Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function `f(x) = 2x^2 - 4x +1`?

Answer 1

To find the vertex and the axis of symmetry, rearrange the equation into vertex form by completing the square.

Explanation:

This equation is for a parabola, because it follows the general form `y= ax^2 + bx+c`
Hence the maximum or minimum value is at the vertex. In this case, because the squared term is positive, it will be a minimum value.

To find the vertex and the axis of symmetry, rearrange the equation into vertex form by completing the square.

`y = 2x^2 -4x +1 = 2(x-1)^2 -2 +1 = 2(x-1)^2 -1`

This expression is at its minimum when `x=1` (the bracketed term is zero) and so the vertex is `(1,-1)`

The axis of symmetry is `x=1`

Answer 2

A slightly 'cheating' sort of way to find that the axis of symmetry is at `x=1`

Explanation:

Given: `" "2x^2-4x+1`

Write as`" "2(x^2-4/2x)+1`

Now consider the `-4/2x`

Apply:`" "(-1/2)xx(-4/2) =+4/4=1`

`color(brown)("This is in fact, part of the process for completing the square")` `color(brown)("but it is in disguise.")`

Now compare this to the graph