# What is the axis of symmetry and vertex for the graph y = x^2+2x-3?

Oct 17, 2016

the axis of symmetry is x=-1
and the vertex is (-1,-4)

#### Explanation:

$y = {x}^{2} + 2 x - 3$
Rewrite the equation in the vertex form
$y = {x}^{2} + 2 x + 1 - 4 = {\left(x + 1\right)}^{2} - 4$
The line of symmetry is when$\left(x + 1 = 0\right)$
And the vertex is on that line$\left(- 1 , - 4\right)$

If you have not yet studied calculus, forget what I write under

Differentiating with respect to x
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 2$
The vertex is when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$2 x + 2 = 0 \implies x = - 1$ and $y = {\left(- 1\right)}^{2} + \left(2 \cdot - 1\right) - 3 = 1 - 5 = - 4$
Differentiating once more
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 2 \left(> 0\right)$ so we have a minimum

Here is a graph of the function
graph{x^2+2x-3 [-10, 10, -5, 5]}