Imibuzo Yezibonelo Exoxa Ngemisebenzi Yesikwele
Imisebenzi ye-quadratic iyisihloko esibalulekile okuxoxwa ngaso ezibalweni, ikakhulukazi ezibalweni zesibili. Lo msebenzi unesimo esijwayelekile \( f(x) = ax^2 + bx + c \), lapho \(a\), \(b\), kanye \(c\) kuyizinto ezingaguquki ezine-\(a \neq 0\). Lesi sihloko sizoxoxa ngezibonelo eziningana zezinkinga ezihlobene nemisebenzi ye-quadratic kanye nezincazelo ezinemininingwane ukusiza abafundi baqonde kangcono lo mqondo.
1. Ukunquma Izimpande Zomsebenzi We-Quadratic
Umbuzo 1: Thola izimpande zomsebenzi olandelayo we-quadratic:
\[ f(x) = 2x^2 – 3x – 5 \]
Ingxoxo:
Ukuze sithole izimpande zomsebenzi we-quadratic, singasebenzisa ifomula ye-quadratic, okungukuthi:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Kumsebenzi we-quadratic \( f(x) = 2x^2 – 3x – 5 \), singabona amanani ka-\(a\), \(b\), kanye no-\(c\):
– \( a = 2 \)
– \( b = -3 \)
– \( c = -5 \)
Izinyathelo zimi kanje:
1. Thola i-discriminant (\( \Delta \)):
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-3)^2 – 4(2)(-5) \]
\[ \Delta = 9 + 40 \]
\[ \Delta = 49 \]
2. Sebenzisa ifomula ye-quadratic ukuthola izimpande:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} \]
\[ x = \frac{3 \pm 7}{4} \]
Ngakho-ke sithola izixazululo ezimbili:
\[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \]
\[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 \]
Ngakho-ke izimpande zomsebenzi ziyi-\( x = 2.5 \) kanye ne-\( x = -1 \).
2. Ukunquma Ama-Vertices Omsebenzi We-Quadratic
Umbuzo 2: Thola i-vertex yomsebenzi olandelayo we-quadratic:
\[ g(x) = -x^2 + 4x – 3 \]
Ingxoxo:
I-vertex yomsebenzi we-quadratic inganqunywa kusetshenziswa ifomula:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
Kumsebenzi we-quadratic \( g(x) = -x^2 + 4x – 3 \), singabona amanani ka-\(a\), \(b\), kanye no-\(c\):
– \( a = -1 \)
– \( b = 4 \)
– \( c = -3 \)
Izinyathelo zimi kanje:
1. Thola inani \( x \) le-vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-4}{2(-1)} \]
\[ x_{\text{vertex}} = \frac{-4}{-2} \]
\[ x_{\text{vertex}} = 2 \]
2. Thola inani lika-\( y \) ngokufaka u-\( x_{\text{vertex}} \) emsebenzini:
\[ y_{\text{vertex}} = g(2) \]
\[ y_{\text{vertex}} = – (2)^2 + 4(2) – 3 \]
\[ y_{\text{vertex}} = -4 + 8 – 3 \]
\[ y_{\text{vertex}} = 1 \]
Ngakho-ke i-vertex yomsebenzi ingu-\( (2, 1) \).
3. Ukudweba Igrafu Yomsebenzi We-Quadratic
Umbuzo 3: Dweba igrafu yomsebenzi olandelayo we-quadratic:
\[ h(x) = x^2 – 2x – 3 \]
Ingxoxo:
Ngaphambi kokudweba igrafu yomsebenzi we-quadratic, sidinga ukwazi amaphuzu amaningana abalulekile, njengezimpande, i-vertex, kanye nesiqondiso se-parabola.
Ukunquma Izimpande
Singasebenzisa ifomula ye-quadratic ukuthola izimpande ze-\( h(x) = x^2 – 2x – 3 \):
\[ a = 1 \]
\[ b = -2 \]
\[ c = -3 \]
1. Bala umehluko:
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-2)^2 – 4(1)(-3) \]
\[ \Delta = 4 + 12 \]
\[ \Delta = 16 \]
2. Bala izimpande:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-2) \pm \sqrt{16}}{2(1)} \]
\[ x = \frac{2 \pm 4}{2} \]
Ngakho-ke sithola izixazululo ezimbili:
\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
\[ x_2 = \frac{2 – 4}{2} = \frac{-2}{2} = -1 \]
Ukunquma Iphuzu Eliphakeme
3. Sebenzisa ifomula ye-vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-(-2)}{2(1)} \]
\[ x_{\text{vertex}} = 1 \]
4. Bala inani le-\( y \):
\[ y_{\text{vertex}} = h(1) \]
\[ y_{\text{vertex}} = (1)^2 – 2(1) – 3 \]
\[ y_{\text{vertex}} = 1 – 2 – 3 \]
\[ y_{\text{vertex}} = -4 \]
Ngakho-ke i-vertex ingu-\( (1, -4) \).
Amagrafu Okudweba
– Izimpande ziku-\( x = 3 \) kanye ne-\( x = -1 \).
– I-vertex iku-\( (1, -4) \).
– Kusukela ku-\( a > 0 \), i-parabola ivuleka phezulu.
Bhala la maphuzu abalulekile kugrafu bese udweba ipharabola edlula kuwo.
Ngokuqonda izimpande, i-vertex, kanye nesiqondiso se-parabola, singadweba igrafu ephelele kakhulu yomsebenzi we-quadratic.
Isiphetho
Umsebenzi we-quadratic ungumqondo oyisisekelo kwizibalo onezinhlelo zokusebenza ezibanzi. Ukuqonda imisebenzi ye-quadratic kusisiza sijulise ukuqonda kwethu eminye imiqondo kwizibalo kanye nesayensi esetshenziswayo. Ngokuzijwayeza ngezibonelo nokuqonda izinyathelo zokuzixazulula, kunethemba lokuthi ukuqonda kwethu imisebenzi ye-quadratic kuzojula futhi kusebenze kakhulu.
