Question 472862
The {{{x-intercept}}} occurs when {{{y=0}}}, make the equation = {{{0}}} and solve for {{{x}}}

{{{y= -x^2/2+3x+8}}}


{{{0= -x^2/2+3x+8}}}......... multiply by {{{-2}}}


{{{0= x^2-6x-8}}}.......or


{{{x^2-6x-8=0}}}.........use quadratic formula


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


{{{x = (-(-6) +- sqrt( (-6)^2-4*1*(-8) ))/(2*1) }}} 


{{{x = (6 +- sqrt(36+32 ))/2 }}} 


{{{x = (6 +- sqrt(68))/2 }}}


 {{{x = (6 +- 8.25)/2 }}}

solutions:

{{{x = (6 +8.25)/2 }}}

{{{x = 14.25/2 }}}

{{{x = 7.125 }}}

or

{{{x = (6 -8.25)/2 }}}

{{{x = -2.25/2 }}}

{{{x =-1.125 }}}


{{{y-intercept}}} occurs when {{{x=0}}}, substitute {{{0}}} for {{{x}}} in the equation, and find {{{y}}}

{{{y= -0^2/2+3*0+8}}}

{{{y= 0+0+8}}}

{{{y= 8}}}

so, the {{{x-intercepts}}} are at points ({{{7.125}}},{{{0}}}) and ({{{-1.125}}},{{{0}}})

the {{{y-intercept}}} is at point ({{{0}}}{{{8}}})


Axis of symmetry can be found using the formula: {{{x = -b/2a}}}

since {{{a=(-1/2)}}} and {{{b=3}}},

{{{x = -b/2a=-3/2(-1/2)=-3/cross(2)(-1/cross(2))=-3/-1=3}}}

so, the axis of symmetry is {{{x=3}}}


Vertex is the x/y values for the max or min and occurs at the axis of symmetry; so Substitute {{{3}}} for {{{x}}} and find the {{{y}}} value: 

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

{{{y= -9/2+9+8}}}

{{{y= -4.5+17}}}

{{{y=12.5}}}

so, the Vertex is at ({{{3}}},{{{12.5}}})


now see it on a graph:

{{{ graph( 500, 500, -10,10, -10, 20,-x^2/2+3x+8) }}}