Question 1128907
{{{f(x)= 5x^2-70x+227}}}.....complete square

{{{f(x)= (5x^2-70x)+227}}}

{{{f(x)= 5(x^2-14x)+227}}}

{{{f(x)= 5(x^2-14x+b^2)-5b^2+227}}}=> coefficient {{{2ab=14}}}, {{{a=1}}}=>{{{b=7)))

{{{f(x)= 5(x^2-14x+7^2)-5*7^2+227}}}

{{{f(x)= 5(x-7)^2-245+227}}}

{{{f(x)=5 (x - 7)^2 - 18}}}


=> {{{h=7}}}, {{{k=-18}}}

vertex is  global minimum at ({{{7}}}, {{{-18}}})


Shifted: original function {{{f(x)=x^2}}} is shifted {{{7}}} units to the right

Range:

{ {{{y}}} element {{{R}}} : {{{y>=-18}}} }



x-Intercept:

{{{0=5 (x - 7)^2 - 18}}}

{{{18=5(x - 7)^2}}}

{{{18/5 =(x - 7)^2}}}

{{{sqrt(18/5) =x - 7}}}

{{{x=7+sqrt(18/5) }}} or {{{x=7-sqrt(18/5)}}}


 approximately {{{x=8.9}}} or {{{x=5.1}}}

x- intercepts are at ({{{10.6}}}, {{{0}}}) and ({{{3.4}}}, {{{0}}})



y-intercept: 

{{{f(x)=5 (0 - 7)^2 - 18}}}

{{{f(x)=5 ( - 7)^2 - 18}}}

{{{f(x)=5 (49) - 18}}}

{{{f(x)=227}}}

y-intercept is at ({{{0}}}, {{{227}}})



{{{drawing( 600, 600, -20,20, -20, 250,
circle(8.9,0,.26),circle(5.1,0,.26),circle(0,227,.26),
 graph( 600, 600, -20,20, -20, 250, 5x^2-70x+227)) }}}