Te Hanga i ngā Mahi Tapawhā: He Aratohu Katoa
Pendahuluan
I roto i te pāngarau, he kaupapa matua ngā mahi tapawhā e hanga ana i te pūtake mō ētahi atu rangahau, tae atu ki te tātaitai me te ararau rārangi. Ka horapa atu te whakamahinga o ngā mahi tapawhā ki tua atu i te ariā, ā, ka kitea tōna ara ki roto i te whānuitanga o ngā tono mahi, mai i te ahupūngao me te miihini ki te ōhanga. Ka matapakihia e tēnei tuhinga te hanga taipitopito i ngā mahi tapawhā, tae atu ki ō rātou whakamāramatanga, te āhua whānui, ngā otinga pūtake, ngā kauwhata, me ngā tono.
Te Mārama ki ngā Mahi Tapawhā
He mahi pūrau tuarua te mahi pūrua, ka taea te whakaatu i te āhua whānui:
\[ f(x) = ax^2 + bx + c \]
ina ko \(a\), \(b\), me \(c\) he tauwehenga pumau, ā, mā \(a \neq 0\) ka whakarite kia tino rite te mahi ki te mahi tapawhā. Ko tēnei āhua te āhua paerewa o te mahi tapawhā.
Ngā Momo Rerekē o ngā Mahi Tapawhā
I mua i tā tātou haere tonu, he mea nui kia mārama he maha ngā huarahi hei whakaatu i tētahi mahi tapawhā i tua atu i te āhua whānui. Anei ētahi atu āhua e rua e whakamahia whānuitia ana:
1. Puka Whakawehewehe
Ka taea hoki te whakaatu i ngā mahi tapawhā i roto i te āhua tauwehe, ina koa mēnā e mōhiotia ana ngā pūtake:
\[ f(x) = a(x – x_1)(x – x_2) \]
ko \(x_1\) me \(x_2\) ngā pūtake o te mahi. He tino whai hua tēnei tikanga tauwehe ina mōhio kē tātou ki te otinga o te mahi.
2. Āhua o te Pito (Tihi)
Ka taea hoki te huri i te mahi tapawhā ki te āhua tihi, arā:
\[ f(x) = a(x – h)^2 + k \]
ko \((h, k)\) ngā taunga o te tihi o te parabola. He tino whai hua tēnei āhua ina hiahia tātou ki te mōhio ki te tūranga me te āhua taketake o te parabola.
Te Whakatau i ngā Mahi Tapawhā
Hei whakaoti, hei kimi rānei i ngā otinga (ngā pūtake) o te mahi tapawhā \(ax^2 + bx + c = 0\), ka taea e tātou te whakamahi i ētahi tikanga, tae atu ki te tauwehenga, te whakaoti i te tapawhā, me te tātai tapawhā.
1. Te Whakawehewehe
Ko te tikanga tauwehewehe ko te tuhi anō i te mahi tapawhā i runga i te hua o ngā tau rua-ira e rua:
\[ ax^2 + bx + c = a(x – x_1)(x – x_2) \]
Hei tauira, ka taea te whakauru i te mahi \(x^2 – 5x + 6 = 0\) ki roto i te \((x – 2)(x – 3) = 0\), nō reira ko ngā pūtake ko \(x = 2\) me \(x = 3\).
2. Te Whakaoti i te Tapawhā
Ko tēnei tikanga ko te tāpiri me te tango i tētahi uara hei huri i te āhua whānui ki tētahi āhua tapawhā tino tika:
1. Tīmata mai i te āhua whānui: \(ax^2 + bx + c\).
2. Wehea te katoa mā te \(a\) (mēnā ko \(a \neq 1\)).
3. Nukuhia te pūmau \(c/a\) ki te taha matau o te whārite.
4. Tāpirihia, tangohia hoki \((b/2a)^2\).
5. Tātaitia te taha maui, ka whakangawaritia te taha matau.
Hei tauira, mō te mahi \(x^2 + 6x + 8 = 0\):
\[x^2 + 6x = -8 \\
x^2 + 6x + 9 = 1
(x + 3)^2 = 1
x + 3 = \pm 1 \]
e homai ana ngā otinga \(x = -2\) me \(x = -4\).
3. Tātai Tapawhā
Ko te tātai tapawhā te huarahi tino noa, tino pono hoki ki te kimi i ngā pūtake o tētahi mahi tapawhā:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Mā te whakamahi i tēnei tātai, ka taea e tātou te kimi i ngā pūtake o tētahi mahi tapawhā, ahakoa kāore e taea te whakatau i te tauwehe, te whakaoti rānei i te tapawhā. Hei tauira, hei whakaoti i te \(2x^2 + 4x – 6 = 0\):
\[ x = \frac{-4 \pm \sqrt{4^2 – 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \\
x = \frac{-4 \pm \sqrt{16 + 48}}{4} \\
x = \frac{-4 \pm \sqrt{64}}{4} \\
x = \frac{-4 \pm 8}{4} \]
Nō reira, e rua ngā otinga ka puta: \(x = 1\) me \(x = -3\).
Kauwhata Mahi Tapawhā
He parabola te kauwhata o tētahi mahi tapawhā. Ka taea e tēnei parabola te whakatuwhera ki runga, ki raro rānei, i runga i te uara o te tauwehenga \(a\):
– Ki te nui ake te \(a > 0\), ka tuwhera te parabola ki runga.
– Mēnā ko \(a < 0\), ka tuwhera te parabola ki raro. 1. Te Puku me te Tuaka Āhuarite Ko te puku o te parabola (\(h, k\)) te pūwāhi mōrahi, mōrahi rānei o te mahi tapawhā. Ka taea te kimi i ngā taunga o te puku \(h\) mā te tātai: \[ h = \frac{-b}{2a} \] Hei whiwhi \(k\), ka whakakapia te uara o \(h\) ki roto i te mahi tapawhā \( f(h) = k \). Hei tauira, mō \(f(x) = 2x^2 - 4x + 1\): \[ h = \frac{-(-4)}{2 \cdot 2} = 1 \] Whakakapia \(x = 1\) ki roto i te mahi: \[ k = f(1) = 2(1)^2 - 4(1) + 1 = -1 \] Nō reira, ko te puku ko \((1, -1)\). 2. Tuaka Āhuarite Ko te tuaka āhuarite o te parabola ko te rārangi poutū e puta ana i te tihi: \[ x = h \] I te tauira i runga ake nei, ko te tuaka āhuarite ko \(x = 1\). 3. Te Kimi i ngā Āputa - Ka taea te kimi i ngā āputa-x (ngā pūtake) mā te whakaoti i te whārite tapawhā. - Ka whiwhihia ngā āputa-y mā te whakakapi i te \(x = 0\) ki roto i te mahi, ka puta ko \(y = c\). Ngā Whakamahinga o ngā Mahi Tapawhā Kāore i te mea he mea nui ngā mahi tapawhā i roto i ngā akomanga pāngarau anake, engari he whānui hoki ngā whakamahinga i te ao tūturu: 1. Ahupūngao I roto i te ahupūngao, he maha ngā wā ka puta ngā whārite tapawhā i roto i ngā ture nekehanga, pērā i te ara o te pere e whakaahuatia ana e te tātai: \[ y = ax^2 + bx + c \] e whakaahua ana i te nekehanga parabolic o tētahi mea i whiua. 2. Ōhanga me te Pūtea Ka whakamahia ngā mahi tapawhā mō te whakatauira pūtea, pērā i te kimi i te utu whakaputa iti rawa mō tētahi kamupene: \[ C(x) = ax^2 + bx + c \] 3. Hangarau Ā-iwi me te Hoahoa I roto i te hoahoa o ngā piriti me ētahi atu hanganga, ka whakamahia ngā parabola hei tātari me te hoahoa i ngā āwhata pakari. 4. Ko ngā rauropi arotautanga hangarau e whakamahia ana i roto i te ako mīhini he maha ngā wā ka uru ki te whakaiti i ngā mahi tapawhā. Whakamutunga He pūkenga nui, he pūkenga whai hua hoki te hanga mahi tapawhā i roto i ngā momo kaupapa ako. Mā te mārama ki te tuhi, te whakaoti rapanga, me te whakairoiro i ngā mahi tapawhā, me te whakamahi i ēnei ariā i roto i ngā āhuatanga mahi, ka taea e tātou te mārama ake me te whakamahi i ngā mātāpono taketake o te pāngarau ki te ao tūturu. Mā te whai i tētahi huarahi whānui ki te mārama ki ngā mahi tapawhā, ka whakatuwherahia e tātou te tatau ki tētahi māramatanga hohonu ake i roto i te whānuitanga o ngā mara ako me ngā tono.
