# What is the axis of symmetry and vertex for the graph y = x^2 - 4x + 1?

Apr 15, 2017

$x = 2$ is the line of symmetry.

$\left(2 , - 3\right)$ is the vertex.

#### Explanation:

Find the axis of symmetry first using $x = \frac{- b}{2 a}$

$y = {x}^{2} - 4 x + 1$

$x = \frac{- \left(- 4\right)}{2 \left(a\right)} = \frac{4}{2} = 2$

The vertex lies on the line of symmetry, so we know $x = 2$
Use the value of $x$ to find $y$

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

$y = 4 - 8 + 1 = - 3$

The vertex is at $\left(2 , - 3\right)$

You can also use the method of completing the square to write the equation in vertex form: $y = a {\left(x + b\right)}^{2} + c$

$y = {x}^{2} - 4 x \textcolor{b l u e}{+ 4 - 4} + 1 \text{ } \left[\textcolor{b l u e}{+ {\left(\frac{b}{2}\right)}^{2} - {\left(\frac{b}{2}\right)}^{2}}\right]$

$y = {\left(x - 2\right)}^{2} - 3$

The vertex is at $\left(- b , c\right) = \left(2 , - 3\right)$