# How do you find the domain and range of y=-x^2-3x-3?

Dec 16, 2017

Domain: $x \in \mathbb{R}$
Range: $\left\{y | y \le - \frac{3}{4}\right\}$

#### Explanation:

The function is defined for all real $x$, so the domain is just the real numbers, $\mathbb{R}$.

To find the range, we will consider the function as a parabola. Since the ${x}^{2}$ term is negative, we know the parabola will be concave down ($\cap$). Therefor the range will be between $- \infty$ and the vertex of the parabola.

The x coordinate of the vertex of a parabola $a {x}^{2} + b x + c$ can be found using the following formula:
$- \frac{b}{2 a}$

Plugging in our numbers, we get:
$\frac{- \left(- 3\right)}{- 2} = - \frac{3}{2}$

Now, we want the $y$ coordinate of the vertex, so we plug in the value into the original function:
$- {\left(- \frac{3}{2}\right)}^{2} - 3 \left(- \frac{3}{2}\right) - 3 = - \frac{9}{4} + \frac{9}{2} - 3 = - \frac{9}{4} + \frac{18}{4} - \frac{12}{4}$

$= - \frac{3}{4}$

So we know the vertex is at $- \frac{3}{4}$. This means that the range of the function is $\left\{y | y \le - \frac{3}{4}\right\}$