Muaj ntau qhov lej ua haujlwm uas siv cov toj siab. Polyhedra muaj lawv, cov kab ke ntawm kev tsis sib xws tuaj yeem muaj ib lossis ntau qhov siab, thiab cov lus piv txwv lossis cov lej sib npaug kuj tseem muaj lawv. Nrhiav lub vertex sib txawv los ntawm qhov xwm txheej, tab sis ntawm no yog cov lus qhia koj yuav tsum paub txog ntawm txhua qhov xwm txheej.
cov kauj ruam
Txoj Kev 1 ntawm 5: Nrhiav Tus Zauv Vertices hauv Polygon
Kauj Ruam 1. Kawm Euler tus qauv
Euler tus qauv, raws li siv hauv kev hais txog geometry thiab duab, hais tias rau ib qho uas tsis cuam tshuam txog polyhedron, tus naj npawb ntawm lub ntsej muag ntxiv rau cov naj npawb ntawm qhov siab rho tawm tus naj npawb ntawm cov npoo yuav ib txwm sib npaug 2.
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Sau ua ib qho zauv, cov qauv tuaj yeem txhais tau tias yog: F + V - E = 2
- F hais txog tus naj npawb ntawm lub ntsej muag.
- V hais txog tus naj npawb ntawm cov toj, lossis kaum.
- Thiab nws hais txog tus naj npawb ntawm cov npoo.
Kauj Ruam 2. Txheeb cov qauv kom pom cov naj npawb ntawm cov ntsug
Yog tias koj paub tias muaj pes tsawg lub ntsej muag thiab cov npoo uas muaj cov polyhedron, koj tuaj yeem suav tus lej ntawm qhov dav siv Euler tus lej. Rho tawm F los ntawm ob sab ntawm qhov sib npaug thiab ntxiv E rau ob qho, cais V los ntawm lwm qhov.
V = 2 - F + E
Kauj Ruam 3. Sau tus lej thiab daws qhov kev ua zauv
Txhua yam koj yuav tsum ua ntawm qhov no yog muab ob sab thiab tus lej zauv tso rau hauv qhov sib npaug ua ntej ntxiv lossis rho tawm. Cov lus teb uas koj tau txais yuav qhia koj tus lej ntawm qhov siab thiab ua tiav qhov teeb meem.
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Piv txwv: Ib lub polyhedron muaj 6 lub ntsej muag thiab 12 ntug.
- V = 2 - F + E
- V = 2 - 6 + 12
- V = -4 + 12
- V = 8
Txoj Kev 2 ntawm 5: Tshawb Nrhiav Vertices hauv Kab Tsis Ncaj Ncees Tshuab
Kauj Ruam 1. Txheeb cov kev daws teeb meem ntawm txoj kab tsis sib txig sib luag
Hauv qee kis, teeb duab cov kev daws teeb meem ntawm txhua qhov tsis sib xws tuaj yeem pom pom koj qhov twg qee qhov, yog tias tsis yog txhua yam, ntawm qhov ncaj ncees yuav yog. Txawm li cas los xij, thaum nws tsis yog, koj yuav tsum pom nws algebraically.
Yog tias koj siv lub tshuab xam zauv, feem ntau nws tuaj yeem nqes mus rau qhov siab thiab nrhiav cov kev tswj hwm txoj hauv kev
Kauj Ruam 2. Hloov qhov tsis sib xws mus rau qhov sib npaug
Txhawm rau daws cov kab ke ntawm kev tsis sib xws, koj yuav tsum hloov pauv qhov tsis sib xws ib yam mus rau qhov sib npaug, tso cai rau koj muaj peev xwm nrhiav qhov txiaj ntsig ntawm x kev thiab y.
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Piv txwv: Hauv qhov tsis sib xws hauv qab no:
- y <x tau
- y> -x + 4x
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Hloov qhov tsis sib xws rau hauv:
- y = x
- y = -x + 4x
Kauj Ruam 3. Hloov qhov hloov pauv nrog lwm tus
Txawm hais tias muaj ob peb txoj hauv kev sib txawv uas koj tuaj yeem daws tau x kev thiab y, kev hloov pauv feem ntau yog qhov yooj yim siv. Sau tus nqi ntawm y los ntawm ib qho kev sib npaug mus rau lwm qhov, muaj txiaj ntsig "hloov pauv" y ntawm lwm qhov nrog qhov muaj nuj nqis x kev ntxiv.
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Piv txwv: Yog:
- y = x
- y = -x + 4x
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Tom qab ntawd, y = -x + 4x tuaj yeem sau ua:
x = -x + 4x
Kauj Ruam 4. Kev daws rau thawj qhov sib txawv
Tam sim no koj tsuas muaj ib qho sib txawv hauv qhov kev ua zauv, koj tuaj yeem daws tau yooj yim rau qhov sib txawv ntawd, x kev, zoo li koj xav tau lwm yam: ntxiv, rho tawm, faib thiab sib faib.
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Piv txwv: x = -x + 4
- x + x = -x + x + 4
- 2 x = 4x
- 4x / x = 2
- x = 2 os
Kauj Ruam 5. Kev daws rau qhov sib txawv ntxiv
Sau tus nqi tshiab rau x kev hauv ib qho ntawm qhov sib npaug qub los nrhiav tus nqi ntawm y.
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Piv txwv: y = x
y = 2 hli
Kauj Ruam 6. Txiav txim siab qhov tseeb
Lub vertex tsuas yog kev sib koom ua ke suav nrog koj qhov txiaj ntsig tshiab. x kev thiab y.
Piv txwv: (2, 2)
Txoj Kev 3 ntawm 5: Nrhiav Lub Vertex ntawm Parabola nrog Kev Sib Koom Tes
Kauj Ruam 1. Ntsuas qhov sib npaug
Rov sau dua qhov kev ua zauv plaub fab hauv nws daim ntawv foos. Muaj ntau txoj hauv kev los ntsuas qhov sib npaug sib npaug, tab sis thaum ua tiav, koj yuav raug tso tseg nrog ob pawg hauv kab ntawv uas, thaum sib npaug, sib npaug rau qhov qub zauv.
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Piv txwv (los ntawm kev decomposition):
- 3 x ygo2 6x - 45 ib
- Nrhiav qhov feem ntau: 3 (x2 -2x -15)
- Muab cov ntsiab lus a thiab c: 1 × -15 = -15
- Nrhiav ob tus lej nrog cov khoom sib npaug rau -15 thiab cov lej sib npaug rau tus nqi b, -2: 3 × -5 = -15; 3 - 5 = -2
- Hloov ob qhov tseem ceeb rau hauv kab zauv: ax2 + kx + hx + c: 3 (x2 3) x 5x - 4x - 15
- Ntsuas cov polynomial los ntawm pab pawg: f (x) = 3 × (x + 3) × (x - 5)
Kauj Ruam 2. Nrhiav lub ntsiab lus uas qhov sib npaug hla tus x-axis
Thaum twg ua haujlwm ntawm x, lossis f (x), sib npaug rau 0, parabola yuav hla tus x-axis. Qhov no yuav tshwm sim thaum ib qho ntawm pawg teeb meem sib npaug 0.
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Piv txwv: x + 3; -3 + 3 = 0
- x - 5; 5 - 5 = 0
- Yog li, cov hauv paus hniav yog: (-3, 0) thiab (5, 0)
Kauj Ruam 3. Xam qhov nruab nrab
Lub axis ntawm symmetry ntawm qhov sib npaug yuav ncaj qha nruab nrab ntawm ob lub hauv paus ntawm qhov sib npaug. Koj yuav tsum nrhiav lub axis ntawm symmetry txij li lub vertex nyob saum nws.
Piv txwv: x = 1; tus nqi no ncaj qha ntawm -3 thiab 5
Kauj Ruam 4. Muab tus nqi x rau hauv qhov qub kev ua zauv
Muab tus nqi x rau lub axis ntawm symmetry rau hauv ib qho ntawm qhov sib npaug rau parabola. Tus nqi y yuav yog tus y rau lub vertex.
Piv txwv: y = 3x2 - 4x - 45 = 3 (1)2 - 6(1) - 45 = -48
Kauj Ruam 5. Sau lub ntsiab lus taw qhia
Txij ntawm no mus, qhov txiaj ntsig kawg rau x thiab y yuav tsum muab rau koj qhov chaw sib koom ua ke.
Piv txwv: (1, -48)
Txoj Kev 4 ntawm 5: Nrhiav Lub Vev Xaib ntawm Parabola Ua tiav Lub Square
Kauj Ruam 1. Sau thawj qhov sib npaug hauv nws daim ntawv vertex
Qhov "vertex" cov duab ntawm qhov sib npaug tau sau ua y = a (x - h)2 + k, thiab lub vertex yuav yog (h, k) tus. Koj qhov kev ua zauv plaub fab tam sim no yuav tsum tau rov sau dua hauv daim ntawv no, thiab txhawm rau ua qhov no koj yuav tsum ua kom tiav cov xwm txheej.
Piv txwv: y = -x2 -8x -15
Kauj Ruam 2. Tshem tus nqi
Muab qhov sib piv ntawm thawj lo lus, a, los ntawm thawj ob nqe lus ntawm qhov sib npaug. Tawm lub sijhawm kawg, c, tam sim no.
Piv txwv: -1 (x2 + 8x) - 15
Kauj Ruam 3. Nrhiav lub sijhawm thib peb rau kab lus
Lub sijhawm thib peb yuav tsum ua kom tiav cov teeb tsa hauv kab ntawv sib dhos kom cov txiaj ntsig nruab nrab ntawm lawv tsim tau ib lub xwmfab zoo meej. Lub sijhawm tshiab no yuav yog tus nqi sib npaug ntawm ib nrab ntawm qhov sib npaug ntawm lub hauv paus nruab nrab.
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Piv txwv: 8 /2 = 4; 4 × 4 = 16; tsis ntev,
-1 (xws2 + 4x + 16)
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Tsis tas li, nco ntsoov tias yam koj ua sab hauv yuav tsum ua sab nrauv:
y = -1 (x2 + 8x + 16) - 15 + 16
Kauj Ruam 4. Ua kom yooj yim dua
Txij li cov kab lus tam sim no tsim ua lub xwmfab zoo tshaj plaws, koj tuaj yeem ua qhov yooj yim ntawm cov niam txiv mus rau qhov ua piv txwv. Ib txhij, nws muaj peev xwm ua kom tsim nyog ntxiv lossis rho tawm mus rau qhov muaj txiaj ntsig sab nraud ntawm kab lus.
Piv txwv: y = -1 (x + 4)2 + 1
Kauj Ruam 5. Txheeb xyuas seb qhov kev tswj hwm twg yog ua raws qhov vertex equation
Nco ntsoov tias lub vertex cov duab ntawm ib qho kev ua zauv tau muab los ntawm y = a (x - h)2 + k, nrog (h, k) tus sawv cev rau kev tswj hwm ntawm lub vertex. Tam sim no koj muaj cov ntaub ntawv txaus los nkag rau qhov tseem ceeb hauv h thiab k chaw thiab ua tiav qhov teeb meem.
- k = 1 hli
- h = -4
- Yog li, lub vertex ntawm qhov sib npaug no tuaj yeem pom hauv: (-4, 1)
Txoj Kev 5 ntawm 5: Nrhiav Lub Vertex ntawm Parabola nrog Cov Qauv Yooj Yim
Kauj Ruam 1. Nrhiav tus x ua haujlwm ntawm lub vertex ncaj qha
Yog tias qhov sib npaug ntawm koj zaj lus piv txwv tuaj yeem sau ua y = ax2 bx + c, x ntawm lub vertex tuaj yeem tshawb pom los ntawm cov mis x = -b / 2a. Tsuas nkag siab a thiab b qhov tseem ceeb los ntawm qhov sib npaug kom nrhiav x.
- Piv txwv: y = -x2 -8x -15
- x = -b / 2a = -(-8) / 2 × (-1) = 8 / (-2) = -4
- x = -4
Kauj Ruam 2. Nkag mus rau tus nqi no mus rau thawj qhov kev ua zauv
Los ntawm kev nkag tus nqi x rau hauv qhov kev ua zauv, koj tuaj yeem daws rau y. Tus nqi y no yuav yog y ua haujlwm ntawm koj lub vertex.
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Piv txwv: y = -x2 - 4x - 15 = - (- 4)2 - 8(-4) - 15 = -(16) - (-32) - 15 = -16 + 32 - 15 = 1
y = 1
Kauj Ruam 3. Sau qhov chaw ua haujlwm ntawm lub vertex
Qhov x thiab y qhov tseem ceeb tau txais yuav yog kev tswj hwm ntawm nws lub vertex point.