Yuav Ua Li Cas Txheeb Ib Leeg Plaub Ntug: 10 Kauj Ruam (nrog Duab)

2024 Tus sau: Robert Blare | [email protected]. Kawg hloov kho: 2024-02-02 02:07

Thaum graphed, sib npaug ntawm plaub fab ntawm daim ntawv taus² bx + c los yog a (x - h)² + k muab tus U-puab du lossis rov qab U-puab nkhaus hu ua parabola. Kev teeb duab plaub fab sib npaug yog qhov teeb meem ntawm kev nrhiav nws qhov chaw siab, qhia, thiab, feem ntau, nws x thiab y cuam tshuam. Hauv cov xwm txheej ntawm qhov sib npaug ua piv txwv yooj yim, nws kuj tseem yuav txaus los ntsaws rau hauv ntau qhov x qhov tseem ceeb thiab npaj ib txoj kab nkhaus raws li cov ntsiab lus tshwm sim. Saib Kauj Ruam 1 hauv qab no kom pib.

Cov kauj ruam

Kauj Ruam 1. Txiav txim seb hom twg ntawm cov zauv sib npaug uas koj muaj

Qhov sib npaug ntawm plaub npaug tuaj yeem sau ua peb daim ntawv sib txawv: daim ntawv tus qauv, daim ntawv vertex, thiab daim ntawv plaub fab. Koj tuaj yeem siv ob daim ntawv los teeb duab ua zauv sib npaug; txheej txheem rau teeb duab txhua qhov sib txawv me ntsis. Yog tias koj tab tom ua teeb meem hauv tsev, feem ntau koj yuav tau txais cov teeb meem hauv ib ntawm ob daim ntawv no - hauv lwm lo lus, 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 txheem.

  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 sib npaug rau xoom.

  Piv txwv li, ob daim qauv sib npaug sib npaug yog f (x) = x² + 2x + 1 and f (x) = 9x² + 10x -8.

• Vertex daim ntawv.

  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 sib npaug xoom. Vertex daim ntawv muaj npe vim tias h thiab k ncaj qha muab koj lub vertex (qhov chaw 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 ib qho ntawm cov kev sib npaug no, peb yuav tsum xub pom lub vertex ntawm parabola, uas yog lub hauv paus taw tes (h, k) ntawm qhov "ntxeev" ntawm txoj kab nkhaus. Kev tswj hwm ntawm lub vertex hauv daim ntawv tus qauv tau muab los ntawm: h = -b/2a thiab k = f (h), thaum nyob rau hauv daim ntawv vertex, h thiab k tau teev nyob hauv kab zauv.

Kauj Ruam 2. Txheeb xyuas koj qhov kev hloov pauv

Yuav kom daws tau 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 nruab nrab yuav muab rau koj ua ib feem plaub npaug nrog qhov hloov pauv tau ua tiav, feem ntau hauv daim qauv, tab sis qee zaum hauv daim ntawv vertex.

• Piv txwv li, rau tus qauv kev ua zauv f (x) = 2x² + 16x + 39, peb muaj a = 2, b = 16, thiab c = 39.
• Rau qhov vertex form equation f (x) = 4 (x - 5)² + 12, peb muaj a = 4, h = 5, thiab k = 12.

Kauj Ruam 3. Xam h

Hauv vertex daim ntawv sib npaug, koj tus nqi rau h twb tau muab lawm, tab sis hauv tus qauv daim ntawv sib npaug, nws yuav tsum tau suav. Nco ntsoov tias, rau tus qauv sib npaug sib npaug, h = -b/2a.

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

Kauj Ruam 4. Xam k

Raws li nrog h, k twb paub lawm hauv vertex form equations. Txog tus qauv sib npaug, nco ntsoov tias k = f (h). Hauv lwm lo lus, koj tuaj yeem pom k los hloov txhua qhov piv txwv ntawm x hauv koj qhov kev ua zauv nrog tus nqi uas koj nyuam qhuav pom rau h.

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

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

  • k = 32 - 64 + 39 =

    Kauj ruam 7.

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

Kauj Ruam 5. Npaj koj lub vertex

Lub vertex ntawm koj parabola yuav yog lub ntsiab lus (h, k) - h qhia qhov x ua haujlwm, thaum k qhia qhov y ua haujlwm. Lub vertex yog lub hauv paus tseem ceeb hauv koj lub parabola - ib qho hauv qab ntawm "U" lossis sab saum toj kawg nkaus ntawm "U." Paub txog lub vertex yog ib feem tseem ceeb ntawm kev teeb tsa lub parabola raug - feem ntau, hauv kev ua haujlwm hauv tsev kawm ntawv, qhia txog lub vertex yuav yog ib feem ntawm cov lus nug.

• Hauv peb tus qauv ua piv txwv, peb lub vertex yuav nyob ntawm (-4, 7). Yog li, peb lub parabola yuav nce 4 qhov chaw mus rau sab laug ntawm 0 thiab 7 qhov chaw saum toj no (0, 0). Peb yuav tsum npaj cov ntsiab lus no ntawm peb daim duab, kom ntseeg tau tias muaj daim ntawv lo koom ua ke.
• Hauv peb daim ntawv piv txwv piv txwv, peb lub vertex nyob ntawm (5, 12). Peb yuav tsum npaj cov ntsiab lus 5 qhov chaw rau sab xis thiab 12 qhov chaw saum toj no (0, 0).

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

Parabola lub axis ntawm symmetry yog kab khiav los ntawm nws nruab nrab uas faib nws zoo kawg nkaus hauv ib nrab. Hla hla txoj kab no, sab laug ntawm parabola yuav tsom iav sab xis. Rau quadratics ntawm daim ntawv ax² + bx + c lossis a (x - h)² + k, lub axis yog kab sib dhos rau y-axis (hauv lwm lo lus, zoo kawg nkaus ntsug) thiab hla dhau lub vertex.

Hauv qhov xwm txheej ntawm peb tus qauv ua 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 nws tus kheej, maj mam kos cov kab no ntawm koj daim duab tuaj yeem pab koj pom qhov yuav ua li cas parabola nkhaus sib npaug

Kauj Ruam 7. Nrhiav cov lus qhia qhib

Tom qab tau txiav txim siab tawm ntawm qhov chaw thiab axis ntawm parabola, tom ntej no peb yuav tsum paub seb lub parabola qhib siab dua lossis qis dua. Luckily, qhov no yog ib qho yooj yim. Yog tias "a" yog qhov zoo, parabola yuav qhib siab dua, thaum yog "a" tsis zoo, parabola yuav qhib qis (piv txwv li, nws yuav tig rov qab.)

• Rau peb tus qauv ua piv txwv (f (x) = 2x² + 16x + 39), peb paub tias peb muaj parabola qhib siab dua 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 peb kuj muaj parabola qhib siab dua vim tias a = 4 (zoo).

Kauj Ruam 8. Yog tias tsim nyog, nrhiav thiab phiaj xwm x cuam tshuam

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

• Cias teeb f (x) = 0 thiab daws cov kab zauv. Txoj hauv kev no tuaj yeem ua haujlwm rau cov lej sib npaug sib npaug yooj yim, tshwj xeeb tshaj yog nyob rau hauv daim ntawv vertex, tab sis yuav ua pov thawj nyuaj rau qhov nyuaj dua. 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
  • SqRt (1) = (x - 12)
  • +/- 1 = x -12. x = 11 thiab 13 yog parabola's x-cuam tshuam.

• Factor koj qhov sib npaug. Qee qhov sib npaug hauv txoj kab² + bx + c daim ntawv 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 kev cuam tshuam x yog qhov tseem ceeb rau x uas ua ob lo lus 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 teeb tsa x sib npaug rau -1 yuav ua rau ib qho ntawm cov ntsiab lus sib txawv hauv kab lus sib npaug 0.

• Siv cov zauv plaub fab. Yog tias koj tsis tuaj yeem daws tau yooj yim rau koj 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 tsim los rau lub hom phiaj no. Yog tias nws tseem tsis tau, tau txais koj qhov kev ua zauv rau hauv daim ntawv ax² + bx + c, tom qab ntawd ntsaws a, b, thiab c rau hauv cov qauv x = (-b +/- SqRt (b² 4a))/2a. Nco ntsoov tias qhov no feem ntau muab rau koj ob lo lus teb rau x, uas yog OK - qhov no tsuas yog txhais tau tias koj parabola muaj ob x cuam tshuam. Saib hauv qab no rau qhov piv txwv:

  • -5x ib² + 1x + 10 tau txuas rau hauv cov lej sib npaug raws li hauv qab no:
  • x = (-1 +/- SqRt (1² - 4(-5)(10)))/2(-5)
  • x = (-1 +/- SqRt (1 + 200))/-10
  • x = (-1 +/- SqRt (201))/-10
  • x = (-1 +/- 14.18)/-10
  • x = (13.18/-10) thiab (-15.18/-10). Parabola's x cuam tshuam yog kwv yees li x = - 1.318 thiab 1.518
  • Peb tus qauv ua piv txwv yav dhau los, 2x² + 16x + 39 tau txuas rau hauv cov lej sib npaug raws li hauv qab no:
  • x = (-16 +/- SqRt (16² - 4(2)(39)))/2(2)
  • x = (-16 +/- SqRt (256- 312))/4
  • x = (-16 +/- SqRt (-56)/-10
  • Vim tias nrhiav cov hauv paus hauv paus ntawm tus lej tsis zoo yog qhov ua tsis tau, peb paub qhov ntawd tsis muaj kev cuam tshuam x muaj nyob rau qhov tshwj xeeb parabola.

Kauj Ruam 9. Yog tias tsim nyog, nrhiav thiab npaj lub y cuam tshuam

Txawm hais tias nws feem ntau tsis tsim nyog los nrhiav qhov sib npaug y cuam tshuam (lub ntsiab lus uas tus parabola dhau los ntawm y axis), tej zaum koj yuav xav tau, tshwj xeeb tshaj yog tias koj nyob hauv tsev kawm ntawv. Cov txheej txheem no yooj yim yooj yim - tsuas yog teeb tsa x = 0, tom qab ntawd daws koj qhov kev ua zauv rau f (x) lossis y, uas muab koj tus nqi y uas koj tus parabola hla dhau y axis. Tsis zoo li x cuam tshuam, tus qauv parabolas tsuas tuaj yeem muaj ib qho y cuam tshuam. Nco tseg - rau cov qauv kev sib npaug, y cuam tshuam yog ntawm y = c.

• Piv txwv li, peb paub peb cov zauv sib npaug sib npaug 2x² + 16x + 39 muaj y cuam tshuam ntawm y = 39, tab sis nws kuj tuaj yeem pom raws li hauv qab no:

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

  • f (x) = 39. Parabola's y cuam tshuam yog ntawm y = 39 hli

    Raws li tau sau tseg saum toj no, y cuam tshuam yog ntawm y = c.

• Peb qhov vertex form equation 4 (x - 5)² + 12 muaj y cuam tshuam uas tuaj yeem pom raws li 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. Parabola's y cuam tshuam yog ntawm y = 112 hli

Kauj Ruam 10. Yog tias tsim nyog, npaj cov ntsiab lus ntxiv, tom qab ntawd teeb duab

Tam sim no koj yuav tsum muaj qhov siab, taw qhia, x cuam tshuam (s), thiab, tej zaum, y cuam tshuam rau koj qhov kev ua zauv. Txog ntawm qhov no, koj tuaj yeem sim kos koj lub parabola siv cov ntsiab lus uas koj muaj raws li cov lus qhia, lossis koj tuaj yeem pom cov ntsiab lus ntxiv rau "sau" koj parabola kom cov nkhaus koj kos tau raug dua. Txoj hauv kev yooj yim tshaj plaws los ua qhov no tsuas yog ntsaws ob peb x qhov tseem ceeb ntawm ob sab ntawm koj lub vertex, tom qab ntawd npaj cov ntsiab lus no siv y qhov txiaj ntsig koj tau txais. Feem ntau, cov kws qhia ntawv yuav xav kom koj tau txais qee cov ntsiab lus ua ntej koj kos koj daim ntawv parabola.

• Cia peb rov saib qhov sib npaug x² + 2x + 1. Peb twb paub nws tsuas yog x cuam tshuam yog ntawm x = -1. Vim tias nws tsuas yog kov lub x cuam tshuam ntawm ib qho taw tes, peb tuaj yeem pom tias nws cov vertex yog nws x cuam tshuam, uas txhais tau tias nws cov vertex yog (-1, 0). Peb ua tau zoo tsuas yog muaj ib lub ntsiab lus rau qhov parabola no - tsis ze li txaus los kos cov parabola zoo. Cia peb nrhiav ob peb ntxiv kom ntseeg tau tias peb kos cov duab kos kom raug.

  • 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).

  • Npaj cov ntsiab lus no rau 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 dag ntawm tus lej, koj feem ntau tuaj yeem txuag koj tus kheej qee qhov haujlwm los ntawm kev yooj yim cuam tshuam txog qhov muab rau hla parabola txoj kab sib luag los nrhiav qhov sib piv ntawm lwm sab ntawm parabola.

Yees duab - Los ntawm kev siv qhov kev pabcuam no, qee cov ntaub ntawv yuav raug muab qhia rau YouTube

Lub tswv yim

• Nco ntsoov tias hauv f (x) = ax² + bx + c, yog b lossis c sib npaug xoom, cov lej ntawd ploj mus. Ej.: 12 x² + 0x + 6 ua 12x² +6 vim 0x yog 0.
• Hloov tus lej lossis siv cov lej faib raws li koj tus kws qhia lej lej qhia koj kom ua. Qhov no yuav pab koj teeb duab koj cov lej sib npaug ua kom raug.