Question

What is the axis of symmetry and vertex for the graph `y= 3x^2 - 4x + 6`?

Answer 1

Axis of symmetry: `x=2/3`
Vertex: `(2/3, 4 2/3)`

Explanation:

Given
`color(white)("XXX")y=3x^2-4x+6`
We will convert this equation into "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)`

Extracting `color(green)(m)`
`color(white)("XXX")y=color(green)3(x^2-4/3x)+6`

Completing the square
`color(white)("XXX")y=color(green)3(x^2-4/3xcolor(magenta)+color(red)((2/3))^2)+6color(magenta)-color(green)3 * (color(red)(2/3)^2)`

`color(white)("XXX")y=color(green)3(x-color(red)(2/3))^2+color(blue)(4 2/3)`

So the vertex is at `(color(red)(2/3),color(blue)(4 2/3))`

The axis of symmetry is a vertical line of the form `x=color(red)(a)` running through the vertex.