Question

How do you write the Vertex form equation of the parabola `y=x^2+4x-7`?

Answer 1

`y=(x-(-2))^2+(-11)`

Explanation:

First recall the general vertex form:
`color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)` with vertex at `(color(red)(a),color(blue)(b))`

Given
`color(white)("XXX")y=x^2+4x-7`

The required value of `color(green)(m)` is clearly `color(green)(1)` and we can ignore it at this point.

We are trying to get a squared binomial: `(x-color(red)(a))^2`

Completing the square in the given equation:
`color(white)("XXX")y=x^2+4xcolor(magenta)(+2^2)-7 color(magenta)(-2^2)`

`color(white)("XXX")y=(x+2)^2-11`

Into proper vertex form:
`color(white)("XXX")y=color(green)(1)(x-color(red)((-2)))^2+color(blue)((-11))`

graph{x^2+4x-7 [-10.84, 9.16, -12.92, -2.92]}