# How do you find the domain and range of y=x^2 +2x -5?

Jul 9, 2017

Domain: $\left(- \infty , + \infty\right)$
Range: $\left[- 6 , + \infty\right)$

#### Explanation:

$y = {x}^{2} + 2 x - 5$

$y$ is defined $\forall x \in \mathbb{R}$
Hence the domain of $y$ is $\left(- \infty , + \infty\right)$

$y$ is a quadratic function of the form $a {x}^{2} + b x + c$

The graph of $y$ is a parabola with vertex where $x = \frac{- b}{2 a}$

Since the coefficient of ${x}^{2} > 0$ the vertex will be the absolute minimum of $y$

At the vertex $x = \frac{- 2}{2 \times 1} = - 1$

$\therefore {y}_{\min} = y \left(- 1\right) = 1 - 2 - 5 = - 6$

Since $y$ has no upper bound the range of $y$ is [-6, +oo)

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

graph{x^2+2x-5 [-16.02, 16.02, -8.01, 8.01]}