Question

3) Seja f uma função quadrática em que f(0) = 2, f(2) = 12 e f(-1) = 6. Determinar a lei de formação dessa função.

Com cálculo!!​

Answer 1

$\Large\boxed{\begin{array}{l}\sf f(x)=ax^2+bx+c\\\sf f(0)=a\cdot 0^2+b\cdot0 +c\\\sf f(0)=c\implies c=2.\\\sf f(2)=a\cdot2^2+b\cdot2+c\\\sf f(2)=4a+2b+2\\\sf 4a+2b+2=12\\\sf 4a+2b=12-2\\\sf 4a+2b=10\div2\\\sf 2a+b=5.\\\sf f(-1)=a\cdot(-1)^2+b\cdot(-1)+c\\\sf f(-1)=a-b+2\\\sf a-b+2=6\\\sf a-b=6-2\\\sf a-b=4\\\underline{\rm montando\,um\,sistema\,temos:}\end{array}}$

$\Large\boxed{\begin{array}{l}\begin{cases}\sf2a+b=5\\\sf a-b=4\end{cases}\\\sf pelo\,m\acute etodo\,da\,adic_{\!\!,}\tilde ao\,temos:\\+\underline{\begin{cases}\sf 2a+\backslash\!\!\!b=5\\\sf a-\backslash\!\!\!b=4\end{cases}}\\\sf 3a=9\\\sf a=\dfrac{9}{3}\\\\\sf a=3\\\sf a-b=4\\\sf 3-b=4\\\sf b=3-4\\\sf b=-1\\\underline{\boldsymbol{ A\,lei\,da\,func_{\!\!,}\tilde ao\,ser\acute a}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf f(x)=3x^2-x+2}}}}\end{array}}$