# How do you write the vertex form equation of the parabola y=x^2+2x-1?

Mar 7, 2016

$y + 2 = {\left(x + 1\right)}^{2}$

#### Explanation:

Step 1:
Send the constant(-1) to the left by doing,
$y + 1 = {x}^{2} + 2 x - 1 + 1 \rightarrow y + 1 = {x}^{2} + 2 x$

Step 2:
Identify the coefficient of $x$
In this case it's $2$

Step 3:
Divide the cofficient by 2.
You get : 2/2 = color(red)1

Step 4:
Add $1$ and then subtract $1$ from the right.
$y + 1 = {x}^{2} + 2 x \rightarrow y + 1 = {x}^{2} + 2 x + \textcolor{red}{1} - \textcolor{red}{1}$

Step 5:
We see that ${x}^{2} + 2 x + 1$ can be factorized to the perfect square ${\left(x + 1\right)}^{2}$

$\implies y + 1 = {\left(x + 1\right)}^{2} - 1$

Step 6:
Send $- 1$ to the left by adding $1$ to both sides.
$\implies y + 2 = {\left(x + 1\right)}^{2}$

And there you've got the vertex form of the equation of a parabola.

Note:
They call it vertex form simply because it gives us the cordinates of the vertex of the parabola that is being represented.

If we write it like this : $y - \left(- 2\right) = {\left(x - \left(- 1\right)\right)}^{2}$
then the vertex can be read as $\left(- 1 , - 2\right)$