Chimiro cheCanonical cheQuadratic Equation
Maequation equadratic ndeimwe yenyaya dzakakosha mualgebra, dzinoonekwa kakawanda mumasvomhu echikoro uye mumashandisirwo esainzi, economics, uye engineering. Kazhinji, quadratic equation iequation yepolynomial yedhigirii rechipiri inogona kunyorwa seizvi:
\[
mbezo^2 + bx + c = 0
\]
apo \(a \neq 0\), uye \(a\), \(b\), uye \(c\) dziri nhamba chaidzo (kana nhamba dzakaoma, zvichienderana nemamiriro ezvinhu). Kunyange zvazvo chimiro ichi chakajairika chichiwanzounzwa, kune chimwe chimiro chinobatsira zvikuru pakunzwisisa hunhu hwema quadratic equations, kureva chimiro che canonical. Chimiro che canonical chinotibatsira "kuverenga" hunhu hwe parabola—senge vertex, maximum/minimum values, uye axis of symmetry—nekukurumidza uye zvakajeka.
Chii chinonzi canonical form?
Chimiro che "canonical" (chinowanzonziwo "vertex form") che "quadratic function" ndeichi:
\[
y = a(xh)^2 + k
\]
ne:
– \(a\) inosarudza gwara uye "kukombama" kweparabola,
– \((h, k)\) ndiwo makoronesheni enzvimbo iri pakati peparabola.
Kana zviri kutaurwa zviri quadratic equation (kwete basa), fomu rinogona kunyorwa:
\[
a(xh)^2 + k = 0
\]
kana kutamisirwa kufomu rebasa kana zvichidikanwa. Fomu iri rinonzi canonical nekuti rinopa ruzivo rwakanyanya pamusoro pechimiro chegirafu uye maitiro ehuwandu hwebasa.
Sei chimiro chemagwaro ekare chichikosha?
Pane zvikonzero zvakawanda nei mafomu ezvinyorwa zvepamutemo achibatsira zvikuru:
1. Tsvaga nzvimbo yepamusoro zviri nyore
Muchimiro chakajairika \(ax^2+bx+c\), tinofanira kutanga taverenga \(x_p = -\frac{b}{2a}\) kuti tiwane vertex. Zvisinei, muchimiro checanonical \(a(xh)^2+k\), vertex inoonekwa pakarepo, kureva \((h, k)\).
2. Ziva kukosha kwakanyanya/kushoma
Kana \(a>0\), parabola inovhurika kumusoro kuitira kuti vertex ive iyo kukosha kudiki . Kana \(a<0\), parabola inovhurika pasi kuitira kuti vertex ive iyo kukosha kukuru . Kukosha kwakanyanya ndi \(k\). 3. Zvinoita kuti zvive nyore kunyora magirafu Nekuziva vertex uye kutungamira kwekuvhurwa kweparabola, tinogona kudhirowa magirafu nekukurumidza, kusanganisira kuona axis ye symmetry \(x=h\). 4. Inobatsira kugadzirisa quadratic equations Mune zvimwe zviitiko, kugadzirisa \(ax^2+bx+c=0\) kunokurumidza kana kutanga kwashandurwa kuita fomu yakakwana yechikwere kuburikidza nefomu yecanonical. Maitiro ekushandura fomu yakajairika kuita fomu yecanonical Kuchinja \(ax^2+bx+c\) kuenda ku \(a(xh)^2+k\) kunoitwa nenzira yekupedzisa sikwere (kupedzisa sikwere). Matanho acho ndeaya anotevera: Zvatapiwa: \[ y = ax^2 + bx + c \] Danho 1: Factor \(a\) kubva pamashoko ane \(x\) \[ y = a\left(x^2 + \frac{b}{a}x\right) + c \] Danho 2: Wedzera uye bvisa nhamba dzakafanana dziri mumabhuraketi kuti uite sikweya yakakwana Kuti tigadzire \(x^2 + \frac{b}{a}x\) muchimiro \((x+p)^2\), tinotora: \[ p = \frac{1}{2}\cdot \frac{b}{a} = \frac{b}{2a} \] Wedzera uye bvisa \(p^2\): \[ y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Danho 3: Gadzira sikweya yakakwana \[ y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Danho rechina: Paradzira \(a\) uye nyore \[ y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c \] Nekuti: \[ a\left(\frac{b}{2a}\right)^2 = a\cdot \frac{b^2}{4a^2} = \frac{b^2}{4a} \] Zvadaro: \[ y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \] Iyi ndiyo fomu yepamutemo ine: \[ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} \] Cherechedza kuti \(h\) inoenderana nefomura ye axis ye symmetry, nepo \(k\) ichipa kukosha kwebasa pa vertex. Muenzaniso wekushandura kuita chimiro checanonical Semuenzaniso: \[ y = 2x^2 - 8x + 3 \] Danho 1: Factor 2 kubva pamashoko maviri ekutanga \[ y = 2(x^2 - 4x) + 3 \] Danho 2: Pedzisa sikweya iri mukati memabhuraketi Tora hafu ye \(-4\), inova \(-2\), wobva waiisa sikweya kuti uwane \(4\): \[ y = 2(x^2 - 4x + 4 - 4) + 3 \] Danho 3: Chimiro chesikweya chakakwana \[ y = 2((x-2)^2 - 4) + 3 \] Danho 4: Nyoresa \[ y = 2(x-2)^2 - 8 + 3 \] \[ y = 2(x-2)^2 - 5 \] Saka chimiro checanonical ndeichi: \[ y = 2(x-2)^2 - 5 \] Kubva pano tinoziva pakarepo kuti vertex iri \((2, -5)\), axis ye symmetry ndi \(x=2\), parabola inovhurika kumusoro (nekuti \(a=2>0\)), uye kukosha kudiki kwebasa iri \(-5\).
Hukama huripo pakati pechimiro chemutemo nemidzi ye equation
Kana tichida kuwana midzi ye quadratic equation:
\[
demo^2+bx+c=0
\]
Tinogona kuchishandura kuita chimiro chepamutemo:
\[
a(xh)^2 + k = 0
\]
Saka:
\[
a(xh)^2 = -k
\]
\[
(xh)^2 = -\frac{k}{a}
\]
Kemudian:
\[
xh = \pm \sqrt{-\frac{k}{a}}
\]
\[
x = h \pm \sqrt{-\frac{k}{a}}
\]
Kubva pane izvi zvinoonekwa kuti mudzi chaiwo uripo kana:
\[
-\frac{k}{a} \ge 0
\]
izvo zvinoenderana nepfungwa yekusarudza. Kusarudza \(D = b^2-4ac\) kunosarudza kana paine midzi miviri chaiyo, mudzi mumwe chete, kana kuti pasina midzi chaiyo. Muchimiro chemutemo, mamiriro aya anoitika nenzira yechisikigo kuburikidza nechiratidzo chekutaura mukati memudzi.
Mafomu eCanonical uye magirafu ekunzwisisa
Girafu yebasa re quadratic i parabola. Iine chimiro che canonical:
\[
y = a(xh)^2 + k
\]
tinogona kunzwisisa kushandurwa kweparabola yakajairwa \(y=x^2\):
– \(h\) inotamisa girafu kurudyi (kana \(h>0\)) kana kuruboshwe (kana \(h<0\)), - \(k\) inotamisa girafu kumusoro (kana \(k>0\)) kana pasi (kana \(k<0\)), - \(a\) inotambanudza kana kudzvanya parabola uye inosarudza divi rekuvhura (kumusoro kana \(a>0\), pasi kana \(a<0\)). Saka, chimiro checanonical hachisi chishandiso chekuverenga chete, asiwo chishandiso chekuona "kuverenga" maitiro ebasa. Mhedziso Chimiro checanonical che quadratic equation kana basa, kureva \(y = a(xh)^2 + k\), chiratidzo chinopa ruzivo rwakawanda nekuti chinoratidza pakarepo vertex ye \((h,k)\), axis ye symmetry, uye kukosha kwakanyanya kana kushoma. Fomu iri rinowanikwa kubva kuchimiro chakajairika \(ax^2+bx+c\) nenzira yekupedzisa sikweya. Kunze kwekubatsira kugadzira parabolas, chimiro checanonical chinoitawo kuti zvive nyore kuongorora midzi uye hunhu hwe quadratic equations. Nekuda kweizvi, kunzwisisa chimiro checanonical idanho rakakosha mukuziva algebra uye mashandisirwo e quadratic equations muminda yakasiyana-siyana.
