Nā nīnau hoʻohālike e kūkākūkā ana i nā hana Quadratic
He kumuhana koʻikoʻi nā hana quadratic i kūkākūkā ʻia ma ka makemakika, ʻoi aku hoʻi ma ka makemakika lua. Loaʻa i kēia hana ke ʻano laulā \( f(x) = ax^2 + bx + c \), kahi \(a\), \(b\), a me \(c\) he mau kūpaʻa me \(a \neq 0\). E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike e pili ana i nā hana quadratic me nā wehewehe kikoʻī e kōkua i nā haumāna e hoʻomaopopo maikaʻi i kēia manaʻo.
1. Ke hoʻoholo nei i nā aʻa o kahi hana Quadratic
Nīnau 1: E huli i nā aʻa o ka hana quadratic ma lalo nei:
\[ f(x) = 2x^2 – 3x – 5 \]
Kūkākūkā:
No ka ʻike ʻana i nā aʻa o kahi hana quadratic, hiki iā mākou ke hoʻohana i ke ʻano quadratic, ʻo ia hoʻi:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Ma ka hana quadratic \( f(x) = 2x^2 – 3x – 5 \), hiki iā kākou ke ʻike i nā waiwai o \(a\), \(b\), a me \(c\):
– \( a = 2 \)
– \( b = -3 \)
– \( c = -5 \)
Eia nā ʻanuʻu:
1. E huli i ka mea hoʻokaʻawale (\( \Delta \)):
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-3)^2 – 4(2)(-5) \]
\[ \Delta = 9 + 40 \]
\[ \Delta = 49 \]
2. E hoʻohana i ke ʻano hana quadratic e ʻike ai i nā aʻa:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} \]
\[ x = \frac{3 \pm 7}{4} \]
No laila, loaʻa iā mākou ʻelua mau hopena:
\[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 \]
\[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 \]
No laila, ʻo nā aʻa o ka hana ʻo \( x = 2.5 \) a me \( x = -1 \).
2. Ke hoʻoholo nei i nā Vertices o kahi Hana Quadratic
Nīnau 2: E hoʻoholo i ka piko o ka hana quadratic aʻe:
g(x) = -x^2 + 4x – 3
Kūkākūkā:
Hiki ke hoʻoholo ʻia ka piko o kahi hana quadratic me ka hoʻohana ʻana i ke ʻano hana:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
Ma ka hana quadratic \( g(x) = -x^2 + 4x – 3 \), hiki iā kākou ke ʻike i nā waiwai o \(a\), \(b\), a me \(c\):
– \( a = -1 \)
– \( b = 4 \)
– \( c = -3 \)
Eia nā ʻanuʻu:
1. E huli i ka waiwai \( x \) o ka piko:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-4}{2(-1)} \]
\[ x_{\text{vertex}} = \frac{-4}{-2} \]
\[ x_{\text{vertex}} = 2 \]
2. E huli i ka waiwai o \( y \) ma ke pani ʻana iā \( x_{\text{vertex}} \) i loko o ka hana:
\[ y_{\text{vertex}} = g(2) \]
\[ y_{\text{vertex}} = – (2)^2 + 4(2) – 3 \]
\[ y_{\text{vertex}} = -4 + 8 – 3 \]
\[ y_{\text{vertex}} = 1 \]
No laila, ʻo ke kihi o ka hana ʻo \( (2, 1) \).
3. Ke kaha kiʻi ʻana i kahi pakuhi hana quadratic
Nīnau 3: E kaha kiʻi i ka pakuhi o ka hana quadratic ma lalo nei:
\[ h(x) = x^2 – 2x – 3 \]
Kūkākūkā:
Ma mua o ke kaha kiʻi ʻana i kahi pakuhi o kahi hana quadratic, pono mākou e ʻike i kekahi mau mea nui, e like me nā aʻa, ka piko, a me ke kuhikuhi o ka parabola.
Ke hoʻoholo nei i nā aʻa
Hiki iā kākou ke hoʻohana i ke ʻano hana quadratic e ʻike ai i nā aʻa o \( h(x) = x^2 – 2x – 3 \):
\[ a = 1 \]
\[ b = -2 \]
\[ c = -3 \]
1. E helu i ka hoʻokaʻawale:
\[ \Delta = b^2 – 4ac \]
\[ \Delta = (-2)^2 – 4(1)(-3) \]
\[ \Delta = 4 + 12 \]
\[ \Delta = 16 \]
2. E helu i nā aʻa:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-(-2) \pm \sqrt{16}}{2(1)} \]
\[ x = \frac{2 \pm 4}{2} \]
No laila, loaʻa iā mākou ʻelua mau hopena:
\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
\[ x_2 = \frac{2 – 4}{2} = \frac{-2}{2} = -1 \]
Ke hoʻoholo nei i ka piko kiʻekiʻe
3. E hoʻohana i ke ʻano vertex:
\[ x_{\text{vertex}} = \frac{-b}{2a} \]
\[ x_{\text{vertex}} = \frac{-(-2)}{2(1)} \]
\[ x_{\text{vertex}} = 1 \]
4. E helu i ka waiwai o \( y \):
\[ y_{\text{vertex}} = h(1) \]
\[ y_{\text{vertex}} = (1)^2 – 2(1) – 3 \]
\[ y_{\text{vertex}} = 1 – 2 – 3 \]
\[ y_{\text{vertex}} = -4 \]
No laila, ʻo ke kihi hope loa ʻo \( (1, -4) \).
Nā Kiʻi Kiʻi
– Aia nā aʻa ma \( x = 3 \) a me \( x = -1 \).
– Aia ke kihi ma \( (1, -4) \).
– No ka mea, ʻo \( a > 0 \), e wehe ana ka parabola i luna.
E kaha kiʻi i kēia mau kiko koʻikoʻi ma ka pakuhi a kahakiʻi i kahi parabola e hele ana ma waena o lākou.
Ma ka hoʻomaopopo ʻana i nā aʻa, ka piko, a me ke kuhikuhi o ka parabola, hiki iā kākou ke kahakiʻi i kahi pakuhi kūpono o kahi hana quadratic.
Ka hopena
He manaʻo nui ka hana quadratic i ka makemakika me nā noi ākea. ʻO ka hoʻomaopopo ʻana i nā hana quadratic e kōkua iā mākou e hoʻoikaika i ko mākou ʻike i nā manaʻo ʻē aʻe i ka makemakika a me nā ʻepekema i hoʻopili ʻia. Ma ka hoʻomaʻamaʻa ʻana me nā laʻana a me ka hoʻomaopopo ʻana i nā ʻanuʻu e hoʻoponopono ai iā lākou, ke manaʻolana ʻia e hohonu aʻe ko mākou ʻike i nā hana quadratic a lilo i mea pili pono.
