Fomu Yovomerezeka ya Quadratic Equation
Ma equation a quadratic ndi amodzi mwa mitu yofunika kwambiri mu algebra, yomwe imawonekera kawirikawiri m'masamu akusukulu komanso mu ntchito za sayansi, zachuma, ndi uinjiniya. Kawirikawiri, equation ya quadratic ndi equation ya polynomial ya digiri yachiwiri yomwe ingalembedwe motere:
\[
nkhwangwa^2 + bx + c = 0
\]
kumene \(a \neq 0\), ndi \(a\), \(b\), ndi \(c\) ndi manambala enieni (kapena manambala ovuta, kutengera zomwe zili mkati). Ngakhale kuti mawonekedwe onsewa nthawi zambiri amayambitsidwa, pali mawonekedwe ena omwe ndi othandiza kwambiri kumvetsetsa makhalidwe a ma quadratic equation, omwe ndi mawonekedwe ovomerezeka. Mawonekedwe ovomerezeka amatithandiza "kuwerenga" makhalidwe a parabola—monga vertex, maximum/minimum values, ndi axis of symmetry—mwachangu komanso momveka bwino.
Kodi mawonekedwe ovomerezeka ndi chiyani?
Fomu yovomerezeka (yomwe nthawi zambiri imatchedwanso mawonekedwe a vertex) ya ntchito ya quadratic ndi:
\[
y = a(xh)^2 + k
\]
ndi:
– \(a\) imatsimikiza njira ndi "kupindika" kwa parabola,
– \((h, k)\) ndi ma coordinates a vertex ya parabola.
Ngati zomwe zikukambidwa ndi quadratic equation (osati ntchito), mawonekedwewo akhoza kulembedwa:
\[
a(xh)^2 + k = 0
\]
kapena kusamutsidwa ku mawonekedwe a ntchito ngati kuli kofunikira. Fomu iyi imatchedwa yovomerezeka chifukwa imapereka chidziwitso chochuluka cha mawonekedwe a graph ndi momwe ntchitoyo imagwirira ntchito.
N’chifukwa chiyani mawonekedwe ovomerezeka ndi ofunikira?
Pali zifukwa zingapo zomwe mitundu yovomerezeka ndi yothandiza kwambiri:
1. Dziwani mosavuta malo otsetsereka
Mu mawonekedwe onse \(ax^2+bx+c\), choyamba tiyenera kuwerengera \(x_p = -\frac{b}{2a}\) kuti tipeze vertex. Komabe, mu mawonekedwe ovomerezeka \(a(xh)^2+k\), vertex imawonekera nthawi yomweyo, yomwe ndi \((h, k)\).
2. Dziwani mtengo wapamwamba/wocheperako
Ngati \(a>0\), parabola imatseguka mmwamba kotero kuti vertex ndiye mtengo wocheperako. Ngati \(a<0\), parabola imatseguka pansi kotero kuti vertex ndiye mtengo wapamwamba kwambiri. Mtengo wokwera kwambiri ndi \(k\). 3. Zimapangitsa kuti zikhale zosavuta kujambula ma graph Podziwa vertex ndi komwe kutseguka kwa parabola kumachitikira, titha kujambula ma graph mwachangu, kuphatikiza kudziwa mzere wofanana \(x=h\). 4. Zimathandiza kuthetsa ma equation a quadratic Nthawi zina, kuthetsa \(ax^2+bx+c=0\) kumakhala kofulumira ngati kaye kwasinthidwa kukhala mawonekedwe angwiro a sikweya kudzera mu mawonekedwe ovomerezeka. Momwe mungasinthire mawonekedwe onse kukhala mawonekedwe ovomerezeka Kusintha \(ax^2+bx+c\) kukhala \(a(xh)^2+k\) kumachitika pogwiritsa ntchito njira yomaliza sikweya (kumaliza sikweya). Masitepe ndi awa: Operekedwa: \[ y = ax^2 + bx + c \] Gawo 1: Factor \(a\) kuchokera ku mawu omwe ali ndi \(x\) \[ y = a\left(x^2 + \frac{b}{a}x\right) + c \] Gawo 2: Onjezani ndikuchotsa manambala omwewo m'mabulaketi kuti mupange sikweya wangwiro Kuti tipange \(x^2 + \frac{b}{a}x\) mu mawonekedwe \((x+p)^2\), timatenga: \[ p = \frac{1}{2}\cdot \frac{b}{a} = \frac{b}{2a} \] Onjezani ndikuchotsa \(p^2\): \[ y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Gawo 3: Gulukani mu sikweya wangwiro \[ y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Gawo 4: Falitsani \(a\) ndikuchepetsa \[ y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c \] Chifukwa: \[ a\left(\frac{b}{2a}\right)^2 = a\cdot \frac{b^2}{4a^2} = \frac{b^2}{4a} \] Kenako: \[ y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \] Iyi ndi njira yodziwika bwino yokhala ndi: \[ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} \] Dziwani kuti \(h\) ikugwirizana ndi fomula ya mzere wozungulira wa symmetry, pomwe \(k\) imapereka mtengo wa ntchitoyo pa vertex. Chitsanzo chosinthira kukhala mawonekedwe ovomerezeka Mwachitsanzo: \[ y = 2x^2 - 8x + 3 \] Gawo 1: Factor 2 kuchokera ku mawu awiri oyamba \[ y = 2(x^2 - 4x) + 3 \] Gawo 2: Malizitsani sikweya mkati mwa mabulaketi. Tengani theka la \(-4\), lomwe ndi \(-2\), kenako lembani sikweya kuti mupeze \(4\): \[ y = 2(x^2 - 4x + 4 - 4) + 3 \] Gawo 3: Fomu yoyenera sikweya \[ y = 2((x-2)^2 - 4) + 3 \] Gawo 4: Chepetsani \[ y = 2(x-2)^2 - 8 + 3 \] \[ y = 2(x-2)^2 - 5 \] Chifukwa chake mawonekedwe ovomerezeka ndi awa: \[ y = 2(x-2)^2 - 5 \] Kuchokera apa tikudziwa nthawi yomweyo kuti vertex ndi \((2, -5)\), mzere wa symmetry ndi \(x=2\), parabola imatseguka mmwamba (chifukwa \(a=2>0\)), ndipo mtengo wocheperako wa ntchitoyo ndi \(-5\).
Ubale pakati pa mawonekedwe ovomerezeka ndi mizu ya equation
Ngati tikufuna kupeza mizu ya equation ya quadratic:
\[
nkhwangwa^2+bx+c=0
\]
Tikhoza kusintha kukhala mtundu wovomerezeka:
\[
a(xh)^2 + k = 0
\]
Kotero:
\[
a(xh)^2 = -k
\]
\[
(xh)^2 = -\frac{k}{a}
\]
Kenako:
\[
xh = \pm \sqrt{-\frac{k}{a}}
\]
\[
x = h \pm \sqrt{-\frac{k}{a}}
\]
Kuchokera pa izi, zitha kuwoneka kuti muzu weniweni ulipo ngati:
\[
-\frac{k}{a} \ge 0
\]
zomwe zikugwirizana ndi lingaliro la kusiyanitsa. Kusiyanitsa \(D = b^2-4ac\) kumatsimikiza ngati pali mizu iwiri yeniyeni, mizu iwiri imodzi, kapena palibe mizu yeniyeni. Mu mawonekedwe ovomerezeka, vutoli limabwera mwachibadwa kudzera mu chizindikiro cha mawu omwe ali mkati mwa mizu.
Mafomu ovomerezeka ndi ma graph omvetsetsa
Chithunzi cha ntchito ya quadratic ndi parabola. Ndi mawonekedwe ovomerezeka:
\[
y = a(xh)^2 + k
\]
Titha kumvetsetsa kusintha kwa parabola yokhazikika \(y=x^2\):
– \(h\) amasuntha graph kumanja (ngati \(h>0\)) kapena kumanzere (ngati \(h<0\)), - \(k\) amasuntha graph mmwamba (ngati \(k>0\)) kapena pansi (ngati \(k<0\)), - \(a\) amatambasula kapena kukanikiza parabola ndikusankha komwe kutseguka (mmwamba ngati \(a>0\), pansi ngati \(a<0\)). Chifukwa chake, mawonekedwe ovomerezeka si chida chowerengera chokha, komanso chida chowonera "chowerengera" machitidwe a ntchitoyo. Mapeto Mtundu wovomerezeka wa equation ya quadratic kapena ntchito, womwe ndi \(y = a(xh)^2 + k\), ndi chithunzi chodziwitsa kwambiri chifukwa nthawi yomweyo umawonetsa vertex ya \((h,k)\), axis of symmetry, ndi mtengo wapamwamba kwambiri kapena wocheperako. Fomu iyi imapezeka kuchokera ku mawonekedwe wamba \(ax^2+bx+c\) pogwiritsa ntchito njira yomaliza sikweya. Kupatula kuthandiza kujambula ma parabola, mawonekedwe ovomerezeka amathandizanso kusanthula mizu ndi makhalidwe a ma quadratic equation. Pachifukwa ichi, kumvetsetsa mawonekedwe ovomerezeka ndi gawo lofunikira pakumvetsetsa algebra ndi kugwiritsa ntchito ma quadratic equation m'magawo osiyanasiyana.
