# How to graph a parabola  y = x^2 - 6x + 7 ?

##### 1 Answer
Jun 10, 2015

graph{x^2-6x+7 [-5, 10, -5, 10]}

#### Explanation:

Find the intersections :
${\Delta}_{x} = 36 - 4 \cdot 7 = 36 - 28 = 12 = {\left(2 \sqrt{3}\right)}^{2}$
${x}_{\text{1,2}} = \frac{6 \pm 2 \sqrt{3}}{2} = 3 \pm \sqrt{3}$
${x}_{1} = 3 - \sqrt{3} \approx 1.27$
${x}_{2} = 3 + \sqrt{3} \approx 4.73$

$x = 0 \to y = 7$
So the intersections are:
$I = \left\{\begin{matrix}0 & 7 \\ 3 - \sqrt{3} & 0 \\ 3 + \sqrt{3} & 0\end{matrix}\right\}$.

Find the vertex :
${x}_{v} = - \frac{b}{2 a} = \frac{6}{2} = 3$
${y}_{v} = - \frac{\Delta}{4 a} = - \frac{12}{4} = - 3.$
The vertex is $V \left(3 , - 3\right)$.

graph{x^2-6x+7 [-5, 10, -5, 10]}