# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y= x^2 -6x + 5?

Mar 1, 2016

The axis of symmetry is $x = 3$.

The vertex is $\left(3 , - 4\right)$.

The zeroes are $\left(1 , 0\right)$ and $\left(5 , 0\right)$

#### Explanation:

In order to be thorough, this is a very long explanation.

In order to graph this parabola, you will need the vertex and the zeroes.

$y = {x}^{2} - 6 x + 5$ is a quadratic equation in standard form , $a {x}^{2} + b x + c$, where $a = 1 , b = - 6 , c = 5$.

The axis of symmetry is an imaginary vertical line that divides a parabola into two equal halves that are mirror images. The formula for the axis of symmetry for an equation in standard form is $x = \frac{- b}{2 a}$.

For the equation $y = {x}^{2} - 6 x + 5$,

$x = \frac{- \left(- 6\right)}{2 \cdot 1}$

$x = \frac{6}{2} = 3$

The axis of symmetry is $x = 3$.

The vertex of a parabola is its maximum or minimum point $\left(x , y\right)$. In the equation, $y = {x}^{2} - 6 x + 5$, the vertex is the minimum point because $a > 0$.

The value of $x$ from the axis of symmetry is also the value of $x$ of the vertex. To find the value of $y$, substitute $3$ for $x$ and solve for $y$.

$y = {x}^{2} - 6 x + 5$

$y = {3}^{2} - 6 \left(3\right) + 5$

Simplify and solve for $y$.

$y = 9 - 18 + 5$

$y = - 4$

The vertex is $\left(3 , - 4\right)$.

In order to graph the parabola, you will also need the zeroes , which are the values of $x$ when $y = 0$.

Substitute $0$ for $y$, factor and solve for $x$.

$0 = {x}^{2} - 6 x + 5$

Find two numbers that when added equal $- 6$ and when multiplied equal $5$. The numbers $- 1$ and $- 5$ meet the requirements.

$0 = \textcolor{red}{\left(x - 1\right)} \textcolor{b l u e}{\left(x - 5\right)}$

$\textcolor{red}{\left(x - 1\right) = 0}$

$\textcolor{red}{x = 1}$

$\textcolor{b l u e}{\left(x - 5\right) = 0}$

$\textcolor{b l u e}{x = 5}$

The zeroes are $\left(1 , 0\right)$ and $\left(5 , 0\right)$.

Plot the vertex and the zeroes. Sketch a parabola making sure it goes through these three points. The graph should be curved. Do not connect the dots.

graph{y=x^2-6x+5 [-10, 10, -5, 5]}