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

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]}