Question

Determine a equação da parábola que passa pelos pontos (1, –1), (2, –3) e (–2, –7).

Answer 1

$\displaystyle \sf \underline{\text{Equa{\c c}{\~a}o da Par{\'a}bola}} : \\\\ f(x) = ax^2+bx+c \\\\ \text{temos os pontos } (1,-1) , \ (2,-3),\ (-2,-7) : \\\\ (1,-1) \to a.(1)^2+b.(1)+c =-1 \to a+b+c = -1 \\\\ (2,-3) \to a.(2)^2+b.(2)+c = -3 \to 4a+2b+c=-3 \\\\ (-2,-7)\to a.(-2)^2+b.(-2)+c = -7 \to 4a-2b+c=-7 \\\\\ Temos : \\\\ \left\{\begin{array}{I}\sf (1) \ a+b+c = -1 \\\\ \sf (2)\ 4a+2b+c=-3 \\\\ \sf (3)\ 4a-2b+c=-7 \end{array}$

$\displaystyle \sf (2)-(1): \\\\ 4a+2b+c -a-b-c = -3-(-1)=-3+1 \\\\ 3a+b=-2 \\\\\\ (3)-(2) : \\\\ 4a-2b+c-4a-2b-c = -7-(-3) = -7+3 \\\\ -4b = -4 \to \boxed{\sf b = 1} \\\\ Da{\'i}}:\\\\ 3a+b=-2 \\\\ 3a+1=-2 \\\\ 3a=-3 \to \boxed{\sf a =-1}\\\\ logo: \\\\ a+b+c= - 1 \\\\ -1+1+c = -1 \to \boxed{\sf c =-1} \\\\ Logo : \\\\ \huge\boxed{\sf f(x) = -x^2+x-1 }\checkmark$