Mienzaniso yeMibvunzo Kukurukurirana Kugadzirisa Matambudziko neMashandiro eQuadratic
Muchinyorwa chino, tichadzidza maitiro ekugadzirisa matambudziko tichishandisa mabasa equadratic nekupa mienzaniso uye matanho ekukurukura akadzama. Basa requadratic ibasa rechipiri repolynomial rine chimiro chakajairika \( ax^2 + bx + c \), apo \( a \), \( b \), uye \( c \) ari ma constants uye \( a \neq 0 \). Mabasa equadratic mumamiriro akasiyana-siyana anowanzoonekwa mufizikisi, economics, uye engineering, zvichiita kuti ive nyaya yakakosha kwazvo yekuziva.
Ngatitangei nekukurukura dzimwe pfungwa huru tobva tatanga nemibvunzo yemuenzaniso.
Pfungwa Dzekutanga dzeMabasa eQuadratic
1. Chimiro Chakajairika: Basa re quadratic rinoratidzwa se \( f(x) = ax^2 + bx + c \).
2. Midzi Yesikweya: Midzi ye quadratic equation \( ax^2 + bx + c = 0 \) inowanikwa uchishandisa fomura ye quadratic, inoti:
\[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
\]
3. Kusiyanisa: Kusiyanisa kwe equation ye quadratic ndi \( D = b^2 – 4ac \). Kukosha kwekusarudza kunoratidza hunhu hwemidzi ye equation ye quadratic:
– Kana \( D > 0 \), ine midzi miviri chaiyo yakasiyana.
– Kana \( D = 0 \), ine mudzi mumwe chaiwo (mudzi wemapatya).
– Kana \( D < 0 \), ine midzi miviri yakaoma yakabatana. 4. Vertex yeParabola: Makorodheni evertex yeparabola inoumbwa nebasa requadratic anogona kuwanikwa uchishandisa fomura: \[ x = -\frac{b}{2a} \] Kune kukosha kwe \( y \) pavertex, inogona kuverengerwa nekutsiva \( x \) mu quadratic function.
– Kana \( a < 0 \), parabola inovhura pasi. Tichifunga nezvepfungwa idzi dzese, ngationei kuti tingadzishandisa sei kugadzirisa matambudziko. Muenzaniso Dambudziko 1: Kutsvaga Midzi yeQuadratic Function Dambudziko: Tsvaga midzi yequadratic equation \( 2x^2 - 3x - 2 = 0 \). Mhinduro: Kuti tiwane midzi yequadratic equation, tinogona kushandisa quadratic formula. Matanho acho ndeaya anotevera: 1. Kuziva ma coefficients \( a \), \( b \), uye \( c \): \[ a = 2, \quad b = -3, \quad c = -2 \] 2. Verenga musiyano: \[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25 \] 3. Sezvo \( D > 0 \), tichava nemidzi miviri chaiyo yakasiyana. Ramba uchiverenga midzi iyi:
\[
x_{1,2} = \frac{-(-3) \pm \sqrt{25}}{2 \cdot 2} = \frac{3 \pm 5}{4}
\]
4. Verengai kukosha kuviri kwe \( x \):
\[
x_1 = \frac{3 + 5}{4} = 2 \quad \text{and} \quad x_2 = \frac{3 – 5}{4} = -\frac{1}{2}
\]
Saka, midzi ye equation \( 2x^2 – 3x – 2 = 0 \) ndi \( x = 2 \) uye \( x = -\frac{1}{2} \).
Muenzaniso Mubvunzo 2: Kutsvaga Makorodheni eVertex yeParabola
Mubvunzo:
Tsvaga ma coordinates e vertex yebasa re quadratic \( f(x) = 3x^2 – 6x + 2 \).
Kukurukurirana:
Kuti uwane ma coordinates e peak, shandisa fomura ye peak coordinate:
1. Ziva ma coefficients \( a \) uye \( b \):
\[
a = 3, \quad b = -6
\]
2. Verenga \( x \) kumusoro:
\[
x = -\frac{b}{2a} = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
3. Verenga \( y \) nekutsiva \( x = 1 \) mu "function" \( f(x) \):
\[
f(1) = 3(1)^2 – 6(1) + 2 = 3 – 6 + 2 = -1
\]
Saka, ma "vertex coordinates" ebasa \( f(x) = 3x^2 – 6x + 2 \) ndiwo \( (1, -1) \).
Muenzaniso Mubvunzo 3: Kuziva Nzira Yekuvhura Parabola
Mubvunzo:
Sarudza kwakanangana nekuvhurwa kweparabola kwebasa requadratic \( f(x) = -x^2 + 4x – 7 \).
Kukurukurirana:
Kuti tizive kwakananga parabola opening, tinongotarisa chiratidzo che coefficient \( a \):
1. Ziva chiyero \( a \):
\[
a = -1
\]
2. Sezvo \( a < 0 \), parabola inovhurika ichidzika. Saka, divi rekuvhurwa kweparabola kwebasa \( f(x) = -x^2 + 4x - 7 \) riri pasi. Muenzaniso 4: Kushandisa Mabasa eQuadratic muMamiriro Ehupenyu Hwechokwadi
Mubvunzo: Bhora rinokandwa kubva pasi ne quadratic equation \( h(t) = -5t^2 + 20t \), apo \( h \) kureba kwebhora mumamita uye \( t \) inguva mumasekonzi. Zvinotora nguva yakareba sei kuti bhora risvike pakukwirira kwaro, uye kukwirira kwaro kwakanyanya chii? Kukurukurirana: 1. Tsvaga nguva iyo kukwirira kwepamusoro kunosvika (makoronisheni epeak): \[ a = -5, \quad b = 20 \] \[ t = -\frac{b}{2a} = -\frac{20}{2(-5)} = \frac{20}{10} = 2 \quad \text{seconds} \] 2. Verenga kukwirira kwepamusoro nekutsiva \( t \) mu equation \( h(t) \): \[ h(2) = -5(2)^2 + 20(2) = -5(4) + 40 = -20 + 40 = 20 \quad \text{meters} \] Saka, nguva inotorwa nebhora kusvika pakukwirira kwepamusoro imasekondi maviri, uye kukwirira kwaro kwepamusoro i20 metres. Mhedziso Muchinyorwa chino, takurukura zvinhu zvakasiyana-siyana zvakakosha zvemabasa equadratic pamwe nemaitiro ekugadzirisa matambudziko ane chekuita nemabasa equadratic kuburikidza nemienzaniso yakati wandei. Kukurukura nezvemidzi ye quadratic equation, kuwana ma coordinates e vertex, kuona kwainobva parabola, uye kushandisa quadratic functions mumamiriro ezvinhu chaiwo, sekutsanangura kufamba kwezvinhu. Nekunzwisisa kwakasimba kwepfungwa idzi dzakakosha, uchakwanisa kutarisana nematambudziko akasiyana-siyana emasvomhu nesainzi ane chekuita ne quadratic functions nechivimbo chikuru. Mabasa e quadratic haasi chete akakosha mudzidziso asiwo anobatsira zvikuru mukushandiswa kwechokwadi uye kugadzirisa matambudziko munzvimbo dzakasiyana-siyana.