Ngā Tauira Pātai e Matapaki ana i ngā Huinga o ngā Huringa Mahi
He kaupapa nui ngā panonitanga mahi i roto i te pāngarau, inā koa i roto i te ako i ngā mahi me ā rātou kauwhata. Ko te whakamahinga o ngā panonitanga mahi e uru ana ki ngā mahi maha pēnei i te whakawhiti, te whakaata, te whakawhānui, me te hurihanga. I roto i tēnei tuhinga, ka tūhuratia e mātou he aha ngā panonitanga mahi whakakotahi me pēhea te whakaoti i aua panoni mā roto i ētahi tauira rapanga.
He aha te Huinga Whakawhitinga Mahi?
Ko te whakarerekētanga mahi he whakarerekē i te tūranga āhuahanga, i te āhua rānei o te kauwhata o te mahi taketake. Ko te whakakotahitanga o ngā whakarerekētanga mahi he whakakotahi i ngā whakarerekētanga e rua, neke atu rānei, o tētahi mahi kotahi.
Ko ētahi momo whakarerekētanga mahi noa ko:
– Whakamāoritanga (Nekehanga):
– Whakapae: \( f(x) \to f(x – h) \) neke ki te taha matau mā \( h \)
– Poutū: \( f(x) \ki f(x) + k \) neke ake i tētahi tawhiti \( k \)
– Whakaaroaro:
– Whakapae (e pā ana ki te tuaka-y): \( f(x) \to f(-x) \)
– Poutū (e pā ana ki te tuaka-x): \( f(x) \to -f(x) \)
– Whakawhanui (Whakatauine):
– Whakapae: \( f(x) \to f(ax) \) me \( a \) hei tauwehenga tauine whakapae
– Poutū: \( f(x) \to kf(x) \) me \( k \) hei tauwehenga tauine poutū
Ngā Pātai Tauira me te Kōrero
Pātai 1:
I runga i te mahi taketake \( f(x) = x^2 \). Whakatauhia te āhua hou o te mahi i muri i te whakamahinga o te huinga panoni e whai ake nei:
1. Te nekehanga whakapae ki te taha matau mā te 3 waeine.
2. Whakawhanui poutū me te tauine tauine o te 2.
Kōrero:
1. Whakamāoritanga Whakapae:
Ko te mahi \( f(x) = x^2 \) mēnā ka nekehia ki te taha matau mā te 3 waeine ka noho hei \( f(x – 3) = (x – 3)^2 \).
Nō reira, ko te mahi hou i muri i te whakamāoritanga whakapae ko \( f_1(x) = (x – 3)^2 \).
2. Whakawhanuitanga Poutū:
I muri i te whakawhanui poutū mā te tauwehenga o te 2, ka noho te āhua hei \( 2 \times f_1(x) = 2(x-3)^2 \).
Ko te hua whakamutunga o te mahi i muri i te whakakotahitanga o ngā panonitanga ko:
\[ g(x) = 2(x – 3)^2 \]
Pātai 2:
I te mea kua hoatu te mahi \( f(x) = \sqrt{x} \). Whakatauhia te āhua hou o te mahi i muri i te hoatutanga o te huinga o ngā panonitanga e whai ake nei:
1. Te whakaaroaro mō te tuaka-y.
2. Te nekehanga poutū ki raro mā te 2 waeine.
Kōrero:
1. Te whakaata i te tuaka-y:
Ka whakaatahia te mahi \( f(x) = \sqrt{x} \) ki te tuaka-y, nō reira ka noho ko \( f(-x) = \sqrt{-x} \).
2. Whakamāoritanga Poutū:
Kātahi ka nekehia te hua o te mahi whakaata ki raro mā te 2 waeine kia noho ko \( \sqrt{-x} – 2 \).
Nā, ko te āhua whakamutunga o te mahi i muri i te panonitanga ko:
\[ g(x) = \sqrt{-x} – 2 \]
Pātai 3:
I te mea kua hoatu te mahi \( f(x) = \frac{1}{x} \). Whakatauhia te āhua hou o te mahi i muri i te huinga o ngā panonitanga e whai ake nei:
1. Te nekehanga whakapae ki te taha maui mā te 4 waeine.
2. Whakawhanuitanga whakapae me te tauine tauine \(\frac{1}{2}\).
Kōrero:
1. Whakamāoritanga Whakapae:
Ko te mahi \( f(x) = \frac{1}{x} \) i muri i te neke ki te taha maui mā te 4 waeine ka huri hei \( f(x + 4) = \frac{1}{x + 4} \).
2. Whakawhanuitanga Whakapae:
Kātahi ka whakawhanuitia te mahi whakamāoritanga hua mā te tauwehe \(\frac{1}{2}\) kia rite ki te \( f\left( \frac{x}{\frac{1}{2}} + 4 \right) = f(2x + 4) = \frac{1}{2x + 4} \).
Nā, ko te āhua whakamutunga o te mahi i muri i te panonitanga ko:
\[ g(x) = \frac{1}{2x + 4} \]
Pātai 4:
I te mea kua hoatu te mahi \( f(x) = \sin x \). Whakatauhia te āhua hou o te mahi i muri i te hoatutanga o te huinga o ngā panonitanga e whai ake nei:
1. Whakawhanui poutū me te tauine tauine o te 3.
2. Te whakaaroaro mō te tuaka-x.
Kōrero:
1. Whakawhanuitanga Poutū:
Ko te mahi taketake \( f(x) = \sin x \) i muri i te whakawhanuitanga poutū me te tauwehenga tauine o te 3 ka noho hei \( 3 \sin x \).
2. Te whakaata i te tuaka-x:
Kātahi ka whakaatahia te hua o te mahi whakawhanui huri noa i te tuaka-x, kia noho ko \( -3 \sin x \).
Ko te hua whakamutunga o te mahi i muri i te whakakotahitanga o ngā panonitanga ko:
\[ g(x) = -3 \sin x \]
Te Whakamahinga i roto i ngā Whakairoiro
He mea tino nui anō hoki te mārama ki te huinga o ngā panonitanga mahi i roto i te ako i ngā kauwhata o aua mahi. Anei ētahi kōrero nui hei maumahara:
1. Raupapa Whakawhiti:
Ko te raupapatanga o ngā panonitanga ka pā ki te hua whakamutunga. Hei tauira, ki te whakahaerehia te whakawhanuitanga i mua i te whakamaoritanga, ka rerekē te hua whakamutunga i tērā mēnā i hurihia te raupapatanga o ngā panonitanga.
2. Whakaahuatanga whakairoiro:
Ka pāngia te āhua o te kauwhata e ia panonitanga i tētahi ara motuhake:
– Ka nekehia te kauwhata e te whakamāoritanga, engari kāore e whakarerekētia tōna āhua.
– Ka hurihia e te whakawhanuitanga te "whānui" te "ngaronga" rānei o te kauwhata.
– Ka whakaatahia te kauwhata e te whakaata huri noa i tētahi rārangi.
3. Te Whakaharatau Tonu:
He huarahi whai hua te tuhi kauwhata i ngā mahi i runga i ngā panonitanga hei mārama ki tēnei ariā. Ka taea e koe te whakamātau ki te tuhi kauwhata i ngā mahi kua hoatuhia i roto i ngā pātai kōrero i runga ake nei kia kite ai koe i te āhua o te panoni o ā rātou kauwhata.
Whakamutunga
He ariā taketake te whakarerekētanga mahi i roto i te pāngarau e whakamahia ana i roto i ngā momo mara, i te taha mātauranga me te taha mahi. Ko te ako i ngā huinga o ngā whakarerekētanga mahi me mārama ki ngā kaupapa matua o ia momo whakarerekētanga. Mā te mahi me ngā tauira, ka taea e tātou te matatau ake ki te tuhi me te tuhi kauwhata i ngā mahi. Mā te mahi tonu ka matatau ake koe ki te mārama ki te hurihanga o ngā mahi me ngā whakarerekētanga rerekē.
