Question 651391
{{{y = x^2 -6x +8}}}

A parabola is of the general form {{{y = ax^2 + bx + c}}}, and you have that.

If {{{a}}} is {{{positive}}} the parabola {{{goes}}}{{{ up}}} (looks like a smiley face), if {{{a}}} is {{{negative}}} the parabola {{{goes}}}{{{ down}}} (looks like a sad face).

It is {{{very}}}{{{ important}}} to note that for your question, you have the opposite {{{y=x}}}, instead of {{{x=y}}}, you goes up. 

In case of {{{x=y}}} will be on {{{its}}}{{{ side}}}, and when this occurs, a positive {{{a}}} value means it will go to the {{{right}}}, so it will look like a letter {{{c}}}.


first, you can do is to find the vertex using {{{ -b/ 2a}}}

{{{ -b/ 2a=-(-6)/2*1=6/2=3}}} ...this is {{{x}}} value...plug it in {{{y = x^2 -6x +8}}} and find {{{y}}} value

{{{y = 3^2 -6*3 +8}}}

{{{y = 9 -18 +8}}}

{{{y = 17 -18 }}}

{{{y = -1 }}}

so, the vertex is at the point ({{{3}}},{{{-1}}})


In this example you can factor to find the {{{roots}}}, the possible values for {{{x}}} in our general form, when {{{y = 0}}}. 

So,
 
{{{y = x^2 -6x +8}}}........substitute {{{-6x}}} with {{{-2x -4x}}} 

{{{y = x^2 -2x -4x +8}}}...group

{{{y = (x^2 -2x) -(4x -8)}}}...factor out {{{x}}} from first group and {{{4}}} from second group

{{{y= x(x -2) -4(x -2)}}}...set {{{y=0}}}

{{{0= (x -4)(x -2)}}}

if {{{(x -4)=0}}}..->...{{{x=4}}}

if {{{(x -2)=0}}}..->...{{{x=2}}}


So the parabola passes through the {{{x-axis}}} on the graph at {{{4}}} and {{{2}}}. As parabola are symmetrical, the {{{axis}}} of {{{symmetry}}} is halfway between these, ie {{{x =3}}}. 

So sub this value into the original equation to find the lowest point of the parabola.

{{{y = x^2 -6x +8}}}...now set {{{x=0}}} and solve for {{{x}}} to find {{{y-intercept}}}

{{{y = 0^2 -6*0 +8}}}

{{{y = 8}}}...so, there is {{{y-intercept}}} at the point ({{{0}}},{{{8}}})

you can also take few other positive and negative values for {{{x}}} , calculate {{{y}}} and plot that points too, and you are ready to draw a graph
 


{{{ graph( 600, 600, -6, 15, -10, 15,x^2 -2x -4x +8) }}}