Question 207627
The min/max is simply the y-coordinate of 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 {{{P(x)=-3x^2+6x+5}}}, we can see that {{{a=-3}}}, {{{b=6}}}, and {{{c=5}}}.



{{{x=(-(6))/(2(-3))}}} Plug in {{{a=-3}}} and {{{b=6}}}.



{{{x=(-6)/(-6)}}} Multiply 2 and {{{-3}}} to get {{{-6}}}.



{{{x=1}}} Divide.



So the x-coordinate of the vertex is {{{x=1}}}. Note: this means that the axis of symmetry is also {{{x=1}}}.



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



{{{P(x)=-3x^2+6x+5}}} Start with the given equation.



{{{P(1)=-3(1)^2+6(1)+5}}} Plug in {{{x=1}}}.



{{{P(1)=-3(1)+6(1)+5}}} Square {{{1}}} to get {{{1}}}.



{{{P(1)=-3+6(1)+5}}} Multiply {{{-3}}} and {{{1}}} to get {{{-3}}}.



{{{P(1)=-3+6+5}}} Multiply {{{6}}} and {{{1}}} to get {{{6}}}.



{{{P(1)=8}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=8}}}.



So the vertex is *[Tex \LARGE \left(1,8\right)].



Since the min/max is the y-coordinate of the vertex, this means that the min/max is 8.



Now the question is: is this value a min or is it a max? Since the leading coefficient of {{{y=-3x^2+6x+5}}} is negative, this means that the value is a max. Graph the equation if you're not sure.