Question

Esboce o gráfico das seguintes funções quadráticas:

Answer 1

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$\tt a)$

$\underline{\rm intersecc_{\!\!,}\tilde ao~com~o~eixo~y\!:}\\\sf x=0\\\sf f(0)=0^2-4\cdot0+3\\\sf f(0)=3\implies A(0,3)\\\underline{\rm intersecc_{\!\!,}\tilde ao~com~o~eixo~x\!:}\\\sf f(x)=0\\\sf x^2-4x+3=0\\\sf \Delta=b^2-4ac\\\sf \Delta=(-4)^2-4\cdot1\cdot3\\\sf\Delta=16-12\\\sf\Delta=4\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\sf x=\dfrac{-(-4)\pm\sqrt{4}}{2\cdot1}\\\sf x=\dfrac{4\pm2}{2}\begin{cases}\sf x_1=\dfrac{4+2}{2}=\dfrac{6}{2}=3\\\sf x_2=\dfrac{4-2}{2}=\dfrac{2}{2}=1\end{cases}\\\sf B(3,0)~C(1,0)$

$\underline{\rm coordenadas~do~v\acute ertice\!:}\\\sf x_V=-\dfrac{b}{2a}\\\sf x_V=-\dfrac{-4}{2\cdot1}\\\sf x_V=2\\\sf y_V=-\dfrac{\Delta}{4a}\\\sf y_V=-\dfrac{4}{4\cdot1}\\\sf y_V=-1\\\sf V(2,-1)\\\boxed{\tt o~gr\acute afico~est\acute a~anexo}$

$\tt b)$

$\underline{\rm intersecc_{\!\!,}\tilde ao~com~o~eixo~y\!:}\\\sf x=0\\\sf f(0)=0^2+0+6\\\sf f(0)=6\implies A(0,6)\\\underline{\rm intersecc_{\!\!,}\tilde ao~com~o~eixo~x\!:}\\\sf f(x)=0\\\sf -x^2+x+6=0\\\sf\Delta=b^2-4ac\\\sf\Delta=1^2-4\cdot(-1)\cdot6\\\sf\Delta=1+24\\\sf\Delta=25\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\sf x=\dfrac{-1\pm\sqrt{25}}{2\cdot(-1)}\\\sf x=\dfrac{-1\pm5}{-2}\begin{cases}\sf x_1=\dfrac{-1+5}{-2}=-\dfrac{4}{2}=-2\\\sf x_2=\dfrac{-1-5}{-2}=-\dfrac{6}{-2}=3\end{cases}$

$\sf B(-2,0)~~C(3,0)\\\underline{\rm }$

$\underline{\rm coordenadas~do~v\acute ertice\!:}\\\sf x_V=-\dfrac{b}{2a}\\\sf x_V=-\dfrac{1}{2\cdot(-1)}\\\sf x_V=\dfrac{1}{2}\\\sf y_V=-\dfrac{\Delta}{4a}\\\sf y_V=-\dfrac{25}{4\cdot(-1)}\\\sf y_V=\dfrac{25}{4}\\\sf V\bigg(\dfrac{1}{2},\dfrac{25}{4}\bigg)$