Question

What is the vertex form of `7y = 3x^2+2x+1`?

Answer 1

Vertex form is:

`y = 3/7(x+1/3)^2+2/21`

or if you prefer:

`y = 3/7(x-(-1/3))^2+2/21`

Explanation:

Given:

`7y = 3x^2+2x+1`

Divide both sides by `7` then complete the square:

`y = 3/7x^2+2/7x+1/7`

`color(white)(y) = 3/7(x^2+2/3x+1/9+2/9)`

`color(white)(y) = 3/7(x+1/3)^2+2/21`

The equation:

`y = 3/7(x+1/3)^2+2/21`

is pretty much vertex form:

`y = a(x-h)^2+k`

with multiplier `a=3/7` and vertex `(h, k) = (-1/3, 2/21)`

Strictly speaking, we could write:

`y = 3/7(x-(-1/3))^2+2/21`

just to make the `h` value clear.