Question

What is the the vertex of `y = 3x^2-2x+4`?

Answer 1

Convert into vertex form to get
vertex at `(1/3,3 2/3)`

Explanation:

General vertex form:
`color(white)("XXX")color(green)(y=m(x-a)^2+b)`
with vertex at `(color(green)(a, b))`

Given
`color(white)("XXX")y=3x^2-2x+4`

Extract `m` from terms including `x`
`color(white)("XXX")y=3(x^2-2/3x)+4`

Complete the square
`color(white)("XXX")y=color(red)(3)(x^2-2/3xcolor(blue)(+(1/3)^2))+4-color(red)(3)*color(blue)((1/3)^2)`

Re-write as a squared binomial and simplify the numeric expression
`color(white)("XXX")y=3(x-1/3)^2+3 2/3`
which is the desired "vertex form"
graph{3x^2-2x+4 [-1.07, 2.775, 2.868, 4.788]}