# 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?

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 $y = a {x}^{2} + b x + c$
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.

$y = 2 {x}^{2} - 4 x + 1 = 2 {\left(x - 1\right)}^{2} - 2 + 1 = 2 {\left(x - 1\right)}^{2} - 1$

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

The axis of symmetry is $x = 1$ Feb 24, 2016

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

#### Explanation:

Given: $\text{ } 2 {x}^{2} - 4 x + 1$

Write as$\text{ } 2 \left({x}^{2} - \frac{4}{2} x\right) + 1$

Now consider the $- \frac{4}{2} x$

Apply:$\text{ } \left(- \frac{1}{2}\right) \times \left(- \frac{4}{2}\right) = + \frac{4}{4} = 1$

$\textcolor{b r o w n}{\text{This is in fact, part of the process for completing the square}}$$\textcolor{b r o w n}{\text{but it is in disguise.}}$

Now compare this to the graph 