Mienzaniso yeMibvunzo Inokurukura nezveKugadzira Mabasa eQuadratic
Kugadzira mabasa equadratic inyaya huru mu algebra inowanzoonekwa muzvidzidzo zvemasvomhu zvepakati nepamusoro. Kunzwisisa mabasa equadratic kwakakosha nekuti anowanzo shandiswa mumamiriro akasiyana-siyana, akadai sekuongorora data, physics modeling, uye economics. Muchinyorwa chino, tichakurukura matambudziko akasiyana-siyana emuenzaniso uye maitiro ekugadzirisa iwo kuvaka mabasa equadratic.
Kunzwisisa Mabasa eQuadratic
Basa requadratic ibasa rechipiri repolynomial rine chimiro chakajairika:
\[ f(x) = ax^2 + bx + c \]
apo \(a\), \(b\), uye \(c\) zviri zvisingaperi, uye \(a \neq 0\).
Girafu yebasa re quadratic igoko rinozivikanwa se parabola. Maparabola ane symmetry uye chimiro chinoenderana nechiratidzo che constant \(a\). Kana \(a > 0\), parabola inovhurika kumusoro. Kusiyana neizvi, kana \(a < 0\), parabola inovhurika pasi. Zvinhu Zvakakosha zveMashandiro eQuadratic - Midzi ye quadratic equation: Makoshero e \(x\) ayo \(f(x) = 0\), ayo anogona kuwanikwa uchishandisa quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). - Vertex: Poindi yepamusoro kana yakaderera yeparabola, inowanikwa uchishandisa fomura \((x, y)\) apo \(x = -\frac{b}{2a}\) uye \(y = f(-\frac{b}{2a})\). - Akisi yekuenzana: Mutsetse wakamira unopatsanura parabola kuita zvikamu zviviri zvakafanana, zviri pa \(x = -\frac{b}{2a}\).
Muenzaniso Mubvunzo 1: Kugadzira Basa reQuadratic kubva paMapoinzi Matatu Mubvunzo: Sarudza fomura yebasa requadratic rinodarika nemapoinzi (1, 2), (2, 5), uye (3, 10). Mhinduro: 1. Tinotanga nechimiro chakajairika chebasa re quadratic: \[ f(x) = ax^2 + bx + c \] 2. Isa poindi (1, 2) mu equation: \[ a(1)^2 + b(1) + c = 2 \] \[ a + b + c = 2 \] (Equation 1) 3. Isa poindi (2, 5) mu equation: \[ a(2)^2 + b(2) + c = 5 \] \[ 4a + 2b + c = 5 \] (Equation 2) 4. Isa poindi (3, 10) mu equation: \[ a(3)^2 + b(3) + c = 10 \] \[ 9a + 3b + c = 10 \] (Equation 3) 5. Iye zvino tine masisitimu matatu eequation dzakatwasuka: \[ \begin{cases} a + b + c = 2 \\ 4a + 2b + c = 5 \\ 9a + 3b + c = 10 \\ \end{cases} \] 6. Kuti tigadzirise, tinobvisa maequation echipiri neekutanga: \[ (4a + 2b + c) - (a + b + c) = 5 - 2 \] \[ 3a + b = 3 \] (Equation 4) 7. Bvisa maequation echitatu neechipiri: \[ (9a + 3b + c) - (4a + 2b + c) = 10 - 5 \] \[ 5a + b = 5 \] (Equation 5) 8. Bvisa Equation 5 neEquation 4: \[ (5a + b) - (3a + b) = 5 - 3 \] \[ 2a = 2 \] \[ a = 1 \] 9. Isa \(a = 1\) muEquation 4: \[ 3(1) + b = 3 \] \[ 3 + b = 3 \] \[ b = 0 \] 10. Isa \(a = 1\) uye \(b = 0\) muEquation 1: \[ 1 + 0 + c = 2 \] \[ c = 1 \] 11. Saka, basa re quadratic ndeiri: \[ f(x) = 1x^2 + 0x + 1 \] \[ f(x) = x^2 + 1 \] Muenzaniso Mubvunzo 2: Kutsvaga Basa reQuadratic kubva kuVertex neOne Other Point Mubvunzo: Sarudza fomura yebasa requadratic rine vertex pa (-1, 4) uye rinopfuura nepakati pepoindi (1, 0). Mhinduro: 1. Chimiro che "quadratic function" chine "vertex" \((h, k)\) ndeichi: \[ f(x) = a(x - h)^2 + k \] 2. Isa "vertex" (-1, 4) muchimiro che "standard form": \[ f(x) = a(x + 1)^2 + 4 \] 3. Isa "point" (1, 0) mu "equation" kuti uwane \(a\): \[ 0 = a(1 + 1)^2 + 4 \] \[ 0 = a(2)^2 + 4 \] \[ 0 = 4a + 4 \] \[ 4a = -4 \] \[ a = -1 \] 4. Saka, basa re quadratic nderinoti: \[ f(x) = -1(x + 1)^2 + 4 \] \[ f(x) = - (x + 1)^2 + 4 \] 5. Kugoverwa kwefomu yakajairwa: \[ f(x) = - (x^2 + 2x + 1) + 4 \] \[ f(x) = -x^2 - 2x - 1 + 4 \] \[ f(x) = -x^2 - 2x + 3 \] Muenzaniso Mubvunzo 3: Kushandura Fomu reVertex kuita Fomu Rakajairwa Mubvunzo: Shandura basa re quadratic \( f(x) = 2(x - 3)^2 + 5 \) kuita fomu yakajairwa \( ax^2 + bx + c \). Mhinduro: 1. Kutanga, tinofanira kuwedzera: \[ f(x) = 2(x - 3)^2 + 5 \] 2. Wedzera binomial: \[ (x - 3)^2 = x^2 - 6x + 9 \] 3. Dzorera basa: \[ f(x) = 2(x^2 - 6x + 9) + 5 \] 4. Govera 2 pachikamu chimwe nechimwe chebinomial: \[ f(x) = 2x^2 - 12x + 18 + 5 \] 5. Sanganisa zvikamu zvese: \[ f(x) = 2x^2 - 12x + 23 \] Saka, chimiro chakajairika chebasa re quadratic ndeichi: \[ f(x) = 2x^2 - 12x + 23 \] Mhedziso Kuvaka mabasa e quadratic kubva paruzivo rwakasiyana-siyana hunyanzvi hwakakosha mumasvomhu. Kuburikidza nekudzidzira nguva dzose nemhando dzakasiyana dzematambudziko, tinogona kuvandudza kunzwisisa kwedu uye kugona kwedu kugadzirisa ma quadratic equations. Pfungwa huru dzekurangarira dzinosanganisira kuwana nekudzidzira matekiniki ekutsvaga ruzivo kubva muchimiro che vertex, kushandura pakati pe vertex nechimiro chakajairwa, uye kuvaka mabasa kubva pamapoinzi akapihwa. Nekunzwisisa kwakasimba kwemisoro iyi, tinogona kugadzirisa matambudziko akaomarara emasvomhu mune ramangwana.