Question 959744
for {{{g(x)=3x^2-12x+6}}}

a.find the values of the {{{x}}} intercepts

{{{3x^2-12x+6=0}}}

{{{3(x^2-4x+2)=0}}}...since {{{3<>0}}},then we need to find for what {{{x}}} is

{{{(x^2-4x+2)=0}}}..........use quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

{{{x = (-(-4) +- sqrt( (-4)^2-4*1*2 ))/(2*1) }}}

{{{x = (4 +- sqrt( 16-8 ))/2 }}}

{{{x = (4 +- sqrt( 8 ))/2 }}}


{{{x = (cross(4)2 +- cross(2)sqrt(2 ))/cross(2) }}}

{{{x = (2 +- sqrt(2 )) }}}

solutions
{{{x = 2 +sqrt( 2 ) }}}-exact solution or {{{x = 3.41 }}} approximately
or
{{{x = 2 -sqrt( 2 ) }}} or {{{x = 0.59 }}} approximately

b.state the coordinates of the {{{y}}} intercept
set {{{x=0}}} to find the {{{y}}} intercept 

{{{y=3*0^2-12*0+6}}}
{{{y=6}}}

the {{{y}}} intercept is at ({{{0}}},{{{6}}})


c.state the vertex

the equation for a parabola can also be written in "vertex form":

{{{y = a(x - h)^2 + k}}}

In this equation, the vertex of the parabola is the point ({{{h}}},{{{ k}}}).

{{{y=(3x^2-12x)+6}}}...complete square

{{{y=3(x^2-4x+_)-3*_+6}}}

{{{y=3(x^2-4x+2^2)-3*2^2+6}}}

{{{y=3(x-2)^2-12+6}}}

{{{y=3(x-2)^2-6 }}}=> {{{h=2}}} and {{{k=-6}}}

so, ({{{h}}},{{{ k}}})=({{{2}}},{{{ -6}}})

d.state the axis of symmetry

The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry. 

here, that is {{{x=2}}}

e.state the direction of the parabola

since {{{a=3}}} and if {{{ a > 0}}}, the parabola opens {{{upward}}}


f.decide whether there is a relative maximum or minimum, then state it

the parabola opens {{{upward}}} have an minimum


g.graph the function accurately using the above information

{{{drawing( 600, 600, -10, 10, -10, 10,
circle(2,-6,.12),circle(3.41,0,.12),circle(0.59,0,.12),circle(0,6,.12),
locate(2,-6,V(2,-6)),locate(3.41,0.5,p(3.41,0)),locate(0.59,0.5,p(0.59,0)),locate(0,6,p(0,6)),
 graph( 600, 600, -10, 10, -10, 10,3(x-2)^2-6)) }}}


h. state the domain

{{{R}}}  (all real numbers)

i.state the range

{ {{{y}}} element {{{R}}} : {{{y>=-6}}} }

j.state the interval of {{{x}}} where the graph is increasing

({{{2}}},{{{infinity}}})

k.state the interval of {{{x}}} where the graph is decreasing

({{{-infinity}}},{{{2}}})