Question

Sa se determine functia de gradul al doilea daca
f(1)=0
f(0)=1
f(-1)=8

Answer 1

Am atasat rezolvarea

Answer 2

$\boxed{\text{Metoda polinomului de interpolare Lagrange}} \\ \\ \\ \text{Graficul functiei trece prin punctele:}\\ ~\quad (1,0);~(0,1);~(-1,8). \\ \\ f(x) =0\cdot \dfrac{(x-0)(x-(-1))}{(1-0)(1-(-1))}+1\cdot \dfrac{(x-1)(x-(-1))}{(0-1)(0-(-1))}+ \\ + 8\cdot \dfrac{(x-1)(x-0)}{(-1-1)(-1-0)} \\ \\ f(x) =0+\dfrac{(x-1)(x+1)}{-1\cdot (0+1)}+8\cdot \dfrac{x(x-1)}{-2\cdot (-1)} \\ \\ f(x) = -(x-1)(x+1)+4x(x-1)\\ \\f(x) = \dfrac{(x-1)(x+1)}{-1}+8\cdot \dfrac{x(x-1)}{2} \\ \\f(x) = -(x^2-1)+ 4x^2-4x$

$f(x) = -x^2+1+4x^2-4x \\ \\ \boxed{f(x) = 3x^2-4x+1}$