Question

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

Answer 1

`y+2= (x+1)^2`

Explanation:

Step 1:
Send the constant(-1) to the left by doing,
`y+1=x^2+2x-1+1 rarr y+1=x^2+2x`

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+2x rarr y+1=x^2+2x+ color(red)1-color(red)1`

Step 5:
We see that `x^2+2x+1` can be factorized to the perfect square `(x+1)^2`

`=>y+1=(x+1)^2-1`

Step 6:
Send `-1` to the left by adding `1` to both sides.
`=>y+2=(x+1)^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-(-2)=(x-(-1))^2`
then the vertex can be read as `(-1,-2)`