Ngā Tauira Pātai Kōrero Te Whakaoti Rapanga me ngā Mahi Tapawhā
I roto i tēnei tuhinga, ka ako tātou me pēhea te whakaoti rapanga mā te whakamahi i ngā mahi tapawhā mā te whakarato tauira me ngā kaupae matapaki taipitopito. Ko te mahi tapawhā he mahi pūrau tuarua-tohu e whai ana i te āhua whānui \( ax^2 + bx + c \), ko \( a \), \( b \), me \( c \) he pūmau, ā, ko \( a \neq 0 \). He maha ngā wā ka puta ngā mahi tapawhā i roto i ngā horopaki rerekē i roto i te ahupūngao, te ōhanga, me te hangarau, ā, he kaupapa tino nui hei ako.
Me tīmata tātou mā te matapaki i ētahi ariā taketake, kātahi ka uru atu ki ētahi tauira rapanga.
Ngā Ariā Taketake o ngā Mahi Tapawhā
1. Āhua Whānui: Ka whakaatuhia te mahi tapawhā penei \( f(x) = ax^2 + bx + c \).
2. Pūtake Tapawhā: Ka kitea ngā pūtake o te whārite tapawhā \( ax^2 + bx + c = 0 \) mā te whakamahi i te tātai tapawhā, arā:
\[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
\]
3. Whakawehewehe: Ko te whakawehewehe o tētahi whārite tapawhā ko \( D = b^2 – 4ac \). Ko te uara o te whakawehewehe e tohu ana i te āhua o ngā pūtake o te whārite tapawhā:
– Mēnā ko \( D > 0 \), e rua ōna pūtake tūturu motuhake.
– Mēnā ko \( D = 0 \), kotahi tōna pūtake tūturu (pūtake māhanga).
– Mēnā ko \( D < 0 \), e rua ōna pūtake matatini honohono. 4. Te Puku o te Parabola: Ka kitea ngā taunga o te piko o te parabola i hangaia e te mahi tapawhā mā te whakamahi i te tātai: \[ x = -\frac{b}{2a} \] Mō te uara o \( y \) i te piko, ka taea te tatau mā te whakakapi i te \( x \) ki te mahi tapawhā.
– Ki te \( a < 0 \), ka tuwhera te parabola ki raro. Me ēnei ariā taketake katoa i roto i te hinengaro, me titiro tātou me pēhea te whakamahi i ēnei ki te whakaoti rapanga. Tauira Rapanga 1: Te Kimi i ngā Pūtake o tētahi Mahi Tapawhā Rapanga: Kimihia ngā pūtake o te whārite tapawhā \( 2x^2 - 3x - 2 = 0 \). Otinga: Hei kimi i ngā pūtake o tētahi whārite tapawhā, ka taea e tātou te whakamahi i te tātai tapawhā. Ko ngā mahi e whai ake nei: 1. Tāutuhia ngā tauwehenga \( a \), \( b \), me \( c \): \[ a = 2, \quad b = -3, \quad c = -2 \] 2. Tātaihia te wehewehe: \[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25 \] 3. Mai i te \( D > 0 \), ka rua ā tātou pūtake tūturu motuhake. Haere tonu mā te tatau i ēnei pūtake:
\[
x_{1,2} = \frac{-(-3) \pm \sqrt{25}}{2 \cdot 2} = \frac{3 \pm 5}{4}
\]
4. Tātaihia ngā uara e rua o \( x \):
\[
x_1 = \frac{3 + 5}{4} = 2 \quad \text{and} \quad x_2 = \frac{3 – 5}{4} = -\frac{1}{2}
\]
Nō reira, ko ngā pūtake o te whārite \( 2x^2 – 3x – 2 = 0 \) ko \( x = 2 \) me \( x = -\frac{1}{2} \).
Tauira Pātai 2: Te Kimi i ngā Taunga o te Puku o te Parabola
Pātai:
Kimihia ngā taunga o te tihi o te mahi tapawhā \( f(x) = 3x^2 – 6x + 2 \).
Kōrero:
Hei kimi i ngā taunga o te tihi, whakamahia te tātai taunga tihi:
1. Tāutuhia ngā taunga tau \( a \) me \( b \):
\[
a = 3, \quad b = -6
\]
2. Tātaihia te \( x \) kei runga:
\[
x = -\frac{b}{2a} = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
3. Tātaihia te \( y \) mā te whakakapi i te \( x = 1 \) ki roto i te mahi \( f(x) \):
\[
f(1) = 3(1)^2 – 6(1) + 2 = 3 – 6 + 2 = -1
\]
Nō reira, ko ngā taunga o te tihi o te mahi \( f(x) = 3x^2 – 6x + 2 \) ko \( (1, -1) \).
Tauira Pātai 3: Te Whakatau i te Aronga Tīmatanga o te Parabola
Pātai:
Whakatauhia te ahunga o te whakatuwheratanga o te parabola o te mahi tapawhā \( f(x) = -x^2 + 4x – 7 \).
Kōrero:
Hei whakatau i te ahunga o te whakatuwheratanga o te parabola, ka titiro noa tātou ki te tohu o te tauwehenga \( a \):
1. Tāutuhia te tauwehenga \( a \):
\[
a = -1
\]
2. Nā te mea \( a < 0 \), ka tuwhera te parabola ki raro. Nō reira, ko te ahunga o te whakatuwheratanga o te parabola o te mahi \( f(x) = -x^2 + 4x - 7 \) kei raro. Tauira 4: Te Whakamahi i ngā Mahi Tapawhā i roto i ngā Horopaki Tūturu
Pātai: Ka whiua he pōro mai i te whenua me te whārite tapawhā \( h(t) = -5t^2 + 20t \), ko \( h \) te teitei o te pōro i roto i ngā mita, ā, ko \( t \) te wā i roto i ngā hēkona. Kia pēhea te roa ka eke te pōro ki tōna teitei mōrahi, ā, he aha tōna teitei mōrahi? Kōrero: 1. Kimihia te wā e tae atu ai te teitei mōrahi (ngā taunga o te tihi): \[ a = -5, \quad b = 20 \] \[ t = -\frac{b}{2a} = -\frac{20}{2(-5)} = \frac{20}{10} = 2 \quad \text{hēkona} \] 2. Tātaihia te teitei mōrahi mā te whakakapi i te \( t \) ki roto i te whārite \( h(t) \): \[ h(2) = -5(2)^2 + 20(2) = -5(4) + 40 = -20 + 40 = 20 \quad \text{mita} \] Nō reira, ko te wā i pau i te pōro ki te tae atu ki te teitei mōrahi he 2 hēkona, ā, ko tōna teitei mōrahi he 20 mita. Whakamutunga I roto i tēnei tuhinga, kua matapakihia e mātou ngā āhuatanga nui o ngā mahi tapawhā me te pehea hoki te whakaoti rapanga e pā ana ki ngā mahi tapawhā mā roto i ētahi tauira. Te matapaki i ngā pūtake o ngā whārite tapawhā, te kimi i ngā taunga o te tihi, te whakatau i te ahunga o te whakatuwheratanga o te parabola, me te whakamahi i ngā mahi tapawhā i roto i ngā horopaki o te ao tūturu, pērā i te whakaahua i te nekehanga o ngā mea. Mā te māramatanga pakari ki ēnei ariā taketake, ka taea e koe te whakatata atu ki ngā raruraru pāngarau me te pūtaiao e pā ana ki ngā mahi tapawhā me te māia ake. Ehara i te mea he mea nui noa iho ngā mahi tapawhā i roto i te ariā engari he tino whai hua hoki i roto i ngā tono o te ao tūturu me te whakaoti rapanga puta noa i te whānuitanga o ngā mara.