Yuav Ua Li Cas Kos Daim Duab Square: 10 Kauj Ruam (nrog Duab)

2024 Tus sau: Jason Gerald | [email protected]. Kawg hloov kho: 2024-02-01 14:13

Thaum sawv cev rau duab, qhov sib npaug ntawm plaub yog ntawm daim ntawv taus² bx + c los yog a (x - h)² + k tsim daim ntawv U lossis rov qab U nkhaus hu ua parabola. Kev teeb duab plaub npaug sib npaug tab tom nrhiav rau qhov chaw siab, qhia, thiab feem ntau x thiab y kev sib tshuam. Hauv cov xwm txheej ntawm kev sib npaug plaub yooj yim yooj yim, nkag mus rau txheej x qhov tseem ceeb thiab phiaj xwm nkhaus raws cov ntsiab lus tshwm sim yuav txaus. Saib Kauj Ruam 1 hauv qab no kom pib.

Kauj ruam

Kauj Ruam 1. Txiav txim siab daim ntawv ntawm cov zauv sib npaug uas koj muaj

Qhov sib npaug ntawm plaub npaug tuaj yeem sau ua peb hom sib txawv: daim ntawv dav, daim ntawv vertex, thiab daim ntawv plaub fab. Koj tuaj yeem siv txhua daim ntawv los teeb duab ua zauv sib npaug; txheej txheem ntawm kev piav qhia txhua daim duab sib txawv me ntsis. Yog tias koj ua homework, koj feem ntau yuav tau txais lus nug hauv ib ntawm ob daim ntawv no - ua lwm yam, koj yuav tsis tuaj yeem xaiv, yog li nws yog qhov zoo tshaj kom nkag siab ob qho no. Ob daim ntawv ntawm kev ua zauv plaub fab yog:

• Daim ntawv dav dav.

  Hauv daim ntawv no, kev ua zauv plaub fab tau sau ua: f (x) = ax² + bx + c qhov twg a, b, thiab c yog tus lej tiag thiab a tsis yog xoom.

  Piv txwv li, ob qhov kev ua zauv plaub fab ntawm daim ntawv dav yog f (x) = x² + 2x + 1 and f (x) = 9x² + 10x -8.

• Ncov zoo

  Hauv daim ntawv no, kev ua zauv plaub fab tau sau ua: f (x) = a (x - h)² + k qhov twg a, h, thiab k yog tus lej tiag thiab a tsis yog xoom. Nws yog hu ua daim ntawv vertex vim tias h thiab k yuav muab lub vertex (nruab nrab) ntawm koj tus parabola ntawm qhov chaw (h, k).

  Ob qhov vertex form equations yog f (x) = 9 (x - 4)² + 18 thiab -3 (x - 5)² + 1

• Txhawm rau teeb tsa txhua hom kev sib npaug, peb yuav tsum xub pom lub vertex ntawm parabola, uas yog qhov nruab nrab (h, k) thaum kawg ntawm txoj kab nkhaus. Kev ua haujlwm ntawm cov peaks hauv daim ntawv dav dav yog xam raws li: h = -b/2a thiab k = f (h), thaum nyob rau hauv daim ntawv siab, h thiab k nyob hauv qhov sib npaug.

Kauj Ruam 2. Txheeb xyuas koj qhov kev hloov pauv

Txhawm rau daws qhov teeb meem plaub npaug, qhov sib txawv a, b, thiab c (lossis a, h, thiab k) feem ntau yuav tsum tau txhais. Qhov teeb meem algebra zoo ib yam yuav muab ib qho zauv sib npaug nrog cov hloov pauv uas muaj, feem ntau hauv daim ntawv dav, tab sis qee zaum hauv daim ntawv siab.

• Piv txwv li, rau qhov sib npaug ntawm daim ntawv f (x) = 2x² + 16x + 39, peb muaj a = 2, b = 16, thiab c = 39.
• Rau qhov siab tshaj daim ntawv zauv f (x) = 4 (x - 5)² + 12, peb muaj a = 4, h = 5, thiab k = 12.

Kauj Ruam 3. Xam h

Hauv qhov vertex form equation, koj h tus nqi twb tau muab lawm, tab sis nyob rau hauv daim ntawv dav dav ua zauv, tus nqi h yuav tsum tau muab xam. Nco ntsoov tias, rau kev sib npaug ntawm daim ntawv dav dav, h = -b/2a.

• Hauv peb daim ntawv piv txwv piv txwv (f (x) = 2x² + 16x + 39), h = -b/2a = -16/2 (2). Tom qab daws qhov teeb meem, peb pom tias h = - 4.
• Hauv peb daim ntawv piv txwv piv txwv (f (x) = 4 (x - 5)² + 12), peb paub tias h = 5 yam tsis tau ua lej.

Kauj Ruam 4. Xam k

Zoo li h, k twb paub lawm hauv kab zauv ntawm daim ntawv ncov. Rau qhov sib npaug ntawm daim ntawv dav dav, nco ntsoov tias k = f (h). Hauv lwm lo lus, koj tuaj yeem pom k los ntawm kev hloov tag nrho x qhov tseem ceeb hauv koj qhov kev ua zauv nrog h qhov tseem ceeb uas koj nyuam qhuav pom.

• Peb twb tau txiav txim siab hauv peb qhov piv txwv piv txwv uas h = -4. Txhawm rau nrhiav k, peb daws peb qhov kev ua zauv los ntawm ntsaws rau hauv peb tus nqi h hauv qhov x:

  • k = 2 (-4)² + 16(-4) + 39.
  • k = 2 (16) - 64 + 39.

  • k = 32 - 64 + 39 =

    Kauj ruam 7.

• Hauv peb qhov ua piv txwv piv txwv, dua, peb paub tus nqi ntawm k (uas yog 12) yam tsis tas yuav ua lej.

Kauj Ruam 5. Kos koj lub ncov

Lub vertex ntawm koj tus parabola yog lub ntsiab lus (h, k)-h sawv cev rau x-koom tes, thaum k sawv cev rau y-koom tes. Lub vertex yog qhov nruab nrab ntawm koj lub parabola - tog twg los hauv qab ntawm U lossis nyob rau sab saum toj ntawm qhov rov qab U. Paub txog qhov nce siab yog ib feem tseem ceeb ntawm kev kos ib qho parabola meej - feem ntau, hauv tsev kawm ntawv, txiav txim siab txog qhov kawg yog ib feem los saib hauv nqe lus nug.

• Hauv peb qhov piv txwv piv txwv, peb qhov siab tshaj yog (-4, 7). Yog li, peb cov parabola yuav ua tiav 4 cov kauj ruam mus rau sab laug los ntawm 0 thiab 7 cov kauj ruam saum toj no (0, 0). Peb yuav tsum piav qhia lub ntsiab lus no hauv peb daim duab, ua kom ntseeg tau tias ua kom pom qhov chaw nyob.
• Hauv peb daim ntawv piv txwv piv txwv, peb lub vertex yog (5, 12). Peb yuav tsum kos tus lej 5 kauj ruam mus rau sab xis thiab 12 kauj ruam saum toj no (0, 0).

Kauj Ruam 6. Kos lub axis ntawm parabola (xaiv tau)

Lub axis ntawm symmetry ntawm parabola yog kab uas hla nws qhov chaw, faib nws raws nraim hauv nruab nrab. Ntawm qhov axis no, sab laug ntawm parabola yuav cuam tshuam rau sab xis. Rau quadratic equations hauv daim ntawv ax² + bx + c lossis a (x - h)² + k, txoj kab sib luag ntawm txoj kab sib dhos yog kab uas yog sib npaug rau y-axis (hauv lwm lo lus, raws nraim) thiab dhau los ntawm lub vertex.

Hauv qhov xwm txheej ntawm peb daim ntawv piv txwv, lub axis yog kab sib dhos rau y-axis thiab hla dhau qhov taw tes (-4, 7). Txawm hais tias nws tsis yog ib feem ntawm parabola, qee qhov cim cov kab no ntawm koj daim duab yuav kawg pab koj pom cov duab sib npaug ntawm parabola txoj kab nkhaus

Kauj Ruam 7. Nrhiav cov lus qhia ntawm qhib parabola

Tom qab paub lub ncov thiab axis ntawm parabola, tom ntej no peb yuav tsum paub yog tias parabola qhib nce lossis nqis. Hmoov zoo, qhov no yooj yim. Yog tias tus nqi ntawm qhov zoo, tus parabola yuav qhib siab dua, qhov twg yog tus nqi ntawm qhov tsis zoo, parabola yuav qhib qis (piv txwv li parabola yuav rov qab).

• Rau peb daim ntawv piv txwv piv txwv (f (x) = 2x² + 16x + 39), peb paub tias peb muaj parabola uas qhib vim tias, hauv peb qhov kev ua zauv, a = 2 (zoo).
• Rau peb daim ntawv piv txwv piv txwv (f (x) = 4 (x - 5)² + 12), peb paub tias peb kuj muaj parabola uas qhib vim tias a = 4 (zoo).

Kauj Ruam 8. Yog xav tau, nrhiav thiab kos x-cuam tshuam

Feem ntau, hauv tsev kawm ntawv, koj yuav raug nug kom pom x-cuam tshuam hauv parabola (uas yog ib lossis ob lub ntsiab lus qhov twg parabola ua tau raws li x-axis). Txawm hais tias koj tsis pom ib qho, ob lub ntsiab lus no tseem ceeb heev rau kev kos tus lej parabola meej. Txawm li cas los xij, tsis yog txhua tus parabolas muaj x-cuam tshuam. Yog tias koj lub parabola muaj lub qhov ncauj uas qhib thiab nws lub vertex yog siab dua x-axis lossis yog nws qhib qis thiab nws cov vertex qis dua x-axis, parabola yuav tsis muaj x-cuam tshuam. Txwv tsis pub, daws koj qhov x-cuam tshuam hauv ib qho ntawm cov hauv qab no:

• Tsuas yog ua f (x) = 0 thiab daws cov kab zauv. Txoj hauv kev no tuaj yeem siv rau kev ua zauv plaub fab yooj yim, tshwj xeeb tshaj yog nyob rau hauv daim ntawv siab, tab sis yuav nyuaj rau kev sib npaug nyuaj. Saib hauv qab no rau qhov piv txwv

  • f (x) = 4 (x - 12)² - 4
  • 0 = 4x (x - 12)² - 4
  • 4 = 4x (x - 12)²
  • 1 = (x -12)²
  • Hauv paus (1) = (x - 12)
  • +/- 1 = x -12. x = 11 thiab 13 yog x-cuam tshuam hauv parabola.

• Factor koj qhov sib npaug. Qee qhov sib npaug hauv daim ntawv ax² + bx + c tuaj yeem yooj yim ua rau hauv daim ntawv (dx + e) (fx + g), qhov twg dx × fx = ax², (dx × g + fx × e) = bx, thiab e × g = c. Hauv qhov no, koj qhov x-cuam tshuam yog x qhov tseem ceeb uas yuav ua rau ib lo lus twg hauv kab lus = 0. Piv txwv:

  • x kev² + 2x + 1 ib
  • = (x + 1) (x + 1)
  • Hauv qhov no, koj tsuas yog x -cuam tshuam yog -1 vim tias ua x sib npaug -1 yuav ua rau txhua lub ntsiab lus hauv kab lus sib npaug 0.

• Siv cov zauv plaub fab. Yog tias koj tsis tuaj yeem yooj yim daws koj qhov x-cuam tshuam lossis cuam tshuam koj qhov kev ua zauv, siv qhov kev ua zauv tshwj xeeb hu ua cov lej sib npaug uas tau tsim los rau lub hom phiaj no. Yog tias nws tseem tsis tau daws, hloov koj qhov sib npaug rau daim ntawv ax² + bx + c, tom qab ntawd nkag mus rau a, b, thiab c rau hauv cov qauv x = (-b +/- sqrt (b)² 4a))/2a. Nco ntsoov tias txoj hauv kev no feem ntau muab koj ob lo lus teb rau tus nqi x, uas yog OK-nws tsuas yog txhais tau tias koj tus parabola muaj ob x-cuam tshuam. Saib hauv qab no rau qhov piv txwv:

  • -5x ib² + 1x + 10 muab tso rau hauv cov lej piv txwv zoo li no:
  • x = (-1 +/- Hauv paus (1.)² - 4(-5)(10)))/2(-5)
  • x = (-1 +/- Hauv paus (1 + 200))/-10
  • x = (-1 +/- Hauv paus (201))/-10
  • x = (-1 +/- 14, 18)/-10
  • x = (13, 18/-10) thiab (-15, 18/-10). X-cuam tshuam hauv parabola yog x = - 1, 318 thiab 1, 518
  • Peb qhov piv txwv yav dhau los ntawm daim ntawv dav dav, 2x² +16x+39 tau muab tso rau hauv cov lej sib npaug raws li hauv qab no:
  • x = (-16 +/- Hauv paus (16² - 4(2)(39)))/2(2)
  • x = (-16 +/- Hauv paus (256- 312))/4
  • x = (-16 +/- Hauv paus (-56)/-10
  • Txij li nws tsis yooj yim sua kom nrhiav tau cov hauv paus cag ntawm tus lej tsis zoo, peb paub tias qhov no parabola tsis muaj x-cuam tshuam.

Kauj Ruam 9. Yog xav tau, nrhiav thiab kos y-cuam tshuam

Thaum nws feem ntau tsis tsim nyog nrhiav y-cuam tshuam hauv qhov sib npaug (qhov chaw uas tus parabola hla dhau y-axis), tej zaum koj yuav tsum tau nrhiav nws, tshwj xeeb tshaj yog tias koj nyob hauv tsev kawm ntawv. Cov txheej txheem yog yooj yim yooj yim-tsuas yog ua x = 0, tom qab ntawd daws koj qhov kev ua zauv rau f (x) lossis y, uas muab tus nqi ntawm y qhov twg koj parabola hla dhau y-axis. Tsis zoo li x-cuam tshuam, ib txwm parabola tuaj yeem muaj tsuas yog ib qho y-cuam tshuam. Nco tseg-rau qhov sib npaug ntawm daim ntawv dav dav, y-cuam tshuam yog ntawm y = c.

• Piv txwv li, peb paub tias peb qhov kev ua zauv sib npaug yog 2x² + 16x + 39 muaj y-cuam tshuam ntawm y = 39, tab sis nws kuj tseem tuaj yeem pom nyob hauv cov hauv qab no:

  • f (x) = 2x² +4x+39
  • f (x) = 2 (0)² + 16(0) + 39

  • f (x) = 39. Y-cuam tshuam ntawm parabola yog ntawm y = 39 hli

    Raws li tau hais los saud, y-cuam tshuam ntawm y = c.

• Daim ntawv ntawm peb qhov sib npaug vertex yog 4 (x - 5)² + 12 muaj y-cuam tshuam uas tuaj yeem pom hauv qab no:

  • f (x) = 4 (x - 5)² + 12
  • f (x) = 4 (0 - 5)² + 12
  • f (x) = 4 (-5)² + 12
  • f (x) = 4 (25) + 12

  • f (x) = 112. Y-cuam tshuam ntawm parabola yog ntawm y = 112 hli

Kauj Ruam 10. Yog xav tau, kos cov ntsiab lus ntxiv, tom qab ntawv kos duab

Tam sim no koj muaj qhov vertex, kev taw qhia, x-cuam tshuam, thiab muaj peev xwm, y-cuam tshuam hauv koj qhov kev ua zauv. Nyob rau theem no, koj tuaj yeem sim kos koj lub parabola siv cov ntsiab lus uas koj muaj raws li phau ntawv qhia, lossis nrhiav lwm cov ntsiab lus los sau rau hauv koj daim ntawv parabola kom cov nkhaus koj kos tau meej dua. Txoj hauv kev yooj yim tshaj plaws los ua qhov no tsuas yog nkag qee qhov x qhov tseem ceeb hauv ib sab ntawm koj lub vertex, tom qab ntawv cov ntsiab lus no siv y-qhov koj tau txais. Feem ntau, cov kws qhia ntawv nug koj kom saib ob peb lub ntsiab lus ua ntej kos koj lub parabola.

• Cia peb tshuaj xyuas qhov sib npaug x² + 2x + 1. Peb twb paub lawm tias x -cuam tshuam tsuas yog ntawm x = -1. Txij li txoj kab nkhaus tsuas yog kov x-cuam tshuam ntawm ib kis, peb tuaj yeem xaus tias lub vertex yog nws x-cuam tshuam, uas txhais tau tias lub vertex yog (-1, 0). Peb ua tau zoo tsuas yog muaj ib lub ntsiab lus rau qhov parabola no - tsis txaus los kos cov parabola zoo. Cia peb saib qee qhov ntsiab lus kom ntseeg tau tias peb kos duab kom ntxaws.

  • Cia peb nrhiav y qhov tseem ceeb rau x qhov tseem ceeb hauv qab no: 0, 1, -2, thiab -3.
  • Rau 0: f (x) = (0)² + 2 (0) + 1 = 1. Peb lub ntsiab lus yog (0, 1).

  • Rau 1: f (x) = (1)² + 2 (1) + 1 = 4. Peb lub ntsiab lus yog (1, 4).

  • Rau -2: f (x) = (-2)² + 2 (-2) + 1 = 1. Peb lub ntsiab lus yog (-2, 1).

  • Rau -3: f (x) = (-3)² + 2 (-3) + 1 = 4. Peb lub ntsiab lus yog (-3, 4).

  • Kos cov ntsiab lus no ntawm daim duab thiab kos koj txoj kab nkhaus U. Nco ntsoov tias parabola zoo sib xws - thaum koj cov ntsiab lus ntawm ib sab ntawm parabola yog cov lej, koj feem ntau tuaj yeem txo txoj haujlwm ntawm tsuas yog xav txog qhov muab rau ntawm parabola txoj kab sib luag kom pom tib lub ntsiab lus ntawm lwm sab ntawm parabola.

Lub tswv yim

• Hloov tus lej lossis siv cov lej faib raws li koj tus kws qhia lej algebra qhov kev thov. Qhov no yuav pab koj ua kom zoo dua kab zauv sib npaug.
• Nco ntsoov tias hauv f (x) = ax² + bx + c, yog b lossis c sib npaug rau xoom, cov lej no yuav ploj mus. Ej.: 12 x² + 0x + 6 ua 12x² +6 vim 0x yog 0.