Question

What is the vertex form `y=x^2+2x-8`?

Answer 1

The vertex form of the equation is `y = (x + 1)^2 - 9`

Explanation:

Changing a quadratic function from standard form to vertex form actually requires that we go through the process of completing the square. To do this, we need the `x^2` and `x` terms only on the right side of the equation.

`y = x^2 + 2x - 8`
`y + 8 = x^2 + 2x - 8 + 8`
`y + 8 = x^2 + 2x - 8 + 8`
`y + 8 = x^2 + 2x`

Now, the right side has the `ax^2 + bx` terms, and we need to find `c`, using the formula `c = (b/2)^2`.

In our prepared equation, `b = 2`, so
`c = (2/2)^2 = 1^2 = 1`

Now, we add `c` to both sides of our equation, simplify the left side, and factor the right side.

`y + 8 + 1 = x^2 + 2x + 1`
`y + 9 = (x +1)^2`

To finish putting the equation in vertex form, subtract `9` from both sides, thus isolating the `y`:

`y + 9 - 9 = (x + 1)^2 - 9`
`y = (x + 1)^2 - 9`