# How do you find the axis of symmetry, and the maximum or minimum value of the function y=x^2-4x-5?

Apr 19, 2018

Axis of Symmetry: $x = 2$
Minimum value: $y = - 9$ at point $\left(2 , - 9\right)$

#### Explanation:

First, factor the equation to find the roots:
$y = {x}^{2} - 4 x - 5$ Set $y = 0$ to find the roots of the equation.
$0 = {x}^{2} - 4 x - 5$ Factor
$\left(x + 1\right) \left(x - 5\right) = 0$ Using the zero products property,
$\left(x + 1\right) = 0$ and $\left(x - 5\right) = 0$ so the roots are:
$x = - 1 , 5$

Since $a$ of $a {x}^{2} + b x + c$ for this equation is positive, it opens upward with a minimum value, which is below $0$ because it has $2$ roots. Since parabolas are symmetric, the axis of symmetry must be in the middle of the two roots:
A.o.S.$= \left(\frac{{x}_{1} + {x}_{2}}{2}\right)$ (adapted average formula)
A.o.S.$= \left(\frac{\left(- 1\right) + 5}{2}\right)$
A.o.S.$= 2$
Axis of Symmetry: $x = 2$

The axis of symmetry will intersect the minimum of the parabola, so we can input $x = 2$ into the equation:
$y = {x}^{2} - 4 x - 5$
$y = {\left(2\right)}^{2} - 4 \left(2\right) - 5$ Combining like terms:
$y = 4 - 8 - 5$ Combining like terms:
$y = - 9$
Therefore, the minimum is $\left(2 , - 9\right)$