# How do you find the axis of symmetry, and the maximum or minimum value of the function y = x^2 - 6x + 4?

Jul 25, 2016

Axis if symmetry: $\textcolor{g r e e n}{x = 3}$
Minimum value: $\textcolor{g r e e n}{y = - 5}$

#### Explanation:

A quadratic equation opens upward if the coefficient of the squared variable is greater than zero (i.e. it has a minimum value).

The minimum value is attained when the slope of the tangent (as given by the derivative) is equal to zero.

That is the minimum occurs when
$\textcolor{w h i t e}{\text{XXX}} \frac{d \left({x}^{2} - 6 x + 4\right)}{\mathrm{dx}} = 2 x - 6 = 0$
or
$\textcolor{w h i t e}{\text{XXX}} x = 3$

The axis of symmetry is a vertical line (when $x$ is the dependent variable) passing though the minimum.
Therefore the axis of symmetry is $\textcolor{g r e e n}{x = 3}$

The value of the function $y = {x}^{2} - 6 x + 4$ at the minimum (i.e. when $x = 3$) is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{y =} {\left(3\right)}^{2} - 6 \left(3\right) + 4 = 9 - 18 + 4 = \textcolor{g r e e n}{- 5}$