Question 253267
Take note that {{{y=-0.5x^2+2x+3}}} is equivalent to {{{y=-(1/2)x^2+2x+3}}} since {{{-1/2=-0.5}}}



To find the optimal value, we need to find the vertex.



In order to find the vertex, we first need to find the x-coordinate of the vertex.



To find the x-coordinate of the vertex, use this formula: {{{x=(-b)/(2a)}}}.



{{{x=(-b)/(2a)}}} Start with the given formula.



From {{{y=-(1/2)x^2+2x+3}}}, we can see that {{{a=-1/2}}}, {{{b=2}}}, and {{{c=3}}}.



{{{x=(-(2))/(2(-1/2))}}} Plug in {{{a=-1/2}}} and {{{b=2}}}.



{{{x=(-2)/(-1)}}} Multiply 2 and {{{-1/2}}} to get {{{-1}}}.



{{{x=2}}} Divide.



So the x-coordinate of the vertex is {{{x=2}}}. Note: this means that the axis of symmetry is also {{{x=2}}}.



Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.



{{{y=-(1/2)x^2+2x+3}}} Start with the given equation.



{{{y=-(1/2)(2)^2+2(2)+3}}} Plug in {{{x=-2}}}.



{{{y=-(1/2)(4)+2(2)+3}}} Square {{{-2}}} to get {{{4}}}.



{{{y=-2+2(2)+3}}} Multiply {{{-1/2}}} and {{{4}}} to get {{{-2}}}.



{{{y=-2+4+3}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}.



{{{y=5}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=5}}}.



So the vertex is *[Tex \LARGE \left(2,5\right)].



Since the max/min occurs at the vertex as the 'y' value, this means that the max is {{{y=5}}}. So the optimal value is is {{{y=5}}}.



Here's a graph to visually confirm the answer:



{{{ drawing(500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-0.5x^2+2x+3)

)}}}


Graph of {{{y=-0.5x^2+2x+3}}} with the vertex of (2,5). Here we can see that the largest y value is y=5.