Tauira o ngā Pātai e Matapaki ana i te Hurihanga Mahi
He ariā matua ngā panonitanga mahi i roto i te pāngarau, inā koa i roto i te arapū me te tātari mahi. Kei roto i ēnei panonitanga ngā mahi maha pēnei i te whakawhiti, te whakaata, te whakawhānui, me te hurihanga i runga i te kauwhata o tētahi mahi. He pūkenga nui te mārama ki te mahi a ngā panonitanga mahi me te āhei ki te whakamahi i aua panonitanga ki ngā raruraru, i roto i ngā horopaki mātauranga me ngā mahi o ia rā.
Ka hipokina e tēnei tuhinga ētahi tauira o ngā raruraru whakawhiti mahi, me ngā whakamārama anō hoki, kia mārama ake ai te whakamahi i ēnei ariā. I mua i te ruku ki ngā tauira, me arotake tātou i ētahi momo whakawhiti mahi noa:
1. Whakamāoritanga (Nekehanga):
– Te nekehanga whakapae: \( f(x) \longrightarrow f(x – h) \) ko \( h \) te nui o te nekehanga ki te matau, ki te maui rānei.
– Te nekehanga poutū: \( f(x) \longrightarrow f(x) + k \) ko \( k \) te nui o te nekehanga ki runga, ki raro rānei.
2. Whakaaroaro (Whakaata):
– Te whakaata i te tuaka \( x \): \( f(x) \longrightarrow -f(x) \).
– Te whakaata i te tuaka \( y \): \( f(x) \longrightarrow f(-x) \).
3. Whakawhanui (Hurihanga o te Tauine):
– Te whakawhanuitanga whakapae: \( f(x) \longrightarrow f(cx) \) ko \( c \) te tauwehenga tauine whakapae.
– Te whakawhanui poutū: \( f(x) \longrightarrow af(x) \) ko \( a \) te tauwehenga tauine poutū.
I runga i tēnei māramatanga taketake, ka neke atu tātou ki ētahi tauira o ngā raruraru whakawhiti mahi.
Tauira Pātai 1: Whakamāoritanga Whakapae
Pātai: Hoatu te mahi \( f(x) = x^2 \). Whakatauhia te āhua o te mahi i muri i tōna whakamāoritanga kia 3 ngā waeine ki te taha matau.
Kōrero:
Ka nekehia te kauwhata o te mahi e te nekehanga whakapae i te tuaka \( x \ ). Ka whakamāoritia te nekehanga ki te taha matau mā te 3 waeine penei:
\[ f(x-3) \]
Nō reira, ka whakakapia ia \( x \) ki te \( x – 3 \) i te mahi taketake:
\[ f(x-3) = (x-3)^2 \]
Nō reira, ko te mahi i whakamāoritia ki te taha matau mā te 3 waeine ko:
\[ (x-3)^2 \]
Tauira Pātai 2: Whakamāoritanga Poutū
Pātai: Homai te mahi \( g(x) = \sqrt{x} \). Whakatauhia te āhua o te mahi i muri i tōna whakamāoritanga kia 4 ngā waeine ki runga.
Kōrero:
Ka nekehia te kauwhata o te mahi e te nekehanga poutū i te tuaka \( y \ ). Ka whakamāoritia te nekehanga ki runga mā te 4 waeine penei:
\[ g(x) + 4 \]
Nō reira, ko te mahi i muri i te whakamāoritanga ki runga ko:
\[ \sqrt{x} + 4 \]
Tauira Pātai 3: Te Whakaaroaro mō te Tuaka \( x \)
Pātai: I hoatu te mahi \( h(x) = \sin(x) \). Whakatauhia te āhua o te mahi i muri i tōna whakaata i te tuaka \( x \).
Kōrero:
Ka huri te tohu o te mahi mā te whakaata i te tuaka \( x \). Nō reira, ka whakareatia te mahi mā te -1:
\[ -h(x) \]
Nō reira, ko te mahi i muri i te whakaata i te tuaka \( x \) ko:
\[ -\sin(x) \]
Tauira Pātai 4: Whakaaroaro mō te Tuaka \( y \)
Pātai: I hoatu te mahi \( j(x) = e^x \). Whakatauhia te āhua o te mahi i muri i tōna whakaata i te tuaka \( y \).
Kōrero:
Ka hurihia te tohu o te taurangi \( x \) e te whakaata i te tuaka \( y \). Nō reira, ka whakakapia ia \( x \) ki te \( -x \):
\[ j(-x) \]
Nō reira, ko te mahi i muri i te whakaata i te tuaka \( y \) ko:
\[ e^{-x} \]
Tauira Pātai 5: Whakawhanuitanga Poutū
Pātai: Ki te hoatu te mahi \( f(x) = \cos(x) \). Whakatauhia te āhua o te mahi ina mahia he whakawhanui poutū me te tauwehe o te 2.
Kōrero:
Ko te whakawhanui poutū ko te whakarea i tētahi mahi mā te tauwehenga tauine poutū. Nō reira ka whakareatia te mahi mā te 2:
\[ 2f(x) \]
Nō reira, ko te mahi i muri i te whakawhanuitanga poutū mā te tauwehenga o te 2 ko:
\[ 2\cos(x) \]
Tauira Pātai 6: Te Huinga o ngā Whakamāoritanga Whakapae me te Whakamāoritanga Poutū
Pātai: Hoatu te mahi \( k(x) = \ln(x) \). Whakatauhia te āhua o te mahi i muri i tōna whakamāoritanga kia 2 ngā waeine ki te taha maui me te 3 ngā waeine ki raro.
Kōrero:
Tuatahi, ka whakamaoritia te whakamaoritanga o ngā waeine e 2 ki te taha maui hei \( k(x+2) \). Tuarua, ka whakamaoritia te whakamaoritanga o ngā waeine e 3 ki raro hei:
\[ k(x+2) – 3 \]
Nō reira, ko te mahi i muri i tēnei huinga whakamāoritanga ko:
\[ \ln(x+2) – 3 \]
Tauira Pātai 7: Te Huinga o te Whakaata me te Whakawhanui
Pātai: I hoatu te mahi \( m(x) = x^3 \). Whakatauhia te āhua o te mahi i muri i te whakaata i te tuaka \( y \) me te whakawhanui poutū me te tauwehenga o te 1/2.
Kōrero:
Tuatahi, ko te whakaata mō te tuaka \( y \) ka whakamaoritia hei \( m(-x) \). Tuarua, ko te whakawhānui poutū mā te tauwehenga o te 1/2 ka whakamaoritia hei:
\[ \frac{1}{2} m(-x) \]
Nō reira, ko te mahi i muri i tēnei huinga o te whakaata me te whakawhānui ko:
\[ \frac{1}{2}(-x)^3 = -\frac{1}{2} x^3 \]
Tauira Pātai 8: Whakawhanuitanga Whakapae
Pātai: Ki te hoatu te mahi \( n(x) = \tan(x) \). Whakatauhia te āhua o te mahi ina mahia he whakawhanuitanga whakapae me te tauwehenga o te 3.
Kōrero:
Ko te whakawhanuitanga whakapae ko te whakarea i te taurangi \( x \) ki te 1/c (ko \( c \) te tauwehenga tauine whakapae). Nō reira ka whakareatia e tātou te taurangi \( x \) ki te 1/3:
\[ n(\frac{x}{3}) \]
Nō reira, ko te mahi i muri i te whakawhanuitanga whakapae mā te tauwehenga o te 3 ko:
\[ \tan(\frac{x}{3}) \]
Te Katinga
He tino whai hua te mārama ki ngā panonitanga mahi, tae atu ki ngā whakawhiti, ngā whakaata, me ngā whakawhānui, i roto i te pāngarau me ōna tono. Mā te whakaharatau me te whakaoti rapanga maha e pā ana ki ngā panonitanga mahi, ka whakapakari koe i tō pūkenga ki te tiro me te matapae i ngā panonitanga o te āhua o ngā kauwhata mahi.
Kei roto i tēnei tuhinga ētahi tauira raruraru hei āwhina i a koe ki te ako mō ngā whakarerekētanga mahi. Whai muri i ia tauira raruraru ka whaihia he matapakinga taipitopito hei whakarite kia mārama koe ki ngā ariā. Mā te mahi tonu ki ngā momo raruraru ka āwhina i a koe kia matatau ake ki te mārama me te whakamahi i ngā whakarerekētanga mahi.
