How do you find the axis of symmetry, vertex and x intercepts for #y=x^2-6x+5#?

#"given a parabola in standard form "y=ax^2+bx+c#

#"then the x-coordinate of the vertex can be found using"#

#•color(white)(x)x_(color(red)"vertex")=-b/(2a)#

#y=x^2-6x+5" is in standard form"#

#"with "a=1,b=-6,c=5#

#rArrx_(color(red)"vertex")=-(-6)/2=3#

#"substitute this value into equation for y-coordinate"#

#y_(color(red)"vertex")=3^2-6(3)+5=-4#

#rArrcolor(magenta)"vertex "=(3,-4)#

#"the axis of symmetry is vertical and passes through the"#
#"vertex with equation"#

#x=3#

#"to find x-intercepts let y = 0"#

#rArrx^2-6x+5=0#

#"the factors of + 5 which sum to - 6 are - 1 and - 5"#

#rArr(x-1)(x-5)=0#

#"equate each factor to zero and solve for x"#

#x-1=0rArrx=1#

#x-5=0rArrx=5#

#rArrx=1" and "x=5larrcolor(red)"x-intercepts"#
graph{(y-x^2+6x-5)(y-1000x+3000)=0 [-10, 10, -5, 5]}

Given:

#y=x^2-6x+5# is a quadratic equation in standard form:

#y=ax^2+bx+c,

where:

#a=1#, #b=-6#, and #c=5#.

Axis of symmetry: vertical line that separates the parabola into two equal halves, designated #x#.

#x=(-b)/(2a)#

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

#x=6/2#

#x=3#

The axis of symmetry is #x=3#.

Vertex: maximum or minimum point of the parabola, #(x,y)#. Since #a>0#, the vertex will be the minimum point and the parabola will open upward.

Substitute #3# for #x# in the equation and solve for #y#.

#y=3^2-6(3)+5=-4#

The vertex is #(3,-4)#

X-intercepts: values of #x# when #y=0#

Substitute #0# for #y#. Solve for #x#.

#0=x^2-6x+5#

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

#0=(x-5)(x-1)#

Set each binomial equal to zero.

#(x-5)=0#

#x=5#

#(x-1)=0#

#x=1#

The x-intercepts are #(5,0)# and #(1,0)#.

Y-intercept: value of #y# when #x=0#.

Substitute #0# for #x# and solve for #y#.

#y=0^2-6(0)+5#

#y=5#

The y-intercept is #(0,5)#.

Plot the points for the vertex, x-intercepts, and y-intercept. Sketch a parabola through the points. Do not connect the dots.

graph{y=x^2-6x+5 [-14.3, 14.17, -9.97, 4.27]}