Question

How do you write `y=2x^2+4x+4` in vertex form?

Answer 1

`y = 2(x + 1)^2 + 2` `" "`The vertex will be at (-1, 2)

Explanation:

Use the same process as for "Completing a square",

Treat the right hand side as an expression, rather than as an equation.

Vertex form is also known as the form `a(x+b)^2 + c`.
From this we can get the vertex (or the turning point) of a parabola as being `(-b,c)`

`y = 2x^2 + 4x + 4`

Step 1. Divide each term by 2 to change `2x^2` to just make `x^2`

`y = 2(x^2 + color(red)(2)x + 2)`

Step 2. Complete the square by adding `(color(red)(2)/2)^2` = 1
If this was an equation, we would add the same to both sides. but as we are working with an expression, we will use additive inverses.
`+1 -1 = 0`

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

The first 3 terms inside the bracket now represent the square of a binomial, so we can write them like that, and simplify the last 2 terms.

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

Step 3. Multiply the 2 into the bracket and the answer is now in vertex form.

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