Question

Determine a lei da função que deu origem a parábola.







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Answer 1

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$\sf f(0)=0\implies A(0,0)\\\sf f(6)=0\implies B(6,0)\\\sf f(3)=9\implies C(3, 9)\\\sf f(x)=ax^2+bx+c\\\sf f(0)=a\cdot 0^2+b\cdot0+c\implies c=0\\\sf f(3)=a\cdot 3^2+b\cdot3+c\\\sf 9a+3b+c=9\\\sf 9a+3b=9\div3\implies 3a+b=3\\\sf x_v=-\dfrac{b}{2a}\\\sf 3=-\dfrac{b}{2a}\implies b=-6a\\\sf 3a+b=3\\\sf 3a-6a=3\\\sf -3a=3\cdot(-1)\\\sf 3a=-3\\\sf a=-\dfrac{3}{3}\\\sf a=-1\\\sf b=-6a\\\sf b=-6\cdot(-1)\\\sf b =6\\\huge\boxed{\boxed{\boxed{\boxed{\sf f(x)=-x^2+6x}}}}$