Ngā Tauira Pātai e Matapaki ana i te Hanga i ngā Mahi Tapawhā
Ko te hanga i ngā mahi tapawhā he kaupapa matua i roto i te pāngarau e puta pinepine ana i roto i ngā marautanga pāngarau waenga me ngā marautanga matatau. He mea nui te mārama ki ngā mahi tapawhā nā te mea he maha ngā whakamahinga o ēnei i roto i ngā horopaki rerekē, pērā i te tātari raraunga, te whakatauira ahupūngao, me te ōhanga. I roto i tēnei tuhinga, ka matapakihia e mātou ngā tauira raruraru me pēhea te whakaoti i aua raruraru hei hanga i ngā mahi tapawhā.
Te Mārama ki ngā Mahi Tapawhā
Ko te mahi pūrua he mahi pūrua tuarua-tohu, ko tōna āhua whānui tēnei:
\[ f(x) = ax^2 + bx + c \]
ko \(a\), \(b\), me \(c\) he pūmau, ā, ko \(a \neq 0\).
Ko te kauwhata o tētahi mahi tapawhā he kōpiko e mōhiotia ana ko te parabola. He ōrite te āhua o ngā parabola, ā, he āhua e whakawhirinaki ana ki te tohu o te pūmau \(a\). Mena \(a > 0\), ka tuwhera te parabola ki runga. I tetahi atu taha, mena \(a < 0\), ka tuwhera te parabola ki raro. Ngā Huānga Hira o Ngā Mahi Tapawhā - Ngā Pūtake o tētahi whārite tapawhā: Ko ngā uara o \(x\) mōna \(f(x) = 0\), ka kitea mā te whakamahi i te tātai tapawhā \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). - Te Poutū: Ko te pūwāhi teitei rawa atu, te pūwāhi iti rawa rānei o te parabola, ka kitea mā te whakamahi i te tātai \((x, y)\) kei reira \(x = -\frac{b}{2a}\) me \(y = f(-\frac{b}{2a})\). - Tuaka ōrite: Ko te rārangi poutū e wehewehe ana i te parabola kia rua ngā wāhanga ōrite, kei \(x = -\frac{b}{2a}\).
Tauira Pātai 1: Te Tito i tētahi Mahi Tapawhā mai i ngā Pūwāhi e Toru Pātai: Whakatauhia te tātai mō te mahi tapawhā e tika ana mā roto i ngā pūwāhi (1, 2), (2, 5), me (3, 10). Otinga: 1. Ka tīmata tātou me te āhua whānui o te mahi tapawhā: \[ f(x) = ax^2 + bx + c \] 2. Tāpirihia te pūwāhi (1, 2) ki roto i te whārite: \[ a(1)^2 + b(1) + c = 2 \] \[ a + b + c = 2 \] (Whārite 1) 3. Tāpirihia te pūwāhi (2, 5) ki roto i te whārite: \[ a(2)^2 + b(2) + c = 5 \] \[ 4a + 2b + c = 5 \] (Whārite 2) 4. Tāpirihia te pūwāhi (3, 10) ki roto i te whārite: \[ a(3)^2 + b(3) + c = 10 \] \[ 9a + 3b + c = 10 \] (Whārite 3) 5. E toru ā tātou pūnaha whārite rārangi: \[ \begin{cases} a + b + c = 2 \\ 4a + 2b + c = 5 \\ 9a + 3b + c = 10 \\ \end{cases} \] 6. Hei whakaoti, ka tangohia e tātou ngā whārite tuarua me te tuatahi: \[ (4a + 2b + c) - (a + b + c) = 5 - 2 \] \[ 3a + b = 3 \] (Whārite 4) 7. Tangohia ngā whārite tuatoru me te tuarua: \[ (9a + 3b + c) - (4a + 2b + c) = 10 - 5 \] \[ 5a + b = 5 \] (Whārite 5) 8. Tangohia te Whārite 5 me te Whārite 4: \[ (5a + b) - (3a + b) = 5 - 3 \] \[ 2a = 2 \] \[ a = 1 \] 9. Tāpirihia te \(a = 1\) ki te Whārite 4: \[ 3(1) + b = 3 \] \[ 3 + b = 3 \] \[ b = 0 \] 10. Tāpirihia te \(a = 1\) me te \(b = 0\) ki te Whārite 1: \[ 1 + 0 + c = 2 \] \[ c = 1 \] 11. Nō reira, ko te mahi pūrua ko: \[ f(x) = 1x^2 + 0x + 1 \] \[ f(x) = x^2 + 1 \] Tauira Pātai 2: Te Whakatau i tētahi Mahi Tapawhā mai i tētahi Poutū me tētahi Pūwāhi kē atu Pātai: Whakatauhia te tātai mō tētahi mahi tapawhā he poutū kei (-1, 4) ā, e haere ana i roto i te pūwāhi (1, 0). Otinga: 1. Ko te āhua paerewa o tētahi mahi tapawhā me te uho \((h, k)\) ko: \[ f(x) = a(x - h)^2 + k \] 2. Whakakapia te uho (-1, 4) ki te āhua paerewa: \[ f(x) = a(x + 1)^2 + 4 \] 3. Whakakapia te pūwāhi (1, 0) ki te whārite hei kimi \(a\): \[ 0 = a(1 + 1)^2 + 4 \] \[ 0 = a(2)^2 + 4 \] \[ 0 = 4a + 4 \] \[ 4a = -4 \] \[ a = -1 \] 4. Nō reira, ko te mahi tapawhā koia tēnei: \[ f(x) = -1(x + 1)^2 + 4 \] \[ f(x) = - (x + 1)^2 + 4 \] 5. Te tohatoha mō te āhua paerewa: \[ f(x) = - (x^2 + 2x + 1) + 4 \] \[ f(x) = -x^2 - 2x - 1 + 4 \] \[ f(x) = -x^2 - 2x + 3 \] Tauira Pātai 3: Te Tahuri i te Āhua Poutū ki te Āhua Paerewa Pātai: Tahurihia te mahi tapawhā \( f(x) = 2(x - 3)^2 + 5 \) ki te āhua paerewa \( ax^2 + bx + c \). Otinga: 1. Tuatahi, me whakawhanui tātou: \[ f(x) = 2(x - 3)^2 + 5 \] 2. Whakawhanuihia te rua-ira: \[ (x - 3)^2 = x^2 - 6x + 9 \] 3. Whakahokia ki roto i te mahi: \[ f(x) = 2(x^2 - 6x + 9) + 5 \] 4. Tohatohahia te 2 ki ia wāhanga o te rua-ira: \[ f(x) = 2x^2 - 12x + 18 + 5 \] 5. Whakakotahitia ngā wāhanga katoa: \[ f(x) = 2x^2 - 12x + 23 \] Nō reira, ko te āhua paerewa o te mahi rua-ira ko: \[ f(x) = 2x^2 - 12x + 23 \] Whakamutunga He pūkenga nui i roto i te pāngarau te hanga mahi rua-ira mai i ngā momo mōhiohio. Mā te mahi tonu me ngā momo raruraru maha, ka taea e tātou te whakapai ake i tō tātou māramatanga me te matatau ki te whakaoti rapanga whārite. Ko ngā mea matua hei maumahara ko te kimi me te matatau ki ngā tikanga mō te tango mōhiohio mai i te āhua o te tihi, te huri i waenga i te tihi me te āhua paerewa, me te hanga mahi mai i ngā pūwāhi kua hoatu. Mā te māramatanga pakari ki ēnei kaupapa, ka taea e tātou te aro atu ki ngā wero pāngarau uaua ake ā muri ake nei.
