Fa'ata'ita'iga o Fesili e Talanoaina ai le Fausiaina o Galuega Fa'atino Fa'atafafā
O le fausiaina o galuega fa'afa'afa o se autu taua i le algebra e masani ona alia'e mai i mataupu fa'a-matematika tulaga lua ma tulaga maualuga. O le malamalama i galuega fa'afa'afa e taua tele aua e masani ona fa'aaogaina i tulaga eseese, e pei o le au'ili'iliga o fa'amaumauga, fa'ata'ita'iga o le fisiki, ma le tamaoaiga. I totonu o lenei tusiga, o le a tatou talanoaina ai fa'ata'ita'iga eseese o fa'afitauli ma le auala e foia ai e fausia ai galuega fa'afa'afa.
Malamalama i Galuega Fa'atino Fa'atafafā
O le galuega fa'atafafā o se galuega fa'apolinomial tikeri lona lua e iai lona foliga lautele:
\[ f(x) = ax^2 + bx + c \]
lea o le \(a\), \(b\), ma le \(c\) o ni tumau, ma le \(a \neq 0\).
O le kalafi o se galuega fa'afa'afa o se pi'o e ta'ua o le parabola. O parabolas e iai o latou uiga tutusa ma foliga e faʻalagolago i le faʻailoga o le tumau \(a\). Afai o le \(a > 0\), e tatala i luga le parabola. I se isi itu, afai o le \(a < 0\), e matala i lalo le parabola. Elemene Tāua i Galuega Fa'atafafā - A'a o se fa'atusatusaga fa'atafafā: O tau o le \(x\) lea e \(f(x) = 0\), lea e mafai ona maua e fa'aaoga ai le fa'atusatusaga fa'atafafā \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). - Vertex: O le tulaga aupito maualuga po'o le tulaga aupito maualalo o le parabola, lea e maua e fa'aaoga ai le fua fa'atatau \((x, y)\) lea \(x = -\frac{b}{2a}\) ma le \(y = f(-\frac{b}{2a})\). - Axis of symmetry: O le laina fa'atulagaina e vaevaeina ai le parabola i ni vaega se lua e tutusa lelei, o lo'o i le \(x = -\frac{b}{2a}\). Fa'ata'ita'iga Fesili 1: Fatufatua'iina o se Galuega Fa'atafafā mai Points e Tolu Fesili: Fuafua le fua fa'atatau mo le galuega fa'atafafā e ui atu i points (1, 2), (2, 5), ma le (3, 10). Tali: 1. Tatou amata i le fa'atulagaga lautele o se galuega fa'afa'atasi: \[ f(x) = ax^2 + bx + c \] 2. Sui le tulaga (1, 2) i totonu o le fua fa'atatau: \[ a(1)^2 + b(1) + c = 2 \] \[ a + b + c = 2 \] (Fua fa'atatau 1) 3. Sui le tulaga (2, 5) i totonu o le fua fa'atatau: \[ a(2)^2 + b(2) + c = 5 \] \[ 4a + 2b + c = 5 \] (Fua fa'atatau 2) 4. Sui le tulaga (3, 10) i totonu o le fua fa'atatau: \[ a(3)^2 + b(3) + c = 10 \] \[ 9a + 3b + c = 10 \] (Fua fa'atatau 3) 5. Ua ia i tatou nei ni faiga se tolu o fua fa'atatau laina: \[ \begin{cases} a + b + c = 2 \\ 4a + 2b + c = 5 \\ 9a + 3b + c = 10 \\ \end{cases} \] 6. Mo le foia, tatou te to'esea le fua fa'atatau lona lua ma le muamua: \[ (4a + 2b + c) - (a + b + c) = 5 - 2 \] \[ 3a + b = 3 \] (Fua fa'atatau 4) 7. To'ese le fua fa'atatau lona tolu ma le lona lua: \[ (9a + 3b + c) - (4a + 2b + c) = 10 - 5 \] \[ 5a + b = 5 \] (Fua fa'atatau 5) 8. To'ese le Fa'atusatusaga 5 ma le Fa'atusatusaga 4: \[ (5a + b) - (3a + b) = 5 - 3 \] \[ 2a = 2 \] \[ a = 1 \] 9. Sui le \(a = 1\) i le Fa'atusatusaga 4: \[ 3(1) + b = 3 \] \[ 3 + b = 3 \] \[ b = 0 \] 10. Sui le \(a = 1\) ma le \(b = 0\) i le Fa'atusatusaga 1: \[ 1 + 0 + c = 2 \] \[ c = 1 \] 11. O lea la, o le galuega fa'atafafā e fa'apea: \[ f(x) = 1x^2 + 0x + 1 \] \[ f(x) = x^2 + 1 \] Fa'ata'ita'iga Fesili 2: Fuafuaina o se Galuega Fa'atafafā mai se Vertex ma se Isi Point Fesili: Fuafua le fua fa'atatau mo se galuega fa'atafafā e iai se vertex i le (-1, 4) ma ui atu i le point (1, 0). Tali: 1. O le fa'atulagaga masani o se galuega fa'atafafā ma le vertex \((h, k)\) o le: \[ f(x) = a(x - h)^2 + k \] 2. Sui le tumutumu (-1, 4) i le faatulagaga masani: \[ f(x) = a(x + 1)^2 + 4 \] 3. Sui le tulaga (1, 0) i totonu o le fua fa'atatau e su'e ai \(a\): \[ 0 = a(1 + 1)^2 + 4 \] \[ 0 = a(2)^2 + 4 \] \[ 0 = 4a + 4 \] \[ 4a = -4 \] \[ a = -1 \] 4. O lea la, o le galuega fa'afa'atasi e fa'apea: \[ f(x) = -1(x + 1)^2 + 4 \] \[ f(x) = - (x + 1)^2 + 4 \] 5. Tufatufaga mo le fa'atulagaga lautele: \[ f(x) = - (x^2 + 2x + 1) + 4 \] \[ f(x) = -x^2 - 2x - 1 + 4 \] \[ f(x) = -x^2 - 2x + 3 \] Fa'ata'ita'iga Fa'afitauli 3: Liliuina o le Fa'atulagaga Vertex i le Fa'atulagaga Masani Fa'afitauli: Liliu le galuega fa'atafafā \( f(x) = 2(x - 3)^2 + 5 \) i le fa'atulagaga masani \( ax^2 + bx + c \). Tali: 1. I le amataga, e manaʻomia ona tatou faʻalauteleina: \[ f(x) = 2(x - 3)^2 + 5 \] 2. Fa'alautele le binomial: \[ (x - 3)^2 = x^2 - 6x + 9 \] 3. Toe sui i le galuega faatino: \[ f(x) = 2(x^2 - 6x + 9) + 5 \] 4. Tufatufaina o le 2 i itu ta'itasi o le binomial: \[ f(x) = 2x^2 - 12x + 18 + 5 \] 5. Tuu faatasi vaega uma: \[ f(x) = 2x^2 - 12x + 23 \] O lea la, o le foliga masani o le galuega fa'afa'atasi e fa'apea: \[ f(x) = 2x^2 - 12x + 23 \] Fa'ai'uga O le fausiaina o galuega fa'afa'atasi mai fa'amatalaga eseese o se tomai taua i le matematika. E ala i le faia pea o faʻataʻitaʻiga i ituaiga eseese o faʻafitauli, e mafai ai ona tatou faʻaleleia atili lo tatou malamalama ma le tulaga lelei i le foʻia o faʻatusatusaga faʻatafafā. O manatu autū e manatua e aofia ai le sailia ma le faʻaleleia atili o metotia mo le mauaina o faʻamatalaga mai foliga o tumutumu, liua i le va o foliga o tumutumu ma foliga masani, ma le fausiaina o galuega faatino mai foliga ua tuʻuina atu. Faatasi ai ma se malamalamaga mautu i lenei mataupu, e mafai ona tatou feagai ma luitau sili atu ona faigata i le matematika i le lumanai.
