Ke Kūkulu ʻana i nā Hana Quadratic: He Alakaʻi Piha
Pendahuluan
I ka makemakika, he kumuhana koʻikoʻi nā hana quadratic e hoʻokumu pinepine ana i ke kumu no ke aʻo hou ʻana, me ka calculus a me ka algebra linear. ʻOi aku ka hoʻohana ʻana i nā hana quadratic ma mua o ke kumumanaʻo a loaʻa i kona ala i loko o kahi ākea o nā noi hana, mai ka physics a me ka ʻenekinia mechanical a hiki i ka hoʻokele waiwai. E kūkākūkā kēia ʻatikala pehea e kūkulu ai i nā hana quadratic me ka kikoʻī, me kā lākou wehewehe ʻana, ke ʻano laulā, nā hopena aʻa, nā kiʻi, a me nā noi.
Ke Hoʻomaopopo ʻana i nā Hana Quadratic
He hana polynomial kekelē lua ka hana quadratic, hiki ke hōʻike ʻia ma ke ʻano laulā:
f(x) = ax^2 + bx + c
kahi ʻo \(a\), \(b\), a me \(c\) he mau coefficients mau, a ʻo \(a \neq 0\) e hōʻoiaʻiʻo ana he hana quadratic maoli ka hana. ʻO kēia ʻano ke ʻano maʻamau o kahi hana quadratic.
Nā ʻAno ʻē aʻe o nā Hana Quadratic
Ma mua o ko kākou hele ʻana aku, he mea nui e hoʻomaopopo he nui nā ʻano e hōʻike ai i kahi hana quadratic ma waho aʻe o ke ʻano maʻamau. Eia ʻelua mau ʻano i hoʻohana pinepine ʻia:
1. Palapala Hoʻohālikelike
Hiki ke hōʻike ʻia nā hana quadratic ma ke ʻano factorized, ʻoiai inā ʻike ʻia nā aʻa:
f(x) = a(x – x_1)(x – x_2)
kahi ʻo \(x_1\) a me \(x_2\) nā aʻa o ka hana. He mea pono loa kēia ʻano factorization ke ʻike mua mākou i ka hopena i ka hana.
2. ʻAno o ke Kikowaena (Piko)
Hiki ke hoʻololi ʻia ka hana quadratic i ke ʻano vertex, ʻo ia hoʻi:
f(x) = a(x – h)^2 + k
kahi ʻo \((h, k)\) nā hoʻonohonoho o ka piko o ka parabola. He mea pono loa kēia ʻano ke makemake mākou e ʻike i ke kūlana a me ke ʻano kumu o ka parabola.
Ka Hoʻoponopono ʻana i nā Hana Quadratic
No ka hoʻoponopono ʻana a i ʻole ka loaʻa ʻana o nā hopena (nā aʻa) o ka hana quadratic \(ax^2 + bx + c = 0\), hiki iā mākou ke hoʻohana i kekahi mau ʻano hana, me ka factorization, ka hoʻopau ʻana i ka huinahā, a me ke ʻano quadratic.
1. Hoʻokaʻawale ʻana
ʻO ke ʻano factorization e pili ana i ke kākau hou ʻana i ka hana quadratic ma ke ʻano o ka huahana o ʻelua mau helu binomial:
\[ ax^2 + bx + c = a(x – x_1)(x – x_2) \]
Eia kekahi laʻana, hiki ke hoʻohui ʻia ka hana \(x^2 – 5x + 6 = 0\) i loko o \((x – 2)(x – 3) = 0\), no laila ʻo nā aʻa \(x = 2\) a me \(x = 3\).
2. Hoʻopau i ka Square
ʻO kēia ʻano hana e pili ana i ka hoʻohui a me ka unuhi ʻana i kahi waiwai e hoʻololi i ke ʻano maʻamau i kahi ʻano huinaha kūpono:
1. E hoʻomaka mai ke ʻano maʻamau: \(ax^2 + bx + c\).
2. E puʻunaue i nā mea āpau ma \(a\) (inā \(a \neq 1\)).
3. E hoʻoneʻe i ke kūpaʻa \(c/a\) i ka ʻaoʻao ʻākau o ka hoohalike.
4. Hoʻohui a unuhi i ka \((b/2a)^2\).
5. E hoʻohālikelike i ka ʻaoʻao hema a hoʻomaʻalahi i ka ʻaoʻao ʻākau.
Eia kekahi laʻana, no ka hana \(x^2 + 6x + 8 = 0\):
\[x^2 + 6x = -8 \\
x^2 + 6x + 9 = 1
(x + 3)^2 = 1
x + 3 = \pm 1 \]
ka mea e hāʻawi ana i nā hopena \(x = -2\) a me \(x = -4\).
3. Formula Quadratic
ʻO ke ʻano quadratic ke ala maʻamau a hilinaʻi hoʻi e ʻike ai i nā aʻa o kahi hana quadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Ma ka hoʻohana ʻana i kēia haʻilula, hiki iā mākou ke loaʻa nā aʻa o kekahi hana quadratic, ʻoiai inā he mea kūpono ʻole ka factoring a i ʻole ka hoʻopau ʻana i ka square. No ka laʻana, no ka hoʻoponopono ʻana i ka \(2x^2 + 4x – 6 = 0\):
\[ x = \frac{-4 \pm \sqrt{4^2 – 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \\
x = \frac{-4 \pm \sqrt{16 + 48}}{4} \\
x = \frac{-4 \pm \sqrt{64}}{4} \\
x = \frac{-4 \pm 8}{4} \]
No laila, loaʻa iā mākou ʻelua mau hopena: \(x = 1\) a me \(x = -3\).
Kiʻi Hana Kuadratika
ʻO ke kiʻikuhi o kahi hana quadratic he parabola. Hiki i kēia parabola ke wehe i luna a i lalo paha ma muli o ka waiwai o ke koina \(a\):
– Inā \(a > 0\), e wehe ana ka parabola i luna.
– Inā \(a < 0\), e wehe ana ka parabola i lalo. 1. Vertex a me ke Axis of Symmetry ʻO ka vertex o ka parabola (\(h, k\)) ke kiko kiʻekiʻe a i ʻole ka liʻiliʻi loa o ka hana quadratic. Hiki ke loaʻa nā hoʻonohonoho o ka vertex \(h\) me ke ʻano hana: \[ h = \frac{-b}{2a} \] No ka loaʻa ʻana o \(k\), hoʻololi mākou i ka waiwai o \(h\) i loko o ka hana quadratic \( f(h) = k \). No ka laʻana, no \(f(x) = 2x^2 - 4x + 1\): \[ h = \frac{-(-4)}{2 \cdot 2} = 1 \] Hoʻololi \(x = 1\) i loko o ka hana: \[ k = f(1) = 2(1)^2 - 4(1) + 1 = -1 \] No laila, ʻo ka vertex ʻo \((1, -1)\). 2. ʻO ke axis o ka symmetry ʻO ke axis o ka symmetry o kahi parabola ka laina kū pololei e hele ana ma o ka vertex:
