Āhua Tūturu o te Whārite Tapawhā
Ko ngā whārite tapawhā tētahi o ngā kaupapa tino nui o te kaupapa arapū, e puta pinepine ana i roto i te pāngarau kura me ngā tono i roto i te pūtaiao, te ōhanga, me te hangarau. I te nuinga o te wā, ko te whārite tapawhā he whārite pūronomial tuarua-tohu ka taea te tuhi penei:
\[
ax^2 + bx + c = 0
\]
ko \(a \neq 0\), me \(a\), \(b\), me \(c\) he tau tūturu (he tau matatini rānei, i runga i te horopaki). Ahakoa e whakaurua pinepinetia ana tēnei āhua whānui, tera anō tētahi atu āhua e tino whai hua ana mō te mārama ki ngā āhuatanga o ngā whārite tapawhā, arā, ko te āhua ā-ture. Mā te āhua ā-ture tātou e āwhina ki te "pānui" i ngā āhuatanga o tētahi parabola—pērā i te tihi, ngā uara mōrahi/iti rawa, me te tuaka o te ōritetanga—kia tere ake, kia mārama ake hoki.
He aha te āhua o te ture ā-ture?
Ko te āhua matua (e kiia ana hoki ko te āhua tihi) o tētahi mahi tapawhā ko:
\[
y = a(xh)^2 + k
\]
me:
– Ka whakatau a \(a\) i te ahunga me te "piko" o te parabola,
– Ko \((h, k)\) ngā taunga o te tihi o te parabola.
Mena he whārite tapawhā te mea e kōrerohia ana (ehara i te mahi), ka taea te tuhi i te āhua:
\[
a(xh)^2 + k = 0
\]
ka whakawhitia rānei ki te puka mahi mēnā e tika ana. Ka kiia tēnei puka he āhua tūturu nā te mea e whakarato ana i te whakaaturanga tino whai kōrero mō te āhua o te kauwhata me te whanonga o ngā uara mahi.
He aha te hiranga o te āhua o te pukapuka?
He maha ngā take e tino whai hua ai ngā āhua tūāpapa:
1. Whakatauhia te tihi mā te ngāwari
I te āhua whānui \(ax^2+bx+c\), me tatau tuatahi tātou \(x_p = -\frac{b}{2a}\) kia kitea ai te tihi. Heoi, i te āhua matua \(a(xh)^2+k\), ka kitea tonutia te tihi, arā, \((h, k)\).
2. Kia mōhio ki te uara mōrahi/iti rawa
Mena ko te \(a>0\), ka tuwhera te parabola ki runga kia noho ko te tihi te uara iti rawa. Mena ko te \(a<0\), ka tuwhera te parabola ki raro kia noho ko te tihi te uara mōrahi. Ko te uara tino nui ko te \(k\). 3. Ka māmā ake te tuhi kauwhata Mā te mōhio ki te tihi me te ahunga o te whakatuwheratanga o te parabola, ka taea e tātou te tuhi kauwhata tere ake, tae atu ki te whakatau i te tuaka o te ōritetanga \(x=h\). 4. Ka āwhina ki te whakaoti i ngā whārite tapawhā I ētahi wā, ka tere ake te whakaoti i te \(ax^2+bx+c=0\) mena ka hurihia tuatahitia hei āhua tapawhā tino tika mā te āhua matua. Me pēhea te huri i te āhua whānui ki te āhua matua Ko te huri i te \(ax^2+bx+c\) ki te \(a(xh)^2+k\) ka mahia mā te tikanga whakaoti i te tapawhā (te whakaoti i te tapawhā). Ko ngā mahi e whai ake nei: E hoatu ana: \[ y = ax^2 + bx + c \] Hipanga 1: Whakakotahitia \(a\) mai i ngā kupu kei roto ko \(x\) \[ y = a\left(x^2 + \frac{b}{a}x\right) + c \] Hipanga 2: Tāpirihia, tangohia hoki ngā tau ōrite i roto i ngā pūwero hei hanga i tētahi tapawhā tino tika Hei hanga i te \(x^2 + \frac{b}{a}x\) ki te āhua \((x+p)^2\), ka tangohia e tātou: \[ p = \frac{1}{2}\cdot \frac{b}{a} = \frac{b}{2a} \] Tāpirihia, tangohia hoki \(p^2\): \[ y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Hipanga 3: Rōpūhia hei tapawhā tino tika \[ y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \] Hipanga 4: Horahia \(a\) ka whakangawari \[ y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c \] Nā te mea: \[ a\left(\frac{b}{2a}\right)^2 = a\cdot \frac{b^2}{4a^2} = \frac{b^2}{4a} \] Kātahi: \[ y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \] Koinei te āhua matua me: \[ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} \] Kia mōhio koe ko \(h\) e rite ana ki te tātai mō te tuaka o te ōritetanga, ko \(k\) ia e homai ana i te uara o te mahi i te tihi. Tauira o te huri ki te āhua matua Hei tauira: \[ y = 2x^2 - 8x + 3 \] Hipanga 1: Tauwehea te 2 mai i ngā kupu tuatahi e rua \[ y = 2(x^2 - 4x) + 3 \] Hipanga 2: Whakaotia te tapawhā i roto i ngā pareneti Tangohia te haurua o \(-4\), arā, \(-2\), kātahi ka tapawhāhia kia whiwhi \(4\): \[ y = 2(x^2 - 4x + 4 - 4) + 3 \] Hipanga 3: Āhua tapawhā tino tika \[ y = 2((x-2)^2 - 4) + 3 \] Hipanga 4: Whakangawaritia \[ y = 2(x-2)^2 - 8 + 3 \] \[ y = 2(x-2)^2 - 5 \] Nō reira, ko te āhua matua ko: \[ y = 2(x-2)^2 - 5 \] Mai i konei ka mōhio tonu tātou ko te tihi ko \((2, -5)\), ko te tuaka o te ōritetanga ko \(x=2\), ka tuwhera te parabola ki runga (nā te mea \(a=2>0\)), ā, ko te uara iti rawa o te mahi ko \(-5\).
Te whanaungatanga i waenga i te āhua matua me ngā pūtake o te whārite
Ki te hiahia tātou ki te kimi i ngā pūtake o tētahi whārite tapawhā:
\[
ax^2+bx+c=0
\]
Ka taea e tātou te huri hei āhua ā-ture:
\[
a(xh)^2 + k = 0
\]
Nā reira:
\[
a(xh)^2 = -k
\]
\[
(xh)^2 = -\frac{k}{a}
\]
Kātahi:
\[
xh = \pm \sqrt{-\frac{k}{a}}
\]
\[
x = h \pm \sqrt{-\frac{k}{a}}
\]
Mai i tēnei ka kitea he pūtake tūturu kei te noho mēnā:
\[
-\frac{k}{a} \ge 0
\]
e hangai ana ki te ariā o te wehewehe. Mā te wehewehe \(D = b^2-4ac\) e whakatau mēnā e rua ngā pakiaka tūturu, kotahi te pakiaka māhanga, kāore rānei he pakiaka tūturu. I roto i te āhua matua, ka puta noa mai tēnei āhuatanga mā te tohu o te kīanga i roto i te pakiaka.
Ngā āhua ā-tikanga me te mārama ki ngā kauwhata
He parabola te kauwhata o tētahi mahi tapawhā. Me te āhua matua:
\[
y = a(xh)^2 + k
\]
ka taea e tātou te mārama ki te panonitanga o te parabola paerewa \(y=x^2\):
– Ka nekehia e te \(h\) te kauwhata ki te taha matau (mēnā \(h>0\)) ki te taha maui rānei (mēnā \(h<0\)), - Ka nekehia e te \(k\) te kauwhata ki runga (mēnā \(k>0\)) ki raro rānei (mēnā \(k<0\)), - Ka totoro, ka pēhi rānei te parabola e te \(a\) ka whakatau i te ahunga o te whakatuwheratanga (ki runga mēnā \(a>0\), ki raro mēnā \(a<0\)). Nō reira, ehara i te mea he taputapu tatau noa te āhua matua, engari he taputapu tirohanga hoki mō te "pānui" i te whanonga o te mahi. Whakamutunga He whakaaturanga tino whai kōrero te āhua matua o te whārite, o te mahi rānei tapawhā, arā, \(y = a(xh)^2 + k\), nā te mea e whakaatu tonu ana i te tihi o \((h,k)\), te tuaka o te ōritetanga, me te uara mōrahi, mōkito rānei. Ka riro mai tēnei āhua mai i te āhua whānui \(ax^2+bx+c\) mā te tikanga whakaoti i te tapawhā. Haunga te āwhina ki te whakatakoto i ngā parabola, ka whakangāwari ake hoki te āhua ā-ture i te tātari i ngā pūtake me ngā āhuatanga o ngā whārite tapawhā. Nā reira, ko te mārama ki te āhua ā-ture he taahiraa nui ki te matatau ki te arapū me ngā whakamahinga o ngā whārite tapawhā i roto i ngā momo mara.
