# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y = x^2 - 4x - 2?

Feb 13, 2018

The line of symmetry is at $x = 2$, the minimum at $y = - 6$, and the maximum at $y = \infty$.

#### Explanation:

The $x$-value of the vertex is the line of symmetry of a parabola.
The $y$-value of the vertex is the minimum value, when the leading coefficient, $c$, is of $c > 0$. When $c < 0$, the $y$-value of the vertex is the maximum value.

The coordinates of the vertex are $\left(h , k\right)$, when the quadratic is of the form $a {\left(x - h\right)}^{2} + k$.

We need to find $h$ for the line of symmetry, and $k$ for the minimum value.

We first write ${x}^{2} - 4 x - 2$ as:

$\left({x}^{2} - 4 x\right) - 2$, where $a = 1$.

We must then complete the square. Add $4$ inside the brackets and subtract it outside.

We get $\left({x}^{2} - 4 x + 4\right) - 6$

Which reduces to:

${\left(x - 2\right)}^{2} - 6$.

So the vertex has the coordinates $\left(2 , - 6\right)$, and the axis of symmetry is at $x = 2$, and the minimum value is $y = - 6$.

When $c > 0$, the maximum value is $\infty$. When $c < 0$, the minimum value is $- \infty$. Here, $c > 0$, so the maximum value is $\infty$.