Question

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

Answer 1

The axis of symmetry is `x=3`.

The vertex is `(3,-4)`.

The zeroes are `(1,0)` and `(5,0)`

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^2-6x+5` is a quadratic equation in standard form , `ax^2+bx+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=(-b)/(2a)`.

For the equation `y=x^2-6x+5`,

`x=(-(-6))/(2*1)`

`x=6/2=3`

The axis of symmetry is `x=3`.

The vertex of a parabola is its maximum or minimum point `(x,y)`. In the equation, `y=x^2-6x+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^2-6x+5`

`y=3^2-6(3)+5`

Simplify and solve for `y`.

`y=9-18+5`

`y=-4`

The vertex is `(3,-4)`.

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^2-6x+5`

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

`0=color(red)((x-1))color(blue)((x-5))`

`color(red)((x-1)=0)`

`color(red)(x=1)`

`color(blue)((x-5)=0)`

`color(blue)(x=5)`

The zeroes are `(1,0)` and `(5,0)`.

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]}