Question

Esboce o gráfico: Y= -x²+4x-3

Answer 1

Segue esboço do gráfico em anexo.

$y=-x^2+4x-3$


$\bullet\;\;$ Interseção com o eixo $y:$

Fazendo $x=0\,,$

$y=-3$

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$\bullet\;\;$ Interseção com o eixo $x:$

Encontrando as raízes da equação:

$y=0\\\\ -x^2+4x-3=0~~~\Rightarrow~~\left\{\! \begin{array}{l}a=-1\\b=4\\c=-3 \end{array} \right.\\\\\\ \Delta=b^2-4ac\\\\ \Delta=4^2-4\cdot (-1)\cdot (-3)\\\\ \Delta=16-12\\\\ \Delta=4$


$x=\dfrac{-4\pm \sqrt{4}}{2\cdot (-1)}\\\\\\ x=\dfrac{-4\pm 2}{-2}\\\\\\ \begin{array}{rcl} x_1=\dfrac{-4+2}{-2}&~\text{ e }~&x_2=\dfrac{-4-2}{-2}\\\\\\ x_1=\dfrac{-2}{-2}&~\text{ e }~&x_2=\dfrac{-6}{-2}\\\\\\ x_1=1&~\text{ e }~&x_2=3 \end{array}$

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$\bullet\;\;$ Coordenadas do vértice:

$\bullet\;\;x_{_V}=-\,\dfrac{b}{2a}\\\\\\ x_{_V}=-\,\dfrac{4}{2\cdot (-1)}\\\\\\ x_{_V}=-\,\dfrac{4}{-2}\\\\\\ \boxed{\begin{array}{c} x_{_V}=2 \end{array}}$

$\bullet\;\;y_{_V}=-\,\dfrac{\Delta}{4a}\\\\\\ y_{_V}=-\,\dfrac{4}{4\cdot (-1)}\\\\\\ y_{_V}=-\,\dfrac{4}{-4}\\\\\\ \boxed{\begin{array}{c} y_{_V}=1 \end{array}}$