# How do you find the range of the equation y = -x^2 – 6x – 13?

Aug 18, 2017

Range of $y = \left[- 4 , - \infty\right)$

#### Explanation:

$y = - {x}^{2} - 6 x - 13$

$y$ is a quadratic function, represented on the $x y -$plane as a parabola of the form: $a {x}^{2} + b x + c$

The vertex of the parabola will be at $x = \frac{- b}{2 a}$

In our case, $b = - 6 , a = - 1$

Hence, ${x}_{v e r t e x} = \frac{6}{- 2} = - 3$

Since $a < 0$ then $y \left({x}_{v e r t e x}\right)$ will be a maximum of $y$

$\therefore {y}_{\max} = y \left(- 3\right) = - {\left(- 3\right)}^{2} + 6 \cdot 3 - 13 = - 9 + 18 - 13 = - 4$

$\therefore$ the greatest value of $y$ is $- 4$

Since $y$ has no lower bounds, the range of $y$ is $\left[- 4 , - \infty\right)$

As can be seen from the graph of $y$ below.

graph{-x^2-6x-13 [-23.18, 22.45, -15.1, 7.71]}