Zitsanzo za Mafunso Okhudza Ntchito za Quadratic
Ntchito za quadratic ndi nkhani yofunika kwambiri yomwe imakambidwa mu masamu, makamaka mu masamu achiwiri. Ntchitoyi ili ndi mawonekedwe onse \( f(x) = ax^2 + bx + c \), pomwe \(a\), \(b\), ndi \(c\) ndi zosasinthika ndi \(a \neq 0\). Nkhaniyi ikambirana zitsanzo zingapo za mavuto okhudzana ndi ntchito za quadratic pamodzi ndi kufotokozera mwatsatanetsatane kuti athandize ophunzira kumvetsetsa bwino lingaliro ili.
1. Kudziwa Mizu ya Ntchito ya Quadratic
Funso 1: Pezani mizu ya ntchito yotsatirayi ya quadratic:
\[ f(x) = 2x^2 – 3x – 5 \]
Kukambirana:
Kuti tipeze mizu ya ntchito ya quadratic, tingagwiritse ntchito njira ya quadratic, yomwe ndi:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Mu ntchito ya quadratic \( f(x) = 2x^2 – 3x – 5 \), titha kuzindikira ma values a \(a\), \(b\), ndi \(c\):
– \( a = 2 \)
– \( b = -3 \)
– \( c = -5 \)
Masitepe ake ndi awa:
1. Pezani chosiyanitsa (\( \Delta \)):
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-3)^2 – 4(2)(-5) \]
\[ \Delta = 9 + 40 \]
\[ \Delta = 49 \]
2. Gwiritsani ntchito njira ya quadratic kuti mupeze mizu:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} \]
\[ x = \frac{3 \pm 7}{4} \]
Kotero tikupeza njira ziwiri:
\[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \]
\[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 \]
Kotero mizu ya ntchito ndi \( x = 2.5 \) ndi \( x = -1 \).
2. Kudziwa Ma Vertices a Ntchito ya Quadratic
Funso 2: Dziwani vertex ya ntchito yotsatirayi ya quadratic:
\[ g(x) = -x^2 + 4x – 3 \]
Kukambirana:
Mutu wa ntchito ya quadratic ukhoza kudziwika pogwiritsa ntchito fomula iyi:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
Mu ntchito ya quadratic \( g(x) = -x^2 + 4x – 3 \), titha kuzindikira ma values a \(a\), \(b\), ndi \(c\):
– \( a = -1 \)
– \( b = 4 \)
– \( c = -3 \)
Masitepe ake ndi awa:
1. Pezani mtengo \( x \) wa 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. Pezani mtengo wa \( y \) posintha \( x_{\text{vertex}} \) mu ntchito:
\[ y_{\text{vertex}} = g(2) \]
\[ y_{\text{vertex}} = – (2)^2 + 4(2) – 3 \]
\[ y_{\text{vertex}} = -4 + 8 – 3 \]
\[ y_{\text{vertex}} = 1 \]
Kotero vertex ya ntchito ndi \( (2, 1) \).
3. Kujambula Chithunzi cha Ntchito ya Quadratic
Funso 3: Jambulani graph ya ntchito yotsatirayi ya quadratic:
\[ h(x) = x^2 – 2x – 3 \]
Kukambirana:
Tisanajambule graph ya ntchito ya quadratic, tiyenera kudziwa mfundo zingapo zofunika, monga mizu, vertex, ndi komwe parabola imalowera.
Kudziwa Mizu
Tingagwiritse ntchito njira ya quadratic kuti tipeze mizu ya \( h(x) = x^2 – 2x – 3 \):
\[a = 1 \]
\[ b = -2 \]
\[ c = -3 \]
1. Werengerani chosiyanitsa:
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-2)^2 – 4(1)(-3) \]
\[ \Delta = 4 + 12 \]
\[ \Delta = 16 \]
2. Werengani mizu:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-2) \pm \sqrt{16}}{2(1)} \]
\[ x = \frac{2 \pm 4}{2} \]
Kotero tikupeza njira ziwiri:
\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
\[ x_2 = \frac{2 – 4}{2} = \frac{-2}{2} = -1 \]
Kudziwa Malo Ofunika Kwambiri
3. Gwiritsani ntchito njira ya vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-(-2)}{2(1)} \]
\[ x_{\text{vertex}} = 1 \]
4. Werengani mtengo wa \( y \):
\[ y_{\text{vertex}} = h(1) \]
\[ y_{\text{vertex}} = (1)^2 – 2(1) – 3 \]
\[ y_{\text{vertex}} = 1 – 2 – 3 \]
\[ y_{\text{vertex}} = -4 \]
Kotero vertex ndi \( (1, -4) \).
Zojambula za Magalafu
– Mizu ili pa \( x = 3 \) ndi \( x = -1 \).
– Mutu uli pa \( (1, -4) \).
– Popeza \( a > 0 \), parabola imatseguka mmwamba.
Lembani mfundo zofunika izi pa graph ndikujambula parabola yomwe ikudutsamo.
Mwa kumvetsetsa mizu, vertex, ndi komwe parabola ili, titha kujambula graph yabwino kwambiri ya ntchito ya quadratic.
Mapeto
Ntchito ya quadratic ndi mfundo yofunika kwambiri mu masamu yomwe imagwiritsidwa ntchito kwambiri. Kumvetsetsa ntchito za quadratic kumatithandiza kumvetsetsa bwino mfundo zina mu masamu ndi sayansi yogwiritsidwa ntchito. Mwa kuchita ndi zitsanzo ndikumvetsetsa njira zothetsera vutoli, tikuyembekeza kuti kumvetsetsa kwathu ntchito za quadratic kudzakula kwambiri ndikukhala koyenera.
