CV #VOB HÌSF GPOLTJZPOVO HSBGJôJ JÀJO BöBôEBLJ QBSBCPMMFJMHJMJBöBôEBWFSJMFOJGBEFMFSEFOIBO- JGBEFMFSEFOIBOHJTJLFTJOMJLMFZBOMöUS HJTJLFTJOMJLMFEPôSVEVS \" (SBGJóJ BOBMJUJLEÐ[MFNJOZBMO[DBпCËMHFTJO- \" 5FQFOPLUBT LPPSEJOBUEÐ[MFNJOJO**CËMHFTJO- EFOHF¿FS EFEJS # 5FQF OPLUBT BOBMJUJL EÐ[MFNJO JLJODJ CËMHFTJO- # ,PMMBSZVLBSZËOMÐEÐS EFEJS $ Z FLTFOJOJ PSEJOBU QP[JUJG PMBO CJS OPLUBEB LF- $ ZFLTFOJOJ FLTFOJOQP[JUJGUBSBGOEBLFTFS TFS % ,PMMBSZVLBSZËOMÐEÐS % 5FQFOPLUBTOOBQTJTJQP[JUJGUJS & 5FQFOPLUBT NBLTJNVNOPLUBTES & 5FQFOPLUBT NJOJNVNOPLUBTES % B \" 22 $ B \"
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"3*** ÷MJöLJMJ,B[BONMBS 11.3.2.1 : öLJODJEFSFDFEFOCJSEFóJõLFOMJGPOLTJZPOVOHSBGJóJOJ¿J[FSFLZPSVNMBS 1BSBCPMÑO5FQF/PLUBTOO,PPSEJOBUMBS ÖRNEK 2 TANIM Z= -Y2 +Y- 3 1BSBCPMÐO UFQF OPLUBTOO LPPSEJOBUMBS HF- QBSBCPMÑOÑO UFQF OPLUBTOO LPPSEJOBUMBSO CVMV- OFMMJLMF5 S L CJ¿JNJOEFHËTUFSJMJS,PPSEJOBU- OV[ MBS QBSBCPMEFOLMFNJOJOWFSJMJõJOFHËSF BõBó- EBLJJLJZËOUFNEFOCJSJLVMMBOMBSBLCVMVOVS b2 r =- =- = 1 Z=B Y-S 2 +LEFOLMFNJZMFWFSJMFOQBSBCP- MÐOUFQFOPLUBT5 S L ES 2a - 2 k = f ( 1 ) = -12 +- 3 = -2 j T ( 1, -2 ) ±SOFóJO r Z= - Y- 2 +QBSBCPMÐOÐOUFQFOPL- ÖRNEK 3 UBT r Z= Y+ 2 -QBSBCPMÐOÐOUFQFOPLUB- Z=Y2 +Y- 1 T - - QBSBCPMÑOÑOUFQFOPLUBTOOLPPSEJOBUMBSOCVMVOV[ r Z= Y- 2QBSBCPMÐOÐOUFQFOPLUBT ES b1 r =- =- Z=BY2 +CY+DEFOLMFNJZMFWFSJMFOQBSBCPM J¿JO 2a 4 r =- b k = fd - 1 n = 2. 1 19 2a - -1 =- 4 16 4 8 FõJUMJóJJMFUFQFOPLUBTOOBQTJTJCFMJSMFOJS Td - 1 , - 9 n %BIBTPOSBSEFóFSJQBSBCPMEFOLMFNJOEFYZF- 48 SJOFZB[MBSBL UFQFOPLUBTOOPSEJOBUPMBO Z=LEFóFSJCVMVOBCJMJS\"ZSDB LEFóFSJJ¿JO ÖRNEK 4 k = 4ac - b2 4a Z= - Y- 2 + 1 QBSBCPMÑOÑOUFQFOPLUBTOOLPPSEJOBUMBSOCVMVOV[ FõJUMJóJOEFOZBSBSMBOMBCJMJS f ( x ) =B Y- r ) 2 +LNPEFMJOFHÌSF 5 L S =5 S G S = T = f - b , 4ac - b2 p r =WFL= 1 j T ( 3, 1 ) 2a 4a ÖRNEK 1 ÖRNEK 5 Z=Y2 -Y+ 2 Z= -Y 2 - 2 QBSBCPMÑOÑOUFQFOPLUBTOOLPPSEJOBUMBSOCVMVOV[ QBSBCPMÑOÑO UFQF OPLUBTOO LPPSEJOBUMBSO CVMV- OV[ y = (x - 2)2 - 2 r=2 , k=-2 b6 T(2, -2) r =- =- = 3 2a 2 k = f ( 3 ) = 32 -+ 2 = -7 T ( 3, - 7 ) ( 3, –7 ) 23 (1, –2) d - 1 , - 9 n (3, 1) (2, –2) 48
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 Z= Y- 2 - 1 Z= -Y2 +NY+N+ 1 QBSBCPMÑOÑO UFQF OPLUBTOO LPPSEJOBUMBSO CVMV- QBSBCPMÑOÑO UFQF OPLUBT Y = N + EPôSVTVOVO OV[ Ñ[FSJOEF PMEVôVOB HÌSF UFQF OPLUBTOO PSEJOBU LBÀUS 12 b mm y = d 2.d x - n n - 1 r=- =- = 2 2a - 2 2 12 1 m y = 4.d x - n - 1 & Td , - 1 n = m + 1 & m = 2m + 2 jN= -2 22 2 y = -x2 - 2x -EFOLMFNJFMEFFEJMJS m -2 = - 1PMEVôVOEBO r= = 22 k = -(-1)2 - -1) - 1 = ÖRNEK 7 ÖRNEK 10 Z= Y+ 2 +L Z= -Y2 -LY+ 1 QBSBCPMÑOÑOUFQFOPLUBTY+ y - 2 =EPôSVTV- OVOÑ[FSJOEFPMEVôVOBHÌSF LLBÀUS QBSBCPMÑOÑO UFQF OPLUBTOO Z FLTFOJOF V[BLMô CJSJN PMEVôVOB HÌSF L OJO BMBCJMFDFôJ EFôFSMFSJO r = -2 j T ( - L PMVS ÀBSQNLBÀUS 2x + y - 2 =j -4 + k - 2 = k=6 b -k k r =- =- =- 2a - 4 4 | r | = 2 j r =WS= 2 kk - = 2 W - = - 2 44 k = -8 k = 8 j -= -64 ÖRNEK 8 ÖRNEK 11 Z= -Y2 +Y-L Z=Y2 -BY+B QBSBCPMÑ Z + 1 = EPôSVTVOB UFôFU PMEVôVOB HÌ- QBSBCPMÑOÑO UFQF OPLUBT FLTFOMFSF FöJU V[BLML- SF LLBÀUS UBPMEVôVOBHÌSF BOOBMBCJMFDFôJGBSLMEFôFSMFSJO UPQMBNLBÀUS r =- -a = a WFk = d a 2 a +a 22 n - a· 22 5FQFOPLUBTZ= -EPôSVTVOVOÑ[FSJOEFEJS T ( r, - EJS k= - 2 + 4a a b =- 4 = 2 & f^ 2 h =-1 4 r =- 2a - 2 5 S L FLTFOMFSFFöJUV[BLMLUBJTF| r | = | k |EJS -22 +- k = -1 r =LWL= -k 4 - k = -1 j k = 5 a - a2 + 4a a a2 - 4a = v= 24 24 a2 - 2a = B2 - 6a = a = B= 2, a = 6 BEFôFSMFSJOJOUPQMBNEJS d 1 , - 1 n 6 5 24 –64 8 2
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 12 %m/*m Z=Y2 +Y+L #JSQBSBCPMYFLTFOJOFUFóFUJTFUFQFOPLUBTY QBSBCPMÐOÐOUFQFOPLUBTOOYFLTFOJOFV[BLMó ZFL- FLTFOJÐ[FSJOEFEJS:BOJQBSBCPMÐOYFLTFOJOF TFOJOFPMBOV[BLMóOEBOEBIBGB[MBES EFóEJóJOPLUB UFQFOPLUBTES#VEVSVNEB UF- #VOBHÌSF LOJOFOHFOJöEFôFSBSBMôOCVMVOV[ QFOPLUBTOOPSEJOBUTGSES±SOFóJO 5FQFOPLUBT5 B C PMTVO Z= Y+ 2 Z= Y- 2 Z= Y+ 2 a = - 4 = - 2 jC= (-2)2 + -2) + k jC= k - 4 QBSBCPMMFSJOJOUFQFOPLUBMBSYFLTFOJÐ[FSJOEF- 2 EJS | k - 4 | >PMNBM ÖRNEK 15 k - 4 >WL- 4 < -2 Z= - Y- 2 +N- 2 k >WL< 2 QBSBCPMÑYFLTFOJOFUFôFUPMEVôVOBHÌSF NLBÀUS k ` (-Þ, 2) b (6, Þ) ÖRNEK 13 N- 2 =jN= 2 Z= Y- B-Y QBSBCPMÑOÑO UFQF OPLUBT BOBMJUJL EÑ[MFNJO ** CÌM- HFTJOEF PMEVôVOB HÌSF B OO FO HFOJö EFôFS BSBM- ôOCVMVOV[ y = -x2 + (a + 2)x -BQBSBCPMÑOÑOUFQFOPLUBTOO ÖRNEK 16 BQTJTJOFHBUJG PSEJOBUQP[JUJGPMNBM Z=Y2 -Y+L QBSBCPMÑYFLTFOJOFUFôFUPMEVôVOBHÌSF LLBÀUS a+2 a+2 < 0 &a <-2 - <0 & -2 2 #VEVSVNEB QBSBCPMÑOZFLTFOJOJLFTUJôJOPLUB Y= JÀJO-BESB< -JLFO-2a >PMVS,PMMBSBöBôZÌO- b - 10 r =- =- =5 2a 2 MÑWFZFLTFOJQP[JUJGOPLUBEBLFTUJôJOEFO UFQFOPLUB- 5FQFOPLUBTOOPSEJOBUTGSPMNBMES TOOPSEJOBULFTJOPMBSBLQP[JUJGUJS²ZMFZTF 52 -+ k =j k = 25 a ` (-Þ, - EJS ÖRNEK 14 ÖRNEK 17 Z=LY2 -Y+QBSBCPMÐOÐOYFLTFOJOJLFTNFEJóJCJ- Z= Y+N 2 -O-N MJOJZPS QBSBCPMÑYFLTFOJOF FLTFOJOQP[JUJGUBSBGOEBUFôFU #VOBHÌSF LOJOFOLÑÀÑLUBNTBZEFôFSJJÀJO QB- PMEVôVOBHÌSF SBCPMÑOYFLTFOJOFFOZBLOOPLUBTOOPSEJOBULBÀ- US * N+O= **N< ***O> y =JÀJOLY2 - 4x + 1=EFOLMFNJOEFD PMNBMES JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS D 2L j 16 < 4k jL PMVS * 5FQFOPLUBTOOPSEJOBUTGSES-O-N= jN+O= &OLÑÀÑLLUBNTBZT L=JÀJOZ= 5x2 - 4x +QBSB- ** 5FQFOPLUBTOOBQTJTJ-NEJSWF-N>PMNB- CPMÑOÑOYFLTFOJOFFOZBLOOPLUBT UFQFOPLUBTES MES#VSBEBON<CVMVOVS r =- -4 = 2 & f^ r h = 5·d 2 2 2 +1= 1 *** N<JTFO>ES 2.5 5 n - 4· 5 55 (–Þ, 2) b (6, Þ) 1 25 2 25 * **WF*** (–Þ, –2) 5
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr %m/*m %m/*m %FOLMFNJ Z = BY2 + D PMBO QBSBCPMÐO UF- %FOLMFNJ Z=BY2PMBOQBSBCPMÐOUFQFOPLUB- QF OPLUBT Z FLTFOJ Ð[FSJOEFEJS ,PPSEJOBUMBS TPSJKJOEJS 5 D EJS | | #VQBSBCPMEF BTBZTCÐZÐEÐL¿F QBSBCPMÐO ÖRNEK 18 | |LPMMBSZFLTFOJOFZBLMBõS BTBZTLпÐMEÐL-==23xx22 Z= -Y2 - O+ Y-O ¿FQBSBCPMÐOLPMMBSZFLTFOJOEFOV[BLMBõSy y y = x2y QBSBCPMÑOÑOUFQFOPLUBTZFLTFOJÑ[FSJOEFPMEVôV- OBHÌSF UFQFOPLUBTOOPSEJOBULBÀUS O x 5FQFOPLUBTOOBQTJTJTGSES b -^ n + 1 h = 0 & n + 1 = 0 jO= -1 r =- =- 2a - 4 y = -2x2 +PMEVôVOEBO 5 EJS yyy===––x–223x2x2 ÖRNEK 19 ÖRNEK 21 Z=BY2 + B- Y-B-QBSBCPMÐYFLTFOJOJBQTJTMF- Z=Y2WFZ= Y- 2 SJUPQMBNTGSPMBOGBSLMJLJOPLUBEBLFTNFLUFEJS QBSBCPMMFSJOJO UFQF OPLUBMBS BSBTOEBLJ V[BLML LBÀ #VOB HÌSF QBSBCPMÑO UFQF OPLUBTOO LPPSEJOBUMB- CJSJNEJS SOCVMVOV[ y = 4x2JÀJOUFQFOPLUBTPSJKJOEJS Y FLTFOJOJ LFTUJôJ OPLUBMBS PSJKJOF HÌSF TJNFUSJL PM- y = ( 4x - 2 )2JÀJOUFQFOPLUBTd 1 , 0 nES EVôVOEBO UFQFOPLUBTZFLTFOJÑ[FSJOEFEJS a =JÀJOZ= 2x2 -PMVQ 5 - ES 2 1 #VJLJOPLUBBSBTOEBLJV[BLML CJSJNEJS 2 ÖRNEK 20 ÖRNEK 22 B<C<DPMNBLÐ[FSF G Y =BY2 +CY+DQBSBCPMÐOÐO (FS¿FLTBZMBSEBUBONMZ=G Y JLJODJEFSFDFEFOGPOL- UFQFOPLUBTZFLTFOJÐ[FSJOEFEJS TJZPOVOVOHSBGJóJ YFLTFOJOFPSJKJOEFUFóFUUJS #VOBHÌSF CVQBSBCPMMFJMHJMJBöBôEBWFSJMFO G = -2 PMEVôVOBHÌSF G LBÀUS * ,PMMBSBõBóEPóSVEVS 5FQFOPLUBTPSJKJOPMBOQBSBCPMÑOEFOLMFNJ ** YFLTFOJOJLFTNF[ *** 0SJKJOEFOHF¿FS f ( x ) = ax2EJS JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS f ( 2 ) =B2 = -2 j 4a = -2 j a = - 1 5FQFOPLUBTOOBQTJTJTGSPMEVôVOEBOC=ES 2 a <<DPMVSB<JTFLPMMBSBöBôZÌOMÑEÑS#VEV- SVNEBYFLTFOJOJJLJOPLUBEBLFTFSD>PMEVôVOEBO f^ x h =- 1 2 & f^ 4 h =- 1 2 = - 8 PSJKJOEFOHFÀNF[ x .4 22 2 m :BMO[* 26 1 –8 2
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*** TEST - 8 Z=Y2 -Y+ \"öBôEBLJQBSBCPMMFSEFOIBOHJTJOJOUFQFOPLUB- QBSBCPMÑOÑOUFQFOPLUBTBöBôEBLJMFSEFOIBO- TZFLTFOJÑ[FSJOEFEJS HJTJEJS \" Z=Y2 +Y # Z= Y- 2 \" # $ - $ Z= - Y+ 2 - % Z=Y2 - 2 % - & & Z= -Y2 -Y+ 2 Z= - Y+ 2 + 1 \"öBôEBLJQBSBCPMMFSEFOIBOHJTJOJOUFQFOPLUB- QBSBCPMÑOÑOUFQFOPLUBTBöBôEBLJMFSEFOIBO- TYFLTFOJÑ[FSJOEFEJS HJTJEJS \" Z= -Y2 +Y+ # Z=Y2 -Y- 36 $ Z= - Y- 2- % Z= - Y+ 2 & Z=Y2 -Y \" # - $ - - % - & - Z=Y2 -WFZ=Y2 -Y Z=Y2 +Y- 5 QBSBCPMMFSJOJOUFQFOPLUBMBSBSBTOEBLJV[BLML QBSBCPMÑOÑOUFQFOPLUBTOOLPPSEJOBUMBSUPQ- LBÀCJSJNEJS MBNLBÀUS \" - # - $ - % - & \" 5 # 2 $ 1 % & 2 1BSBNFUSJLEFOLMFNJ Y=U- 3 Z= -Y2 +NY+O- Z= U+ -U QBSBCPMÑOÑOUFQFOPLUBT - PMEVôVOBHÌ- PMBO QBSBCPMÑO UFQF OPLUBT BöBôEBLJMFSEFO SF N+OUPQMBNLBÀUS IBOHJTJEJS \" - # - $ % & \" - # - $ - % - & - \" % \" B 27 % % % \"
TEST - 9 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*** G Y = -Y2 -Y+ 5 Z=Y2 - O+ Y+O GPOLTJZPOVOVO B[BMBO PMEVôV FO HFOJö BSBML QBSBCPMÐOÐOUFQFOPLUBTZFLTFOJÐ[FSJOEFEJS BöBôEBLJMFSEFOIBOHJTJEJS #VOBHÌSF QBSBCPMÑOYFLTFOJOJLFTUJôJOPLUB- \" -Þ -> # -Þ > $ <- > MBSBSBTOEBLJV[BLMLLBÀCJSJNEJS % <- Þ & < Þ \" # $ % & G Y =BY2 - B+C Y+C+ 4 L`3PMNBLÐ[FSF QBSBCPMÑOÑOUFQFOPLUBTPSJKJOPMEVôVOBHÌSF G Y = - Y+L- 2 +L G C -G B LBÀUS \" # $ % & QBSBCPMÐOÐOUFQFOPLUBTZFLTFOJÐ[FSJOEFEJS #VOBHÌSF CVQBSBCPMÑOFLTFOMFSJLFTUJôJOPL- UBMBS CJSMFöUJSJMFSFL FMEF FEJMFO ÑÀHFOJO BMBO LBÀCJSJNLBSFEJS \" # 4 2 $ % 8 2 & G Y =Y2 - B+C Y+C- 1 Z= N- Y2 - N2 - Y-N+ 4 GPOLTJZPOVOVOHSBGJôJ YFLTFOJOF - OPL- QBSBCPMÐOÐOYFLTFOJOJLFTUJóJOPLUBMBSPSJKJOFHËSF UBTOEBUFôFUPMEVôVOBHÌSF G B LBÀUS TJNFUSJLUJS \" # $ % & #VOB HÌSF QBSBCPMÑO UFQF OPLUBTOO LPPSEJ- OBUMBSUPQMBNLBÀUS \" - # $ % & Z=Y2 +Y+Q Z=Y2 -LY-L QBSBCPMÑOÑOUFQFOPLUBTY- 2y =EPôSVTV- QBSBCPMÑOÑOUFQFOPLUBTOOZFLTFOJOFV[BL- OVOÑ[FSJOEFPMEVôVOBHÌSF QLBÀUS MôCJSJNPMEVôVOBHÌSF YFLTFOJOFV[BLMô LBÀCJSJNPMBCJMJS \" # $ % & \" # $ % & % % $ & 28 % B & B
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*** TEST - 10 y = 1 x2 - nx Z=LY2 -Y+L- 4 4 QBSBCPMÑYFLTFOJOFUFôFUPMEVôVOBHÌSF LOJO BMBCJMFDFôJEFôFSMFSJOUPQMBNLBÀUS QBSBCPMÑOÑOUFQFOPLUBTOOYFLTFOJOFV[BLM- ôLBÀCJSJNEJS # - n2 $ n2 % O2 \" # $ % & 22 \" -O2 & O2 Z= -Y2 -Y+L BáPMNBLÐ[FSF QBSBCPMÑZ- 9 =EPôSVTVOBUFôFUPMEVôVOB Z=Y2 -BY+B HÌSF LLBÀUS \" # $ % & QBSBCPMÑOÑO UFQF OPLUBT FLTFOMFSF FöJU V[BL- MLUBPMEVôVOBHÌSF BLBÀUS \" # $ % & Z=Y2 -Y+ 5 QBSBCPMÑOÑOYFLTFOJOFFOZBLOOPLUBTOOPS- Z=Y2 +OY+ 36 EJOBULBÀUS \" 3 # $ 9 % 5 & 11 QBSBCPMÐ Y FLTFOJOF FLTFOJO OFHBUJG UBSBGOEB UF- 2 4 2 4 óFUPMEVóVOBHËSF OLBÀUS \" - # - $ % & N`3PMNBLÑ[FSF NâWFOâPMNBLÑ[FSF Z= -Y2 -NY+N G Y =N2Y2 -NOY+O QBSBCPMMFSJOJO UFQF OPLUBMBSOO Ñ[FSJOEF CV- GPOLTJZPOVOVOHSBGJôJYFLTFOJOFUFôFUPMEVôV- MVOEVôV QBSBCPMÑO EFOLMFNJ BöBôEBLJMFSEFO OBHÌSF OLBÀUS IBOHJTJEJS \" Z=Y2 -Y # Z= -Y2 -Y \" 1 # 1 $ 1 8 4 2 $ Z= -Y2 +Y % Z=Y2 +Y % & & Z=Y2 + 1 % B & \" 29 \" B % \"
TEST - 11 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*** ôFLJMEFG Y =BY2 +CY+DGPOLTJZPOVOVOHSBGJóJ (FS¿FLTBZMBSEBUBONM JLJODJEFSFDFEFOCJS WFSJMNJõUJS Z=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMÐOUFQF y OPLUBTPSJKJOEJS #VGPOLTJZPOJÀJO x r G -B =G -B O r G B- =B FöJUMJLMFSJTBôMBOEôOBHÌSF G LBÀUS y = f(x) \" # $ % & #VOBHÌSF * B C D ** C2 BD *** C D JGBEFMFSJOEFOIBOHJMFSJEBJNBEPôSVEVS G Y = -O Y2 - O2 - Y+O \" :BMO[* # :BMO[** $ *WF** GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMJMFJMHJMJ r YFLTFOJOJLFTUJóJOPLUBMBSPSJKJOFHËSFTJNFUSJL- % **WF*** & * **WF*** UJS r 5FQFOPLUBTNJOJNVNOPLUBTES Z=Y2 -BY+C CJMHJMFSJWFSJMJZPS QBSBCPMÐOÐOUFQFOPLUBTBOBMJUJLEÐ[MFNJO**CËM- HFTJOEFEJS #VOBHÌSF G O LBÀUS #VOBHÌSF BöBôEBLJQBSBCPMMFSEFOIBOHJTJOJO \" - # - $ % & UFQF OPLUBT BOBMJUJL EÑ[MFNJO *7 CÌMHFTJOEF- EJS (FS¿FLTBZMBSEBUBONM JLJODJEFSFDFEFOCJS \" Z= -B Y-B 2 -BC # Z=C Y-B 2 -C $ Z=B Y+B 2 -C % Z=BY2 -C Z=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMÐOUFQF OPLUBT5 S L EJS & Z=BY2 -CY Z= B- Y2 -BY- #VGPOLTJZPOJ¿JO QBSBCPMÐOÐOYFLTFOJOJLFTNFEJóJCJMJOJZPS r Y1áY2PMNBLÐ[FSF G Y1 =G Y2 = r rY` Y1 Y2 J¿JOG Y r Y1 <<Y2WF|Y1 | > |Y2 | CJMHJMFSJWFSJMJZPS #VOBHÌSF BOOBMBCJMFDFôJFOCÑZÑLUBNTBZ #VOB HÌSF BöBôEBLJMFSEFO IBOHJTJ LFTJOMJLMF EFôFSJJÀJOQBSBCPMÑOYFLTFOJOFFOZBLOOPL- EPôSVEVS UBTOOPSEJOBULBÀUS \" S> # L> % - & - 1 $ G L <G S % G Y2 -Y1 > 2 \" -4 # - $ - & G Y1Y2 > $ $ B & & B
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"3*7 ÷MJöLJMJ,B[BONMBS 11.3.2.1 : öLJODJEFSFDFEFOCJSEFóJõLFOMJGPOLTJZPOVOHSBGJóJOJ¿J[FSFLZPSVNMBS 1BSBCPM¦J[JNJ ÖRNEK 1 TANIM G3Z3 G Y =Y2 -Y+ 5 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ öLJODJEFSFDFEFOZ=G Y =BY2 +CY+DGPOL- TJZPOVOVOHSBGJóJPMBOQBSBCPM¿J[JMJSLFOBõBó- y a > EBLJJõMFNBENMBSVZHVMBOS T(2, 1) Y2MJUFSJNJOLBUTBZTPMBOBOOJõBSFUJOFCBL- 5 MBSBLQBSBCPMÐOLPMMBSOOZËOÐCFMJSMFOJS r B>JLFOQBSBCPMÐOLPMMBSZVLBSZBEPóSV r B<JLFOQBSBCPMÐOLPMMBSBõBóZBEPóSV ¿J[JMJS 1T x O2 5FQFOPLUBTOOLPPSEJOBUMBS S L CVMVOVS r Z=BY2 +CY+DEFOLMFNJJ¿JO ÖRNEK 2 r = - b WFL=G S G3Z3 G Y = -Y2 +Y- 12 2a GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ FõJUMJLMFSJOEFOZBSBSMBOMS y T3 a < r Z=B Y-S 2 +L 2 5/2 x Td 5 , 1 n EFOLMFNJJ¿JOUFQFOPLUBT S L ES 1/2 r Z=B Y-S 2 O 22 EFOLMFNJJ¿JOUFQFOPLUBTYFLTFOJÐ[FSJO- -12) EF S OPLUBTES r Z=BY2 +D EFOLMFNJJ¿JOUFQFOPLUBTZFLTFOJÐ[FSJO- EF D OPLUBTES –12 r Z=BY2EFOLMFNJJ¿JOUFQFOPLUBTPSJKJOEJS ÖRNEK 3 1BSBCPMÐOFLTFOMFSJLFTUJóJOPLUBMBSCFMJSMFOJS r %FOLMFNEF Y=J¿JOQBSBCPMÐOZFLTFOJ- G3Z3 G Y = Y- 2 + 5 OJLFTUJóJOPLUBCVMVOVS GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ r Z=J¿JOQBSBCPMÐOWBSTBYFLTFOJOJLFT- UJóJOPLUBMBSCVMVOVS &MEFFEJMFOWFSJMFSJOUÐNÐLVMMBOMBSBLQBSBCPM ¿J[JMJS y a > 9 T(2, 5) 5 T x O2 31
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr ÖRNEK 4 ÖRNEK 7 G3Z3 G Y = - 1 _ x + 2 i2 - 3 G3Z3 G Y =Y2 - 1 2 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ y y a > 5 -1) a < –2 O x T(-2, -3) -5) O f 3 ,0p x3 3 T –3 –3 3 f- 3 ,0p –5 3 T(0,–1) 3 ÖRNEK 5 ÖRNEK 8 G3Z3 G Y = -Y2 +Y G3Z3 G Y =Y2 + 6 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ y y 4T a > 5 O2 a < T(2, 4) T(0,6) O 4 x x ÖRNEK 6 ÖRNEK 9 G3Z3 G Y =Y2 -Y G3Z3 G Y = 2 x2 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ 3 y GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ O 1/4 x a > y 1/2 Td 1 , - 1 n a > 5 –1/8 T 48 x d 1 ,0n O 2 32
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 10 ÖRNEK 13 G[- ] Z3 G Y = -Y2 +Y G - Z3 G Y = - 1 _ x - 6 i2 + 4 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ 4 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ y y 1/4 T a < –4 3 a < 1 Td 1 , 1 n x –3 2 24 O 24 T(6, 4) O 1/2 x –5 -5) –2 (-3, -12) (2, -2) (-4, -21) –12 –21 (4, 3) ÖRNEK 11 ÖRNEK 14 G[- Z3 G Y =Y2 -Y- 3 G - ] Z3 G Y =Y2 +Y- 8 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ y a > 21 T(1, -4) -3) 1 3 (- y a > –4 –1 O T(-1, -9) (-4, 21) –2 –1 O 1 -8) –3 x –5 x –4 –8 (- –9 T (-2, -8) T (1, -5) ÖRNEK 12 ÖRNEK 15 G - ] Z3 G Y =Y2 -Y+ 3 G[- Þ Z3 G Y = 8 -Y2 GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ GPOLTJZPOVOVOHSBGJôJOJÀJ[JOJ[ y a > y a < 15 T(2, -1) T(0, 8) 5 3 6 (- –2 O 1 (-1, 6) –1 O 2 (-2, 15) x x 33
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr 1BSBCPMÑO4JNFUSJ<FOJ ÖRNEK 17 TANIM Z= Y+ 2 + 4 QBSBCPMÑOÑOTJNFUSJFLTFOJOJOEFOLMFNJOJCVMVOV[ 1BSBCPMÐOLPMMBS UFQFOPLUBTOEBOHF¿FOWFY FLTFOJOFEJLPMBOEPóSVZBHËSFTJNFUSJLUJS#V r = - TJNFUSJFLTFOJY= -3 EPóSVZB QBSBCPMÐOTJNFUSJFLTFOJEFOJS 5FQFOPLUBT5 S L PMBOQBSBCPMÐOTJNFUSJFL- TFOJY=SEPóSVTVEVS x=r x=r y y O rx O rx ÖRNEK 18 Simetri ekseni Simetri ekseni G Y = -Y2 +Y+B QBSBCPMÐOÐOTJNFUSJFLTFOJ B- B OPLUBTOEBOHF¿- NFLUFEJS #VOBHÌSF G B LBÀUS 1BSBCPMÐ[FSJOEFPSEJOBUFõJUPMBOJLJOPLUB TJ- 11 3 NFUSJFLTFOJOFFõJUV[BLMLUBES r =- =a-1 & =a-1 j a= -2 2 2 Z = G Y QBSBCPMÐ JMF Z = L EPóSVTVOVO LFTJ- õJNOPLUBMBSOCJSMFõUJSFOEPóSVQBS¿BTOOPS- f^ x h = - x2 + x + 2 ise fd 3 n = 9 UBOPLUBT TJNFUSJFLTFOJÐ[FSJOEFEJS 3 24 y x = r y= f(x) ÖRNEK 19 K y=k AB Z= Y+ 2 -1 O r x2 x QBSBCPMÑOÑO TJNFUSJ FLTFOJ \" L - 3, k - OPLUB- x1 TOEBO HFÀUJôJOF HÌSF \" OPLUBTOO PSJKJOF V[BLMô LBÀCJSJNEJS :VLBSEBLJ õFLJMEF HËTUFSJMEJóJ HJCJ Y FLTFOJ 4JNFUSJFLTFOJY= -3 k - 3 = -3 j k =j\"( -3, -2 ) Ð[FSJOEFLJSOPLUBT Y1WFY2OPLUBMBSOOPSUB OPLUBTPMEVóVOEBO AO = ^ - 3 h2 + ^ - 2 h2 = 13 br r = x1 + x2 EJS ÖRNEK 20 2 N`3PMNBLÐ[FSF #VEVSVNEB G Y =LEFOLMFNJOJO WBSTB HFS- \" N O OPLUBT G Y = -Y2 + N- Y+NQBSBCPMÐ- OÐOTJNFUSJFLTFOJOJOÐ[FSJOEFEJS ¿FLLËLMFSJOJOUPQMBNY1 +Y2 =SEJS #VOBHÌSF QBSBCPMÑOYFLTFOJOJOLFTUJôJOPLUBMBSO ÖRNEK 16 BQTJTMFSJUPQMBNLBÀUS Z=Y2 -Y+ 1 QBSBCPMÑOÑOTJNFUSJFLTFOJOJOEFOLMFNJOJCVMVOV[ b1 1 m-2 r = - = TJNFUSJFLTFOJ x = UÑS - = m & 2m = m - 2 & m = - 2 2a 4 4 -2 f(x) = -x2 - 4x - 2 =EFOLMFNJOEFO -4 x + x = - = - 4 CVMVOVS 12 -1 Ym 34 9 13 –4 Y 4
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 21 ÖRNEK 24 G Y =BY2 + B- Y- 3 Z= - Y- 2 + 1 QBSBCPMÐ Ð[FSJOEF CVMVOBO WF PSEJOBUMBS FõJU PMBO IFS- QBSBCPMÑOÑOTJNFUSJFLTFOJJMFY+ 2y =EPôSVTV- IBOHJJLJOPLUB Y- 1=EPóSVTVOBFõJUV[BLMLUBES OVO LFTJöJN OPLUBTOO Y FLTFOJOF V[BLMô LBÀ CJ- SJNEJS #VOBHÌSF G Y =BEFOLMFNJOJOLÌLMFSJOJOUPQMBN LBÀUS 5FQFOPLUBTOOBQTJTJ S=PMVQTJNFUSJFLTFOJ x =EPôSVTVEVS 4JNFUSJFLTFOJY- 1 =EPôSVTV5FQFOPLUBTOOBQ- 1 1 11 + 2y = 5 j y = j = sisi EJS 2 2 22 3a - 1 1 1 - = & - 6a + 2 = 2a & a = 2a 2 4 f^ x h = 1 x2 - 1 x - 3 = 1 44 4 x2 - x - 12 = 1 j x2 - x - 13 = ÖRNEK 25 -1 Z=NY2-Y x1 + x2 = - 1 = 1 QBSBCPMÑOÑOTJNFUSJFLTFOJY+ 3y =EPôSVTVJMFY FLTFOJÑ[FSJOEFLFTJöUJôJOFHÌSF NLBÀUS ÖRNEK 22 3 4JNFUSJFLTFOJ x = ,PPSEJOBUEÑ[MFNJOEF Z= x2 + 6x -QBSBCPMÑJMF y =EPôSVTVOVOLFTJöJNOPLUBMBSOBFöJUV[BLMLUB 2m CVMVOBO OPLUBMBSO Ñ[FSJOEF CVMVOEVôV EPôSVOVO EFOLMFNJOJCVMVOV[ 5 y =j x = 1BSBCPM Ñ[FSJOEF PSEJOBU BZO PMBO JLJ OPLUBZB FöJU V[BLMLUB CVMVOBO OPLUBMBS TJNFUSJ FLTFOJ Ñ[FSJOEFLJ 4 UÑNOPLUBMBSES 53 6 6 = & m= 5FQFOPLUBTOOBQTJTJ r = - = - 3 PMVQTJNFUSJFL- 4 2m 5 2 ÖRNEK 26 TFOJY= -EPôSVTVEVS G Y = -Y2 -Y+ 12 ÖRNEK 23 GPOLTJZPOVOVO HSBGJóJ PMBO QBSBCPM Y FLTFOJOJ \" WF # Z=Y2 -BY+B OPLUBMBSOEBLFTJZPS QBSBCPMÑOÑO Ñ[FSJOEF PMVQ TJNFUSJ FLTFOJOF B CS #V QBSBCPMÐO TJNFUSJ FLTFOJ Ð[FSJOEFLJ CJS 1 OPLUBT V[BLMLUB CVMVOBO OPLUBMBSO LPPSEJOBUMBS UPQMBN- J¿JO \"MBO( A&PB ) = CS2 PMEVóVOB HËSF 1 OPLUBT- OOBDJOTJOEFOFöJUJOJCVMVOV[ OOUFQFOPLUBTOBV[BLMôFOB[LBÀCJSJNPMBCJMJS -x2 - 4x + 12 = x = -WFY2 =PMVQ|\"#| =CSEJS 1 5FQFOPLUBTOOBQTJTJ r = - - 2a = a ES4JNFUSJFL- \"MBO ( A&PB ) = 8.h = 52 & h = 13 br 2 2 TFOJY=BEPôSVTVEVS#VEPôSVZBBCSV[BLMLUBCVMV- 1OPLUBTOOPSEJOBU-WFZBPMBCJMJS5FQFOPLUB- -4 OBOOPLUBMBSOBQTJTMFSJY1 =WFZBY2 =BPMNBMES x1 =JÀJOZ1 =2 -B+ a = a TOOPSEJOBUJTF r = - = - 2 PMVQ -2 x2=BJÀJOZ2= (2a)2 -BB+ a =BES f ( 2 ) = -4 + 8 + 12 =ES1OPLUBTOOUFQFOPLUBT- /PLUBMBS B WF B B PMVQLPPSEJOBUMBSUPQMBNB OBV[BLMô- 13 =CSZBEB- (-13) =CSPMB- ES CJMJS0IBMEFV[BLMLFOB[CSEJS 1 Y 4a 35 1 6 3 2 5
TEST - 12 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*7 G Y =Y2 +Y- 3 G Y = -Y2 - 4 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS EJS A) y B) y A) y B) y O O x x –2 4 Ox O x –4 –1 3 –3 1 –3 –3 C) y D) y C) y D) y x O 3 3 4 O1 O –4 x x –3 O1 x E) y E) y 3 4 –2 2 x O –1 O 3 x G Y = Y+ 2 - 3 G Y = -Y2 +Y GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS EJS A) y B) y A) y B) y 3 2 2 –2 O 1 O x –2 –1 Ox 12 x –1 x C) y D) y –2 O 4 –3 C) y D) y 1 x –2 2 x O O 2 1 x 1 x –2 –2 2 E) y O O –3 –3 x E) y –1 x –2 O 3 –2 1 O2 B $ 36 % \"
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*7 TEST - 13 G Y = -Y2 +Y- 5 y y = ax2 y = bx2 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS A) y 1 x B) y x –1 x O O O –4 y = cx2 –5 –4 Z=BY2 Z=CY2WFZ=DY2QBSBCPMMFSJOJOHSBGJL- –5 MFSJõFLJMEFWFSJMNJõUJS C) y D) y 5 5 #VOB HÌSF B C D TBZMBS BSBTOEBLJ TSBMBNB 4 4 BöBôEBLJMFSEFOIBOHJTJEJS O1 x –1 O x \" B<C<D # D<B<C $ D<C<B E) y % C<B<D & B<D<C 5 x 4 O1 BC>PMNBLÐ[FSF | | | | C<<DWF C > DPMNBLÐ[FSF G Y =BY2 +CY G Y = Y+C 2 -D2 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ PMBCJMJS PMBCJMJS A) y B) y A) y B) y Ox Ox x x O O C) y D) y C) y D) y Ox O O x Ox x E) y x E) y O x O \" % 37 $ %
TEST - 14 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*7 Z=Y2 +Y+ 3 öLJODJ EFSFDFEFO Z = G Y GPOLTJZPOVOVO TJNFUSJ QBSBCPMÑOÑOTJNFUSJFLTFOJOJOEFOLMFNJBöBô- FLTFOJY=EPóSVTVEVS EBLJMFSEFOIBOHJTJEJS #VOBHÌSF G Y =EFOLMFNJOJOLÌLMFSJOJOUPQ- \" Y- 4 = # Y- 2 = $ Y+ 2 = MBNLBÀUS % Y+ 4 = & Y+ 1 = \" 1 # $ % & 2 \"öBôEBEFOLMFNMFSJWFSJMFOQBSBCPMMFSEFOIBO- \"öBôEBLJ EFOLMFNMFSEFO IBOHJTJ Y FLTFOJOJ HJTJOJOTJNFUSJFLTFOJPSJKJOEFOHFÀFS \" -O WF# O- CJÀJNJOEFJLJOPLUBEB LFTFOCJSQBSBCPMFBJUPMBCJMJS \" Z=Y2 +Y # Z=Y2 + 2 $ Z= Y- 2 + % Z= -Y2 -Y+ 1 \" Z=Y2-Y+B # Z=Y2 +Y+C & Z= Y- -Y $ Z=Y2 +Y+D % Z= Y+ 2 -E & Z=Y2 + 3 \"öBôEBLJ TFÀFOFLMFSJO IBOHJTJOEF WFSJMFO JLJ \" - OPLUBT QBSBCPMÑOTJNFUSJFLTFOJBZOEPôSVEVS y = 1 _ 2x + a i2 - a 4 \" Z= -Y2 +YWFZ=Y2 +Y- 1 # Z=Y2 -Y+WFZ= - Y+ 2 QBSBCPMÑOÑO TJNFUSJ FLTFOJOJO Ñ[FSJOEF PMEV- $ Z= Y- 2WFZ= Y+ 2 + 2 % Z=Y2 -WFZ=Y2 + 1 ôVOBHÌSF BLBÀUS & Z=Y2WFZ=Y2 +Y \" - # - $ 1 % & 3 Z=Y2 -LY+L y = - 1 x2 - kx + 1 QBSBCPMÑOÑOTJNFUSJFLTFOJPMBOEPôSV 3 \" L+ OPLUBTOOHFÀUJôJOFHÌSF LLBÀUS QBSBCPMÑOÑOTJNFUSJFLTFOJY- 5y =EPôSV- \" - 4 # - 2 $ 1 % TVJMFYFLTFOJÑ[FSJOEFLFTJöUJôJOFHÌSF k kaç- 3 33 US & # - 2 $ - 1 % 1 3 93 \" - & $ B % \" 38 % $ & $
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS*7 TEST - 15 G Y = -Y2 + B+ Y+C f_ x i = 1 x2 - 4x + 16 QBSBCPMÐOÐOTJNFUSJFLTFOJ YFLTFOJOJOFHBUJGUBSB- 4 GOEBLFTNFLUFEJS QBSBCPMÑOÑOÑ[FSJOEFPMBOWFQBSBCPMÑOTJNFU- f ( 1 ) = PMEVôVOB HÌSF C OJO BMBCJMFDFôJ FO SJFLTFOJOFCJSJNV[BLMLUBCVMVOBOOPLUBMBSO LÑÀÑLUBNTBZEFôFSJLBÀUS PSEJOBUMBSUPQMBNLBÀUS \" # $ % & \" # $ % & G Y =Y2 -BY+ 12 y QBSBCPMÐ Ð[FSJOEFLJ 1 C - D WF 3 C + D OPLUBMBS Y+ 2 =EPóSVTVOBFõJUV[BLMLUBES #VOBHÌSF B+CUPQMBNLBÀUS & \" - # - $ - % AO x B ôFLJMEF G Y = -Y2 +Y+DGPOLTJZPOVOVOHSBGJóJ Z= -Y2 +NY+ 12 WFSJMNJõUJS | | | |0# = 2 AO PMEVôVOBHÌSF #OPLUBTOOBQ- QBSBCPMÐ JMF Y FLTFOJ BSBTOEB LBMBO CËMHFOJO J¿J- OF¿J[JMFCJMFDFLпHFOMFSEFOBMBOFOCÐZÐLPMBOO TJTJLBÀUS ZÐLTFLMJóJ Y- 1 =EPóSVTVOVOÐ[FSJOEFEJS \" # $ % & #VOBHÌSF CVÑÀHFOJOBMBOLBÀCJSJNLBSFEJS \" 25 # 100 $ % 125 & 46 4 Z= -Y2 +Y-BQBSBCPMÑJMFZ= 3x +BEPôSV- TVOVOLFTJöJNOPLUBMBSOEBOCJSJ QBSBCPMÑOTJ- NFUSJFLTFOJÑ[FSJOEFPMEVôVOBHÌSF BLBÀUS \" - # - 1 $ % & | | B C D`3 B<C<DWF C >DPMNBLÐ[FSF 2 G Y =B Y-C Y+D GPOLTJZPOVOVOHSBGJóJY+ 1 =EPóSVTVOBHËSF TJ- NFUSJLUJS Z=Y2 + N- Y+OQBSBCPMÐOÐOTJNFUSJFLTFOJ #VGPOLTJZPOMBJMHJMJBöBôEBWFSJMFO * D=C+ 2 Ð[FSJOEFLJCJS Ad m , 2 nOPLUBTOOQBSBCPMÐOUF- ** G B 2 *** G C- =G -D QFOPLUBTOBPMBOV[BLMóCJSJNEJS #VOBHÌSF OOJOBMBCJMFDFôJEFôFSMFSJOUPQMBN JGBEFMFSJOEFOIBOHJMFSJLFTJOPMBSBLEPôSVEVS LBÀUS \" :BMO[* # :BMO[** $ *WF** \" - # - $ % & % **WF*** & * **WF*** & $ \" & 39 $ \" % &
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"37 ÷MJöLJMJ,B[BONMBS 11.3.2.1 : öLJODJEFSFDFEFOCJSEFóJõLFOMJGPOLTJZPOVOHSBGJóJOJ¿J[FSFLZPSVNMBS ÷LJODJ %FSFDFEFO 'POLTJZPOMBSO &O #ÑZÑL WF ÖRNEK 2 &O,ÑÀÑL%FôFSJ G3Z3 G Y = -Y2 +Y- 11 %m/*m GPOLTJZPOVOVOFOCÑZÑLEFôFSJLBÀUS G3Z3 G Y =BY2 +CY+DGPOLTJZPOVJ¿JO -1 < - 10 B > JLFO QBSBCPMÐO LPMMBS ZVLBS EPóSVEVS 5FQF OPLUBT QBSBCPMÐO NJOJNVN OPLUBTES r= =5 5FQFOPLUBTOOPSEJOBUJTFGPOLTJZPOVOBMBCJ- -2 MFDFóJFOLпÐLEFóFSEJS f(5) = -25 +- 11 =UÑS T( r , k ) 1BSBCPMÐONJOJNVNOPLUBT .BLTJNVNEFôFSUÑS 5 S L GPOLTJZPOVOVO FO LÐ- ¿ÐL EFóFSJ JTF L = G S EJS #V ÖRNEK 3 GPOLTJZPOVOVONBLTJNVNOPL- UBT WF NBLTJNVN EFóFSJ ZPL- G3Z3 G Y = Y- Y+ UVS GPOLTJZPOVOVOFOLÑÀÑLEFôFSJLBÀUS B < JLFO QBSBCPMÐO LPMMBS BõBó EPóSVEVS f ( x ) = 2x2 + 8x - > 5FQF OPLUBT QBSBCPMÐO NBLTJNVN OPLUBT- 8 ES 5FQF OPLUBTOO PSEJOBU JTF GPOLTJZPOVO BMBCJMFDFóJFOCÐZÐLEFóFSEJS r =- =-2 4 T( r , k ) 1BSBCPMÐO NBLTJNVN OPLUBT 5 S L GPOLTJZPOVOVO FO CÐ- f ( -2 ) =--= -98 ZÐL EFóFSJ JTF L = G S EJS #V .JOJNVNEFôFS-EJS GPOLTJZPOVOVONJOJNVNOPLUB- TWFNJOJNVNEFóFSJZPLUVS ÖRNEK 4 #JS QBSBCPMÐO UBONM PMEVóV FO HFOJõ BSBML- G3Z3 G Y =Y2 -NY+1 UBUFQFOPLUBTOOPSEJOBU GPOLTJZPOVONBLTJ- GPOLTJZPOVOVOFOLÑÀÑL EFôFSJ-PMEVôVOBHÌSF NVNZBEBNJOJNVNEFóFSJEJS N`3+ kBÀUS ÖRNEK 1 3 > k = -9 G3Z3 G Y = - Y+B 2 + 8 4.3.1 - m2 = - 9 jN2 = GPOLTJZPOVOVOFOCÑZÑLEFôFSJLBÀUS 4.3 -2 <jUFQFOPLUBTOOPSEJOBUGPOLTJZPOVOVOFOCÑ- m = 2 30 , m = - 2 30 ZÑLEFôFSJEJS T(-B PMVQFOCÑZÑLEFôFSJEJS 8 14 –98 2 30
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 5 ÖRNEK 8 G3Z3 G Y =BY2 -Y-B Y`3PMNBLÐ[FSF 46 GPOLTJZPOV Y = - JÀJO FO CÑZÑL EFôFSJOJ BMEôOB HÌSF CVEFôFSLBÀUS x2 + 5x + 12 JGBEFTJOJOBMBCJMFDFôJFOCÑZÑLEFôFSLBÀUS -1 1 1 r =- = =-3 & a =- 2a 2a 6 1 > f^ x h =- 1 2 - x + 1 4.1.12 - 25 23 46 =8 k= =& x 62 4.1 4 23 f^ - 3 h = - 1 ·9 + 3 + 1 = 2 4 62 ÖRNEK 6 ÖRNEK 9 G3Z3 G Y =BY2 -BY B C D`3PMNBLÑ[FSF B= 4 -D GPOLTJZPOVOVOFOCÑZÑLEFôFSJPMEVôVOBHÌSF D= 2 -C BLBÀUS PMEVôVOBHÌSF BCÀBSQNOOFOLÑÀÑLEFôFSJLBÀ- US r =- - 6a =3 a = 4 -D 2a C= 2 -DjBC= ( 4 -D -D G D =D2 -D+ 8 f ( 3 ) =B-B= 36 4.3.8 - 100 - 1 -9a = 36 j a = -4 k= = 4.3 3 ÖRNEK 7 ÖRNEK 10 G Y =Y2 -Y+ 3 #JSJEJôFSJOJOLBUOEBOFLTJLPMBOJLJSFFMTBZOO GPOLTJZPOVOVOHSBGJôJÑ[FSJOEFLJCJSOPLUBOOLPPS- ÀBSQNOOFOLÑÀÑLEFôFSJLBÀUS EJOBUMBSUPQMBNOOBMBCJMFDFôJFOLÑÀÑLEFôFSLBÀ- US 4BZMBSYWFY-PMTVO f ( x ) = x ( 2x - 3 ) f ( x ) = x2- 5x +Ñ[FSJOEFCJS Y Z OPLUBTOOLPPS- f ( x ) = 2x2 -GPOLTJZPOVJÀJO > EJOBUMBSUPQMBN x + y = x + x2- 5x + 3 = x2 - 4x +PMVS 4.2.0 - 9 - 9 k= = 1 > 4.1.3 - ^ - 4 h2 4.2 8 k = =-1 4.1 2 –4 –1 41 1 9 8 - - 38
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr ÷LJODJ%FSFDFEFO'POLTJZPOMBSO(ÌSÑOUÑ %m/*m ,ÑNFTJ G[Q R] Z3 G Y =BY2 +CY+DCJ¿JNJOEF [Q R] BSBMóOEB UBONM JLJODJ EFSFDFEFO CJS %m/*m GPOLTJZPOVOVOUFQFOPLUBT5 S L PMTVO5F- öLJODJ EFSFDFEFO GPOLTJZPOMBSO HËSÐOUÐ LÐ- QFOPLUBTOOBQTJTJPMBOSEFóFSJOJO GPOLTJZP- NFTJ CVMVOVSLFO ËODF UBON LÐNFTJOJO IBO- OVOVO UBON BSBMóOEB PMVQ PMNBEó JODFMF- HJ BSBML PMEVóV JODFMFOJS 5BONM PMEVóV BSB- OJS MLUBGPOLTJZPOVOHSBGJóJPMBOQBSBCPM¿J[JMFSFL HËSÐOUÐLÐNFTJOJOV¿OPLUBMBSCFMJSMFOJS r S` [Q R]WFB>JLFO G3Z3 G Y =BY2 +CY+DCJ¿JNJOEF UÐN Þy Þy SFFMTBZMBSEBUBONMJLJODJEFSFEFOCJSGPOLTJ- f(q) f(p) ZPOEBUFQFOPLUBT5 S L PMNBLÐ[FSF yy Or x O r x Þ pq p q k k k x (ËSÐOUÐLÐNFTJ (ËSÐOUÐLÐNFTJ xO O [L G R ] WFZB[L G Q ] k mÞ r S` [Q R]WFB<JLFO B> B< Þy Þy k k (ËSÐOUÐLÐNFTJ (ËSÐOUÐLÐNFTJ [L Þ -Þ L] Op rq xO pr qx f(p) f(q) ÖRNEK 11 (ËSÐOUÐLÐNFTJ (ËSÐOUÐLÐNFTJ G3Z3 G Y =Y2 +Y+ 45 [G Q L]WFZB[G R L] GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJCVMVOV[ 4 > k= 4.4.45 - 2 & k=9 24 4.4 (ÌSÑOUÑLÑNFTJ[ ß ÖRNEK 13 G[- ] Z3 G Y =Y2 -Y+ 5 ÖRNEK 12 GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJCVMVOV[ G3Z3 G Y = -Y2 -Y GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJCVMVOV[ r = - - 6 = 3 j 3 ` (-2, 4]PMEVôVOEBO -1 < 2 r =- -4 =-2 & k = f^ -2 h =-4 + 8 = 4 f ( 3 ) = 32 -+ 5 = -FOLÑÀÑLEFôFSEJS x = -JÀJOG -2) = 21 -2 x =JÀJOG = -3 FOCÑZÑLEFôFSEJS (ÌSÑOUÑLÑNFTJ -ß ] (ÌSÑOUÑLÑNFTJ[-4, 21] < ß mß > 42 [–4, 21]
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 14 %m/*m G[- ] Z3 G Y = -Y2 -Y+ 4 G[Q R] Z3 G Y =BY2 +CY+DQBSBCPMÐOÐO GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOEFLBÀUBNTBZEF- ôFSJWBSES UFQFOPLUBTOOBQTJTJUBONBSBMóOOEõOEB -2 JLFOGPOLTJZPOVOHËSÐOUÐLÐNFTJOJOV¿OPLUB- r = - = - 1 ! [- 3, 2]FOCÑZÑLEFôFSG -1 ) =UJS MBS UBON LÐNFTJOJO V¿ OPLUBMBSOO HËSÐOUÐ- -2 x =JÀJOG = -4 - 4+ 4 = -4 MFSJEJS y FOLÑÀÑLEFôFS-4 y -4 # f ( x ) # 5 UBOFUBNTBZEFôFSJWBSES f(p) f(p) O qr q x p x O rp f(q) f(q) ÖRNEK 15 ÖRNEK 17 G[ Z3 G Y = -Y2 +Y- G[- ] Z3 G Y =Y2 +Y+ 1 GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOEFLJ FO CÑZÑL WF GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJCVMVOV[ FOLÑÀÑLUBNTBZEFôFSMFSJOJOUPQMBNLBÀUS r = - 4 = - 2 g 7 - 1, 3 A 18 2 r = - = 3 ! [2, 5) f ( -1 ) = -2 -6 f ( 3 ) = 22 f ( 3 ) = 17 f(5) = 5 (,= [-2, 22] 5 < f ( x ) # 17 j 6 + 17 = 23 ÖRNEK 16 ÖRNEK 18 G ] Z3 G Y =Y2 +Y- G>- n , 3n H Z3 G Y =Y2 -OY+ 1 2 GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOEFLBÀUBOFUBNTB- ZWBSES GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJ [- N] PMEVôVOB r = - 3 b ^ 2, 5 A HÌSF NLBÀUS 2 r = - - n = n ! d - n, 3n G PMEVôVOEBO UFQF OPLUB- f(2) = 3 22 2 f ( 5 ) = 33 TNJOJNVNOPLUBES 2 < x # 5 j 3 < f ( x ) #PMVQ jUBNTBZ 22 fd n n = n n - + 1 = - 8 jO= 6 2 42 f : [-6, 9] Z3 G Y = x2 - 6x + 1 x = -JÀJOG -6 ) = 73 jN 23 73 43 [–2, 22]
TEST - 16 'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS7 3FFMTBZMBSEBUBONM G Y =Y2 - G Y =Y2 -Y+ 1 GPOLTJZPOVJÀJOG [ -3, - LÑNFTJOEFLBÀUB- GPOLTJZPOVIBOHJYEFôFSJJÀJOFOLÑÀÑLEFôFSJ- OFUBNTBZWBSES OJBMS \" # $ % & \" 1 # 5 $ 6 % 5 & 3 5 12 5 3 2 G3Z3 G Y = -Y2 -Y+ 2 G3Z3 G Y = -Y2 +Y GPOLTJZPOVOVOFOCÑZÑLEFôFSJLBÀUS QBSBCPMÐUBONMBONõUS \" # $ % & #VOB HÌSF G [ LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS \" - > # > $ - % <- > & - > G Y =Y2 + N+ Y+N- 1 GPOLTJZPOV FO LÑÀÑL EFôFSJOJ Y = - BQTJTMJ G3Z3 G Y =Y2 +Y+ 1 OPLUBEB BMEôOB HÌSF G Y GPOLTJZPOVOVO Z– FLTFOJOJLFTUJôJOPLUBOOPSEJOBULBÀUS GPOLTJZPOVOVOBMBCJMFDFôJFOLÑÀÑLEFôFSLBÀ- US \" - # - $ - % & \" 1 # 1 $ 3 % & 3 4 2 4 2 (FSÀFLTBZMBSEBUBONM G[- ] Z3 G Y = -Y2 +Y+ 5 G Y = -BY2 +Y+B2 GPOLTJZPOVOVOFOCÑZÑLWFFOLÑÀÑLEFôFSMF- SJOJOUPQMBNLBÀUS GPOLTJZPOVOVOFOCÑZÑLEFôFSJG B PMEVôVOB HÌSF G B LBÀUS \" # $ % & \" # $ % & $ % \" B 44 & & $ B
'POLTJZPOMBSMB÷MHJMJ6ZHVMBNBMBS7 TEST - 17 G[ - ] Z3 G Y =Y2 -Y+ 5 #VÑSÑOÑ Y- 5-ZFBMQ Y2 - 7x + 5-ZF GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOEFLJ FMFNBO- TBUBOCJSUÑDDBSOL»SFOB[LBÀ5-EJS MBSEBOLBÀUBOFTJUBNTBZES \" # $ % & \" # $ % & Y`3PMNBLÐ[FSF CJS\"LÐNFTJOJOFMFNBOMBS G3Z3 G Y =Y2 -LY+L Y- 2 + Y+ 2 GPOLTJZPOVOVOFOLÑÀÑLEFôFSJPMEVôVOBHÌ- SF LOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJPMBCJ- MJS JGBEFTJJMFIFTBQMBONBLUBES \" # $ % & #VOBHÌSF\"LÑNFTJOJOFOLÑÀÑLFMFNBOLBÀ- US \" # $ % & YHFSÀFLTBZPMNBLÑ[FSF Y`3PMNBLÐ[FSF Y- 2 + Y+ 2 + 50 _ 2 - x i_ x + 3 i JGBEFTJOJOBMBCJMFDFôJFOLÑÀÑLEFôFSLBÀUS JGBEFTJOJO BMBCJMFDFôJ FO LÑÀÑL QP[JUJG EFôFS LBÀUS \" 1 # 1 25 5 $ % & \" # $ % & B C`3PMNBLÐ[FSF Z=Y2 -Y+ 8 a+1= b-1 QBSBCPMÑ Ñ[FSJOEFLJ CJS OPLUBOO LPPSEJOBUMBS 2 UPQMBNFOB[LBÀUS FöJUMJôJOJTBôMBZBOBWFCTBZMBSOOÀBSQNFO B[LBÀUS \" - 9 # - 3 $ - 1 \" - # - $ - % & 8 4 2 % - 3 & - 1 8 4 $ B % \" 45 $ % % &
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr '0/,4÷:0/-\"3-\"÷-(÷-÷6:(6-\".\"-\"37* ÷MJöLJMJ,B[BONMBS 11.3.2.1 : öLJODJEFSFDFEFOCJSEFóJõLFOMJGPOLTJZPOVOHSBGJóJOJ¿J[FSFLZPSVNMBS 1BSBCPMÑO%FOLMFNJOJ#VMNB ÖRNEK 2 %m/*m ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMWF- SJMNJõUJS 1BSBCPMÐOUFQFOPLUBTOOLPPSEJOBUMBS5 S L WFSJMJSTF y Z=B Y-S 2 +L 5 CBóOUTOEBOZBSBSMBOMS #VCBóOUEBLJBLBUTBZTOCVMNBLJ¿JO UFQF 1 x OPLUBTEõOEBCJSOPLUBEBIBWFSJMNJõPMNBM- –2 O ES y y = a(x – r)2 + k y #VOBHÌSF CVQBSBCPMÑOZFLTFOJOJLFTUJôJOPLUBOO y = a(x – r)2 PSEJOBULBÀUS k x Or x T(-2, 5) j y =B Y+ 2)2 +WF OPLUBTEFOLMFN- Or EFZFSJOFZB[MSTB =B + 2)2 + 5 j a = - 5 y = ax2 + k y y y = ax2 9 y = - 5 ^ x + 2 h2 + 5 9 5 25 x =JÀJO y = - ·4 + 5 = 99 k x x O O ÖRNEK 3 y ÖRNEK 1 AO x Z=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMõFLJMEFWF- B SJMNJõUJS y 2 2 x f_ x i = - 1 _ x + a i2 + b - 2GPOLTJZPOVOVOHSBGJóJPMBO O 8 –1 QBSBCPMWFSJMNJõUJS 1BSBCPM\"OPLUBTOEBYFLTFOJOFUF- #VOBHÌSF G - EFôFSJLBÀUS óFUWF3 | OA | = 2 |0#| PMEVôVOBHÌSF a +CUPQMB- T ( 2, -1 ) j y =B Y- 2)2 -WF OPLUBTZFSJ- OFZB[MSTB NLBÀUS 2 =B - 2)2 - 1 j 4a = 3 j a = 3 %FOLMFNF HÌSF UFQF OPLUBT -B C - EJS /PLUB Y 4 f^ x h = 3 ^ x - 2 h2 - 1 FLTFOJÑ[FSJOEFPMEVôVOEBO C- 2=jC=EJS 4 |0\"| =BCSj |OB| = 3a CS Bd 0, - 3a n f^ - 2 h = 3 ·^ - 4 h2 - 1 = 11 22 4 y = - 1 ·^ x + a h2 j 3a =- 1 2 jB EJS - 8 28 ·a a +C= 14 11 46 25 14 9
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 .0%·- 11. SINIF ÖRNEK 4 ÖRNEK 5 Z = G Y WF Z = H Y GPOLTJZPOMBSOOO HSBGJLMFSJ PMBO ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMWF- QBSBCPMMFSZFLTFOJÐ[FSJOEFCJS,OPLUBTOEBLFTJõNFL- SJMNJõUJS UFEJS y y 3 y = g(x) y = f(x) K –2 6 O x x –1 O 2 Z=G Y QBSBCPMÐYFLTFOJOFUFóFUUJS #VOBHÌSF CVQBSBCPMÑOEFOLMFNJOJCVMVOV[ G +H - = PMEVôVOBHÌSF G LBÀUS f(x) =B Y+ Y- EFOLMFNJOEF OPLUBTZF- SJOFZB[MSTB 3 =B -6) j a = - 1 f(x) =B Y-2)2 j x =JÀJOZ=BPMVS, B 1 4 1 H Y =N Y+ 1)2+LEFOLMFNJOEF B ZFSJOFZB[MSTB f^ x h =- ·^ x + 2 h.^ x - 6 h =- 2 44 x +x+3 4a =N+LCVMVOVS f(6) +H m =B2 +N+ k =B=j a = 1 2 1 1 f^ x h = ·^ x - 2 h2 & f^ 8 h = 2 = 18 22 ·6 ÖRNEK 6 ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMWF- SJMNJõUJS y 2 3 x O1 %m/*m #VOBHÌSF CVQBSBCPMÑOEFOLMFNJOJCVMVOV[ 1BSBCPMÐOYFLTFOJOJLFTUJóJOPLUBMBSOBQTJTMF- SJY1WFY2CJMJOEJóJOEF YFLTFOJOJLFTUJôJOPLUBMBSY1 =WFY2JMFCVJLJOPL- UBOOPSUBOPLUBTUÑS Z=B Y-Y1 Y-Y2 CBóOUTOEBOZBSBSMBOMBSBLEFOLMFNCVMVOVS 1+x #VCBóOUEBLJBLBUTBZTOCVMNBLJ¿JO YFL- 2 TFOJOJLFTUJóJOPLUBMBSEõOEBCJSOPLUBOOEB- IB CJMJONFTJ HFSFLJS #V OPLUBOO LPPSEJOBUMB- =3 & x =5 SEFOLMFNEFZFSJOFZB[MBSBLBLBUTBZTCVMV- 22 OVS y =B Y- Y- EFOLMFNJOEF OPLUBTZFSJ- OFZB[MSTB 2 f^ x h = 2 ·^ x - 1 h.^ x - 5 h 2 =B - -5) j a = 55 18 47 - 1 2 + x + 3 2 x 2x 12x - + 2 4 55
11. SINIF .0%·- '0/,4÷:0/-\"3%\"6:(6-\".\"-\"3 www.aydinyayinlari.com.tr ÖRNEK 7 ÖRNEK 9 y ôFLJMEFLJG Y =BY2 +CY+DGPOLTJZPOVOVOHSBGJóJPMBO 6 QBSBCPMÐOTJNFUSJFLTFOJY=EPóSVTVEVS y Ox C B x k –2 2n 6 2n f_ x i = 1 x2 + bx + c 9 AO –2 n n GPOLTJZPOVOVOHSBGJóJPMBOQBSBCPMõFLJMEFWFSJMNJõUJS | | | | | |6 OA = 2 0# = 3 OC PMEVôVOB HÌSF a + C + D #VOBHÌSF L+DUPQMBNLBÀUS UPQMBNLBÀUS OPLUBT EFOLMFNEF ZFSJOF ZB[MSTB D = CVMV- OVS a +C+D=G EJS f(x) =B Y-L Y+ 2) = 1 x2 +CY+ 6 f ( x ) =B Y+ 2 ) (x - 6) 9 B Y2 + (2 - k Y- 2k) = 1 x2 +CY+ 6 j a = 1 4 =B + - 6) 1 99 4PMWFTBôUBSBGOTBCJUMFSJFöJUMFOJSTF - =a 3 2k - = 6 & k = - 27 , k +D= -27 + 6 = -21 f^ x h =- 1 ·^ x + 2 h.^ x - 6 h 3 9 f^ 1 h = - 1 ·3.^ - 5 h j f ( 1 ) = 5 3 ÖRNEK 8 ôFLJMEF G Y = -Y2 + CY - GPOLTJZPOVOVO HSBGJóJ PMBOQBSBCPMWFSJMNJõUJS y A B %m/*m O x #JSJ Z FLTFOJ Ð[FSJOEF PMNBL Ð[FSF QBSBCP- | | | |\"# = 2 OA PMEVôVOBHÌSF CLBÀUS MÐO Ð[FSJOEFLJ п OPLUB CJMJOJZPSTB QBSBCP- MÐOEFOLMFNJOJCVMNBLJ¿JOZFLTFOJÐ[FSJOEFLJ |0\"| =Oj |OB| =OPMVS\" O WF# O OPLUB Z=BY2 +CY+D -2x+2 +CY- 24 ==-E2F4OL=MF1N2JO=JO3nL2ÌL&MFnSJ=Y12=!ORW+F EFOLMFNJOEF ZFSJOF ZB[MS WF TBCJU UFSJN D .x -2 CFMJSMFOJS x2=OEJS x %BIB TPOSB EJóFS JLJ OPLUB EFOLMFNEF ZFSMF- 2 SJOF ZB[MBSBL JLJ CJMJONFZFOMJ JLJ EFOLMFNEFO 1 PMVõBOCJSEFOLMFNTJTUFNJFMEFFEJMJS#VTJTUF- NJO¿Ë[ÐNÐZMFBWFCLBUTBZMBSCFMJSMFOJS bb b x + x = - = EJS = 4n & b = 16 1 2 -2 2 2 –21 16 48 5
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