How do you find the axis of symmetry, graph and find the maximum or minimum value of the function f(x) = 2x^2 - 4x +1f(x)=2x24x+1?

2 Answers
Feb 24, 2016

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+cy=ax2+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 -1y=2x24x+1=2(x1)22+1=2(x1)21

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

The axis of symmetry is x=1x=1
SketchSketch

Feb 24, 2016

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

Explanation:

Given: " "2x^2-4x+1 2x24x+1

Write as" "2(x^2-4/2x)+1 2(x242x)+1

Now consider the -4/2x42x

Apply:" "(-1/2)xx(-4/2) =+4/4=1 (12)×(42)=+44=1

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

Now compare this to the graph
Tony BTony B