Mafunso Ofotokoza za Makhalidwe a Ntchito za Quadratic
Ntchito ya quadratic ndi mutu wofunikira kwambiri mu masamu womwe nthawi zambiri umapezeka mu maphunziro a sekondale. Mtundu wamba wa ntchito ya quadratic ndi \( f(x) = ax^2 + bx + c \), pomwe \( a \), \( b \), ndi \( c \) ndi zosasinthika ndi \( a \neq 0 \). Kukambirana za makhalidwe a ntchito za quadratic kumaphatikizapo mbali zosiyanasiyana monga axis of symmetry, vertex, maximum kapena minimum values, ndi malangizo a parabola. Nkhaniyi ikambirana za mavuto angapo a zitsanzo ndi mayankho awo kuti timvetse bwino makhalidwe a ntchito za quadratic.
1. Funso: Kudziwa Mzere wa Symmetry ndi Vertex
Chitsanzo cha mavuto:
Popeza ntchito ya quadratic \( f(x) = 2x^2 – 4x + 1 \). Dziwani mzere wa symmetry ndi vertex ya ntchitoyo.
Kukambirana:
Kuti tidziwe mzere wofanana wa ntchito ya quadratic \( ax^2 + bx + c \), timagwiritsa ntchito fomula iyi:
\[ x = -\frac{b}{2a} \]
Mu ntchito yoperekedwa \( f(x) = 2x^2 – 4x + 1 \), ma values \( a = 2 \) ndi \( b = -4 \). Sinthani ma values awa mu fomula iyi:
\[ x = -\frac{-4}{2 \cdot 2} \]
\[ x = \frac{4}{4} \]
\[x = 1 \]
Kotero, mzere wa kusinthasintha kwa ntchito ndi \( x = 1 \).
Kuti tipeze vertex, timasintha mtengo wa axis of symmetry kukhala ntchito:
\[ f(1) = 2(1)^2 – 4(1) + 1 \]
\[ f(1) = 2 – 4 + 1 \]
\[ f(1) = -1 \]
Kotero, vertex ya ntchito ndi \( (1, -1) \).
2. Funso: Kudziwa Njira ya Parabola
Chitsanzo cha mavuto:
Dziwani komwe parabola ya ntchito ya quadratic ikupita \( f(x) = -3x^2 + 6x – 2 \).
Kukambirana:
Kuwongolera kwa parabola ya ntchito ya quadratic kumatsimikiziridwa ndi mtengo wa coefficient \( a \).
– Ngati \( a > 0 \), parabola imatseguka mmwamba.
– Ngati \( a < 0 \), parabola imatseguka pansi.
Mu ntchito yoperekedwa \( f(x) = -3x^2 + 6x - 2 \), mtengo \( a = -3 \). Popeza \( a < 0 \), parabola imatseguka pansi.
3. Funso: Kupeza Mizu ya Ntchito ya Quadratic Chitsanzo Funso: Pezani mizu ya ntchito ya quadratic \( f(x) = x^2 - 5x + 6 \).
Kukambirana: Mizu ya ntchito ya quadratic ingapezeke mwa kuyika zinthu m'malo mwa factoring kapena kugwiritsa ntchito njira ya quadratic. Tidzawerengera izi: \[ x^2 - 5x + 6 = 0 \] Pezani manambala awiri omwe akachulukitsidwa amafanana ndi 6 ndipo akawonjezedwa amafanana ndi -5. Manambala ndi -2 ndi -3.
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \] Motero, mizu yake ndi iyi: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = 3 \] 4. Funso: Mtengo Waukulu Kapena Wochepa Chitsanzo Funso: Dziwani mtengo wochepa wa ntchito ya quadratic \( f(x) = 2x^2 - 4x + 5 \).
Kukambirana: Kuti tidziwe mtengo wocheperako wa ntchito ya quadratic \( ax^2 + bx + c \), tifunika kuwona ngati parabola imatseguka mmwamba kapena pansi. Ngati \( a > 0 \), parabola imatseguka mmwamba ndipo ili ndi mtengo wocheperako; ngati \( a < 0 \), parabola imatseguka pansi ndipo ili ndi mtengo wapamwamba kwambiri.
Mu ntchito yoperekedwa \( f(x) = 2x^2 - 4x + 5 \), mtengo wa \( a = 2 \), kotero parabola imatseguka mmwamba ndipo ili ndi mtengo wocheperako.
Mtengo wocheperako umapezeka pamalo okwera kwambiri. Tikudziwa kale mzere wa symmetry \( x = -\frac{b}{2a} \). Pa ntchito iyi: \[ x = -\frac{-4}{2 \cdot 2} \] \[ x = \frac{4}{4} \] \[ x = 1 \] Sinthani \( x = 1 \) mu ntchito kuti mupeze mtengo wocheperako: \[ f(1) = 2(1)^2 - 4(1) + 5 \] \[ f(1) = 2 - 4 + 5 \] \[ f(1) = 3 \] Chifukwa chake, mtengo wocheperako wa ntchitoyo ndi 3.
5. Funso: Chithunzi cha Ntchito ya Quadratic Chitsanzo Funso: Pangani chithunzi cha ntchito ya quadratic \( f(x) = -x^2 + 4x - 3 \).
Kukambirana: Kuti tijambule graph ya ntchito ya quadratic, tifunika kupeza makhalidwe angapo ofunikira monga mzere wa symmetry, vertex, ndi mizu ya ntchitoyo, komanso komwe parabola imalowera.
1. Axis of Symmetry: \[ x = -\frac{b}{2a} \] Mu ntchito \( f(x) = -x^2 + 4x - 3 \), ma values \( a = -1 \) ndi \( b = 4 \).
\[ x = -\frac{4}{2(-1)} \] \[ x = -\frac{4}{-2} \] \[ x = 2 \] 2. Vertex : Lowetsani \( x = 2 \) mu ntchito kuti mupeze vertex: \[ f(2) = -(2)^2 + 4(2) - 3 \] \[ f(2) = -4 + 8 - 3 \] \[ f(2) = 1 \] Vertex ndi \( (2, 1) \).
3. Njira ya Parabola: Popeza \( a = -1 \), parabola imatseguka pansi.
4. Mizu ya Ntchito ya Quadratic: \[ -x^2 + 4x - 3 = 0 \] Tingagwiritse ntchito njira ya quadratic: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Pa \( a = -1 \), \( b = 4 \), ndi \( c = -3 \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} \] \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \] \[ x = \frac{-4 \pm \sqrt{4}}{-2} \] \[ x = \frac{-4 \pm 2}{-2} \] Mayankho awiri: \[ x = \frac{-4 + 2}{-2} = 1 \] \[ x = \frac{-4 - 2}{-2} = 3 \] Mizu ndi \( x = 1 \) ndi \( x = 3 \).
Ndi chidziwitso chonsechi, titha kujambula ntchito ya quadratic. Parabola iyi ili ndi vertex pa \( (2, 1) \), imatseguka pansi, ndi mizu pa \( x = 1 \) ndi \( x = 3 \).
Kutsiliza Kudzera mu zitsanzo za mafunso omwe afotokozedwa, titha kumvetsetsa bwino makhalidwe osiyanasiyana a ntchito za quadratic. Kudziwa momwe mungadziwire mzere wa symmetry, vertex, mtengo wapamwamba kapena wocheperako, komwe parabola imatsogolera, ndi mizu ya ntchito ya quadratic ndikofunikira kwambiri pofotokoza mawonekedwe ndi makhalidwe a parabola. Kumvetsetsa bwino mfundo zimenezi kudzapereka maziko olimba kwa ophunzira kuti afufuze mitu ina ya masamu.
