# Question #abef5

May 9, 2017

Min: $\left(- 3.5 , - 3.25\right)$
Roots: $x \cong - 5.3028$ and $x \cong - 1.6972$

#### Explanation:

This polynomial is in the form

$y = a {x}^{2} + b x + c$

Where
$a = 1$
$b = 7$
$c = 9$

The $x$-value of the minimum point of the parabola is given by

${x}_{\min} = - \frac{b}{2 a} = - \frac{7}{2} \cong - 3.5$

The minimum $y$-value occurs when you plug that ${x}_{\min}$ into $y$

${y}_{\min} = {\left(- \frac{7}{2}\right)}^{2} + 7 \left(- \frac{7}{2}\right) + 9 = - \frac{13}{4} \cong - 3.25$

So the minimum point of the parabola is at the point $\left(- 3.5 , - 3.25\right)$

Finally, you find where the parabola crosses the $x$-axis by plugging the given equation into the quadratic formula using $a$, $b$, and $c$ from above.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$x = \frac{- 7 \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(1\right) \left(9\right)}}{2 \left(1\right)}$
$x = - 7 \pm \frac{1}{2} \sqrt{13}$
So, $x \cong - 5.3028$ and $x \cong - 1.6972$

Graphing gives:
graph{x^2+7x+9 [-6.4, 0.1, -3.49, 0.408]}