Nā nīnau hoʻohālike e kūkākūkā ana i nā ʻano o nā hana Quadratic

Nā nīnau hoʻohālike e kūkākūkā ana i nā ʻano o nā hana Quadratic

He kumuhana koʻikoʻi ka hana quadratic i ka makemakika i ʻike pinepine ʻia ma ka papa hana kula kiʻekiʻe. ʻO ke ʻano maʻamau o kahi hana quadratic ʻo \( f(x) = ax^2 + bx + c \), kahi ʻo \( a \), \( b \), a me \( c \) he mau kūpaʻa me \( a \neq 0 \). ʻO ke kūkākūkā ʻana i nā ʻano o nā hana quadratic e pili ana i nā ʻano like ʻole e like me ke axis of symmetry, vertex, nā waiwai kiʻekiʻe a i ʻole ka palena iki, a me ke kuhikuhi o ka parabola. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā e hoʻomaopopo maikaʻi i nā ʻano o nā hana quadratic.

1. Nīnau: Ke hoʻoholo nei i ka Axis o Symmetry a me Vertex

Laʻana pilikia:
Hāʻawi ʻia kahi hana quadratic \( f(x) = 2x^2 – 4x + 1 \). E hoʻoholo i ke axis of symmetry a me ka vertex o ka hana.

Kūkākūkā:
No ka hoʻoholo ʻana i ke axis of symmetry o ka hana quadratic \( ax^2 + bx + c \), hoʻohana mākou i ke ʻano hana:
\[ x = -\frac{b}{2a} \]

Ma ka hana i hāʻawi ʻia \( f(x) = 2x^2 – 4x + 1 \), nā waiwai \( a = 2 \) a me \( b = -4 \). E pani i kēia mau waiwai i loko o ke ʻano hana:
\[ x = -\frac{-4}{2 \cdot 2} \]
\[ x = \frac{4}{4} \]
\[ x = 1 \]

No laila, ʻo ke axis of symmetry o ka hana ʻo \( x = 1 \).

No ka loaʻa ʻana o ka vertex, hoʻololi mākou i ka waiwai o ke axis of symmetry i loko o ka hana:
f(1) = 2(1)^2 – 4(1) + 1
\[ f(1) = 2 – 4 + 1 \]
\[ f(1) = -1 \]

No laila, ʻo ke kihi o ka hana ʻo \( (1, -1) \).

2. Nīnau: Ke hoʻoholo nei i ke kuhikuhi o kahi Parabola

Laʻana pilikia:
E hoʻoholo i ke kuhikuhi o ka parabola o ka hana quadratic \( f(x) = -3x^2 + 6x – 2 \).

Kūkākūkā:
Hoʻoholo ʻia ke kuhikuhi o ka parabola o kahi hana quadratic e 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. Ma ka hana i hāʻawi ʻia \( f(x) = -3x^2 + 6x - 2 \), ʻo ka waiwai o \( a = -3 \). ʻOiai \( a < 0 \), e wehe ana ka parabola i lalo. 3. Pilikia: Ke ʻimi nei i nā aʻa o kahi hana Quadratic Laʻana Pilikia: E ʻimi i nā aʻa o ka hana quadratic \( f(x) = x^2 - 5x + 6 \). Hoʻonā: Hiki ke loaʻa nā aʻa o kahi hana quadratic ma ka factoring a i ʻole ka hoʻohana ʻana i ke ʻano quadratic. E factor mākou iā ia: \[ x^2 - 5x + 6 = 0 \] E ʻimi i ʻelua mau helu e hoʻonui e hāʻawi i ka 6 a hoʻohui e hāʻawi i ka -5. ʻO kēia mau helu he -2 a me -3. \[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \] No laila, ʻo nā aʻa: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = 3 \] 4. Nīnau: Ka Waiwai Kiʻekiʻe a i ʻole ka Waiwai Haʻahaʻa Laʻana Nīnau: E hoʻoholo i ka waiwai haʻahaʻa o ka hana quadratic \( f(x) = 2x^2 - 4x + 5 \).

Kūkākūkā: No ka hoʻoholo ʻana i ka waiwai liʻiliʻi loa o ka hana quadratic \( ax^2 + bx + c \), pono mākou e nānā inā wehe ka parabola i luna a i lalo paha. Inā \( a > 0 \), wehe ka parabola i luna a loaʻa iā ia kahi waiwai liʻiliʻi loa; inā \( a < 0 \), wehe ka parabola i lalo a loaʻa iā ia kahi waiwai nui loa. Ma ka hana i hāʻawi ʻia \( f(x) = 2x^2 - 4x + 5 \), ʻo ka waiwai o \( a = 2 \), no laila wehe ka parabola i luna a loaʻa iā ia kahi waiwai liʻiliʻi loa. Loaʻa ka waiwai liʻiliʻi loa ma ka piko. Ua ʻike mua mākou i ke axis of symmetry \( x = -\frac{b}{2a} \). No kēia hana: \[ x = -\frac{-4}{2 \cdot 2} \] \[ x = \frac{4}{4} \] \[ x = 1 \] E hoʻololi iā \( x = 1 \) i loko o ka hana e ʻike ai i ka waiwai liʻiliʻi loa: \[ f(1) = 2(1)^2 - 4(1) + 5 \] \[ f(1) = 2 - 4 + 5 \] \[ f(1) = 3 \] No laila, ʻo ka waiwai liʻiliʻi loa o ka hana he 3. 5. Nīnau: Ke kaha kiʻi ʻana i nā hana Quadratic Laʻana Nīnau: E hana i kahi kiʻi o ka hana quadratic \( f(x) = -x^2 + 4x - 3 \). Kūkākūkā: No ke kaha kiʻi ʻana i kahi hana quadratic, pono mākou e ʻike i kekahi mau ʻano koʻikoʻi e like me ke axis of symmetry, ka vertex, a me nā aʻa o ka hana, a me ke kuhikuhi o ka parabola. 1. Axis of Symmetry: \[ x = -\frac{b}{2a} \] Ma ka hana \( f(x) = -x^2 + 4x - 3 \), nā waiwai \( a = -1 \) a me \( b = 4 \). \[ x = -\frac{4}{2(-1)} \] \[ x = -\frac{4}{-2} \] \[ x = 2 \] 2. Vertex: E hoʻololi iā \( x = 2 \) i loko o ka hana e ʻike ai i ka vertex: \[ f(2) = -(2)^2 + 4(2) - 3 \] \[ f(2) = -4 + 8 - 3 \] \[ f(2) = 1 \] ʻO ke vertex ʻo \( (2, 1) \). 3. Ke kuhikuhi o ka Parabola: No ka mea, \( a = -1 \), e wehe ana ka parabola i lalo. 4. Nā Aʻa o nā Hana Quadratic: \[ -x^2 + 4x - 3 = 0 \] Hiki iā mākou ke hoʻohana i ke ʻano quadratic: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] No \( a = -1 \), \( b = 4 \), a me \( c = -3 \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} \] \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \] \[ x = \frac{-4 \pm \sqrt{4}}{-2} \] \[ x = \frac{-4 \pm 2}{-2} \] ʻElua mau hopena: \[ x = \frac{-4 + 2}{-2} = 1 \] \[ x = \frac{-4 - 2}{-2} = 3 \] ʻO nā aʻa \( x = 1 \) a me \( x = 3 \). Me kēia ʻike a pau, hiki iā mākou ke kaha kiʻi i ka hana quadratic. Loaʻa i kēia parabola kahi vertex ma \( (2, 1) \), wehe i lalo, a loaʻa nā aʻa ma \( x = 1 \) a me \( x = 3 \). Hopena Ma o nā laʻana i kūkākūkā ʻia, hiki iā mākou ke hoʻomaopopo maikaʻi i nā ʻano like ʻole o ka hana quadratic. ʻO ka ʻike pehea e hoʻoholo ai i ke axis o symmetry, vertex, ka waiwai nui a liʻiliʻi paha, ke kuhikuhi o ka parabola, a me nā aʻa o ka hana quadratic he mea nui ia e wehewehe i ke ʻano a me nā waiwai o ka parabola. ʻO ka hoʻomaopopo maikaʻi ʻana i kēia mau manaʻo e hāʻawi i kahi kahua paʻa no nā haumāna e ʻimi i nā kumuhana holomua i ka makemakika.