Ngā Tauira Pātai e Matapaki ana i ngā Āhuatanga o ngā Mahi Tapawhā
He kaupapa nui te mahi tapawhā i roto i te pāngarau e kitea pinepinetia ana i roto i te marautanga kura tuarua. Ko te āhua whānui o te mahi tapawhā ko \( f(x) = ax^2 + bx + c \), ko \( a \), \( b \), me \( c \) he pūmau me \( a \neq 0 \). Kei roto i te matapakinga o ngā āhuatanga o ngā mahi tapawhā ngā āhuatanga maha pēnei i te tuaka o te ōritetanga, te tihi, ngā uara mōrahi, mōkito rānei, me te ahunga o te parabola. Ka matapakihia e tēnei tuhinga ētahi tauira raruraru me ā rātou otinga hei mārama ake i ngā āhuatanga o ngā mahi tapawhā.
1. Pātai: Te Whakatau i te Tuaka o te Āhuarite me te Pito
Tauira raruraru:
Hoatu he mahi tapawhā \( f(x) = 2x^2 – 4x + 1 \). Whakatauhia te tuaka ōrite me te tihi o te mahi.
Kōrero:
Hei whakatau i te tuaka ōrite o te mahi tapawhā \( ax^2 + bx + c \), ka whakamahia e mātou te tātai:
\[ x = -\frac{b}{2a} \]
I roto i te mahi kua hoatu \( f(x) = 2x^2 – 4x + 1 \), ko ngā uara \( a = 2 \) me \( b = -4 \). Tāpirihia ēnei uara ki te tātai:
\[ x = -\frac{-4}{2 \cdot 2} \]
\[ x = \frac{4}{4} \]
\[ x = 1 \]
Nō reira, ko te tuaka ōrite o te mahi ko \( x = 1 \).
Hei kimi i te tihi, ka whakakapia te uara o te tuaka ōrite ki roto i te mahi:
\[ f(1) = 2(1)^2 – 4(1) + 1 \]
\[ f(1) = 2 – 4 + 1 \]
\[ f(1) = -1 \]
Nō reira, ko te pito o te mahi ko \( (1, -1) \).
2. Pātai: Te Whakatau i te Aronga o te Parabola
Tauira raruraru:
Whakatauhia te ahunga o te parabola o te mahi tapawhā \( f(x) = -3x^2 + 6x – 2 \).
Kōrero:
Ka whakatauhia te ahunga o te parabola o tētahi mahi tapawhā e te uara o te tauwehenga \( a \).
– Ki te mea ko te \( a > 0 \), ka tuwhera te parabola ki runga.
– Ki te mea ko te \( a < 0 \), ka tuwhera te parabola ki raro.
I roto i te mahi kua hoatu \( f(x) = -3x^2 + 6x - 2 \), ko te uara \( a = -3 \). Nā te mea ko \( a < 0 \), ka tuwhera te parabola ki raro.
3. Pātai: Te Kimi i ngā Pūtake o tētahi Mahi Tapawhā Tauira Pātai: Kimihia ngā pūtake o te mahi tapawhā \( f(x) = x^2 - 5x + 6 \).
Kōrero: Ka taea te kimi i ngā pūtake o tētahi mahi tapawhā mā te tauwehe, mā te whakamahi rānei i te tātai tapawhā. Ka tauwehea e tātou: \[ x^2 - 5x + 6 = 0 \] Kimihia ngā tau e rua, ina whakareatia ka ōrite ki te 6, ā, ina tāpirihia ka ōrite ki te -5. Ko ngā tau ko -2 me -3.
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \] Nō reira, ko ngā pūtake koia ēnei: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = 3 \] 4. Pātai: Uara Mōrahi, Uara Iti rānei Tauira Pātai: Whakatauhia te uara iti rawa o te mahi tapawhā \( f(x) = 2x^2 - 4x + 5 \).
Kōrero: Hei whakatau i te uara iti rawa o te mahi tapawhā \( ax^2 + bx + c \), me tirotiro tātou mēnā ka tuwhera te parabola ki runga, ki raro rānei. Mena ko te \( a > 0 \), ka tuwhera te parabola ki runga, ā, he iti rawa te uara; mena ko te \( a < 0 \), ka tuwhera te parabola ki raro, ā, he mōrahi te uara.
I roto i te mahi kua hoatu \( f(x) = 2x^2 - 4x + 5 \), ko te uara o \( a = 2 \), nō reira ka tuwhera te parabola ki runga, ā, he uara iti rawa tōna.
Ka puta te uara iti rawa i te pūwāhi tihi. E mōhio ana tātou ki te tuaka o te ōritetanga \( x = -\frac{b}{2a} \). Mō tēnei mahi: \[ x = -\frac{-4}{2 \cdot 2} \] \[ x = \frac{4}{4} \] \[ x = 1 \] Whakakapia \( x = 1 \) ki roto i te mahi hei kimi i te uara iti rawa: \[ f(1) = 2(1)^2 - 4(1) + 5 \] \[ f(1) = 2 - 4 + 5 \] \[ f(1) = 3 \] Nō reira, ko te uara iti rawa o te mahi ko te 3.
5. Pātai: Kauwhata o te Mahi Tapawhā Tauira Pātai: Waihangahia he kauwhata o te mahi tapawhā \( f(x) = -x^2 + 4x - 3 \).
Kōrero: Hei tuhi i tētahi kauwhata o tētahi mahi tapawhā, me kimi e tātou ētahi āhuatanga nui pērā i te tuaka ōrite, te tihi, me ngā pūtake o te mahi, tae atu ki te ahunga o te parabola.
1. Tuaka o te Āhuarite: \[ x = -\frac{b}{2a} \] I roto i te mahi \( f(x) = -x^2 + 4x - 3 \), ko ngā uara \( a = -1 \) me \( b = 4 \).
\[ x = -\frac{4}{2(-1)} \] \[ x = -\frac{4}{-2} \] \[ x = 2 \] 2. Te Tūranga: Whakakapia te \( x = 2 \) ki roto i te mahi hei kimi i te tūranga: \[ f(2) = -(2)^2 + 4(2) - 3 \] \[ f(2) = -4 + 8 - 3 \] \[ f(2) = 1 \] Ko te tūranga ko \( (2, 1) \).
3. Te Aronga o te Parabola: Mai i te mea \( a = -1 \), ka tuwhera te parabola ki raro.
4. Ngā Pūtake o tētahi Mahi Tapawhā: \[ -x^2 + 4x - 3 = 0 \] Ka taea e tātou te whakamahi i te tātai tapawhā: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Mō \( a = -1 \), \( b = 4 \), me \( c = -3 \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} \] \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \] \[ x = \frac{-4 \pm \sqrt{4}}{-2} \] \[ x = \frac{-4 \pm 2}{-2} \] E rua ngā otinga: \[ x = \frac{-4 + 2}{-2} = 1 \] \[ x = \frac{-4 - 2}{-2} = 3 \] Ko ngā pūtake ko \( x = 1 \) me \( x = 3 \).
Me ēnei mōhiotanga katoa, ka taea e tātou te tuhi i tētahi kauwhata o tētahi mahi tapawhā. Kei tēnei parapara tētahi pito kei \( (2, 1) \), e tuwhera ana ki raro, me ngā pakiaka kei \( x = 1 \) me \( x = 3 \).
Whakamutunga Mā roto i ngā tauira o ngā pātai kua matapakihia, ka taea e tātou te mārama ake ki ngā āhuatanga rerekē o ngā mahi tapawhā. He mea tino nui te mōhio ki te whakatau i te tuaka ōrite, te tihi, te uara mōrahi, mōkito rānei, te ahunga o te parabola, me ngā pakiaka o te mahi tapawhā hei whakaahua i te āhua me ngā āhuatanga o te parabola. Mā te mārama pai ki ēnei ariā ka waiho hei tūāpapa pakari mō ngā ākonga ki te tūhura i ngā kaupapa pāngarau i tua atu.
