Mehlala ea Lipotso tse Buisanang ka ho Haha Mesebetsi ea Quadratic
Ho theha mesebetsi ea quadratic ke sehlooho sa bohlokoa ho algebra se hlahang khafetsa lithutong tsa lipalo tse mahareng le tse tsoetseng pele. Ho utloisisa mesebetsi ea quadratic ho bohlokoa hobane e sebelisoa khafetsa maemong a fapaneng, joalo ka tlhahlobo ea data, mohlala oa fisiks le moruo. Sehloohong sena, re tla buisana ka mathata a fapaneng a mehlala le mokhoa oa ho a rarolla ho haha mesebetsi ea quadratic.
Ho utloisisa Mesebetsi ea Quadratic
Mosebetsi oa quadratic ke mosebetsi oa polynomial oa boemo ba bobeli o nang le sebopeho se akaretsang:
\[ f(x) = selepe^2 + bx + c \]
moo \(a\), \(b\), le \(c\) e leng di-constant, le \(a \neq 0\).
Grafo ea mosebetsi oa quadratic ke curve e tsejoang e le parabola. Li-parabola li na le symmetry le sebopeho se itšetlehileng ka letšoao la kamehla \(a\). Haeba \(a > 0\), parabola e buleha ho ea holimo. Ka lehlakoreng le leng, haeba \(a < 0\), parabola e buleha ho ea tlase. Likarolo tsa Bohlokoa tsa Mesebetsi ea Quadratic - Metso ea equation ea quadratic: Litekanyetso tsa \(x\) tseo \(f(x) = 0\), tse ka fumanoang ho sebelisoa foromo ea quadratic \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). - Vertex: Ntlha e phahameng ka ho fetisisa kapa e tlase ka ho fetisisa ea parabola, e fumanoang ho sebelisoa foromo \((x, y)\) moo \(x = -\frac{b}{2a}\) le \(y = f(-\frac{b}{2a})\). - Axis ea symmetry: Mola o otlolohileng o arolang parabola ka likarolo tse peli tse symmetry, e leng ho \(x = -\frac{b}{2a}\).
Mohlala Potso ea 1: Ho qapa Mosebetsi oa Quadratic ho tsoa ho Lintlha tse Tharo Potso: Fumana foromo ea mosebetsi oa quadratic o fetang lintlheng (1, 2), (2, 5), le (3, 10). Tharollo: 1. Re qala ka sebopeho se akaretsang sa mosebetsi wa quadratic: \[ f(x) = ax^2 + bx + c \] 2. Kenya ntlha (1, 2) sebakeng sa equation: \[ a(1)^2 + b(1) + c = 2 \] \[ a + b + c = 2 \] (Equation 1) 3. Kenya ntlha (2, 5) sebakeng sa equation: \[ a(2)^2 + b(2) + c = 5 \] \[ 4a + 2b + c = 5 \] (Equation 2) 4. Kenya ntlha (3, 10) sebakeng sa equation: \[ a(3)^2 + b(3) + c = 10 \] \[ 9a + 3b + c = 10 \] (Equation 3) 5. Jwale re na le ditsamaiso tse tharo tsa diequation tse otlolohileng: \[ \begin{cases} a + b + c = 2 \\ 4a + 2b + c = 5 \\ 9a + 3b + c = 10 \\ \end{cases} \] 6. Ho rarolla, re ntsha di-equation tsa bobedi le tsa pele: \[ (4a + 2b + c) - (a + b + c) = 5 - 2 \] \[ 3a + b = 3 \] (Equation 4) 7. Ntša diekabone tsa boraro le tsa bobedi: \[ (9a + 3b + c) - (4a + 2b + c) = 10 - 5 \] \[ 5a + b = 5 \] (Equation 5) 8. Ntša Equation 5 le Equation 4: \[ (5a + b) - (3a + b) = 5 - 3 \] \[ 2a = 2 \] \[ a = 1 \] 9. Nkela \(a = 1\) sebaka ho Equation 4: \[ 3(1) + b = 3 \] \[ 3 + b = 3 \] \[ b = 0 \] 10. Nkela \(a = 1\) le \(b = 0\) sebaka ho Equation 1: \[ 1 + 0 + c = 2 \] \[ c = 1 \] 11. Kahoo, mosebetsi wa quadratic ke: \[ f(x) = 1x^2 + 0x + 1 \] \[ f(x) = x^2 + 1 \] Mohlala Potso ya 2: Ho Fumana Mosebetsi wa Quadratic ho tswa ho Vertex le Potso e 'Ngoe e 'Ngoe: Fumana foromo ya mosebetsi wa quadratic o nang le vertex ho (-1, 4) mme o feta ka ntlha (1, 0). Tharollo: 1. Sebopeho se tloaelehileng sa mosebetsi wa quadratic o nang le vertex \((h, k)\) ke: \[ f(x) = a(x - h)^2 + k \] 2. Kenya vertex (-1, 4) sebakeng sa foromo e tloaelehileng: \[ f(x) = a(x + 1)^2 + 4 \] 3. Kenya ntlha (1, 0) sebakeng sa equation ho fumana \(a\): \[ 0 = a(1 + 1)^2 + 4 \] \[ 0 = a(2)^2 + 4 \] \[ 0 = 4a + 4 \] \[ 4a = -4 \] \[ a = -1 \] 4. Kahoo, mosebetsi wa quadratic ke: \[ f(x) = -1(x + 1)^2 + 4 \] \[ f(x) = - (x + 1)^2 + 4 \] 5. Kabo bakeng sa foromo e tloaelehileng: \[ f(x) = - (x^2 + 2x + 1) + 4 \] \[ f(x) = -x^2 - 2x - 1 + 4 \] \[ f(x) = -x^2 - 2x + 3 \] Mohlala wa Potso ya 3: Ho Fetolela Foromo ya Vertex ho ya ho Foromo e Tloaelehileng Potso: Fetolela mosebetsi wa quadratic \( f(x) = 2(x - 3)^2 + 5 \) ho ya ho foromo e tloaelehileng \( ax^2 + bx + c \). Tharollo: 1. Taba ea pele, re hloka ho atolosa: \[ f(x) = 2(x - 3)^2 + 5 \] 2. Atolosa binomial: \[ (x - 3)^2 = x^2 - 6x + 9 \] 3. Kenya sebaka mosebetsing: \[ f(x) = 2(x^2 - 6x + 9) + 5 \] 4. Abela tse 2 karolong e 'ngoe le e 'ngoe ea binomial: \[ f(x) = 2x^2 - 12x + 18 + 5 \] 5. Kopanya likarolo tsohle: \[ f(x) = 2x^2 - 12x + 23 \] Kahoo, sebopeho se tloaelehileng sa mosebetsi oa quadratic ke: \[ f(x) = 2x^2 - 12x + 23 \] Qetello Ho theha mesebetsi ea quadratic ho tsoa tlhahisoleseling e fapaneng ke bokhoni ba bohlokoa lipalo. Ka ho itloaetsa ka mokhoa o tsitsitseng ka mefuta e fapaneng ea mathata, re ka ntlafatsa kutloisiso ea rona le boqhetseke ba rona ho rarolleng li-quadratic equation. Lintlha tsa bohlokoa tseo re lokelang ho li hopola li kenyelletsa ho fumana le ho tseba mekhoa ea ho ntša tlhahisoleseling ho tsoa sebopehong sa vertex, ho fetola pakeng tsa vertex le sebopeho se tloaelehileng, le ho haha mesebetsi ho tsoa lintlheng tse fanoeng. Ka kutloisiso e tiileng ea lihlooho tsena, re ka sebetsana le liphephetso tse rarahaneng tsa lipalo nakong e tlang.