Question 677751


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=-3x^2+9x+2}}}, we can see that {{{a=-3}}}, {{{b=9}}}, and {{{c=2}}}.



{{{x=(-(9))/(2(-3))}}} Plug in {{{a=-3}}} and {{{b=9}}}.



{{{x=(-9)/(-6)}}} Multiply 2 and {{{-3}}} to get {{{-6}}}.



{{{x=3/2}}} Reduce.



So the x-coordinate of the vertex is {{{x=3/2}}}. Note: this means that the axis of symmetry is also {{{x=3/2}}}.



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



{{{f(x)=-3x^2+9x+2}}} Start with the given equation.



{{{f(3/2)=-3(3/2)^2+9(3/2)+2}}} Plug in {{{x=3/2}}}.



{{{f(3/2)=-3(9/4)+9(3/2)+2}}} Square {{{3/2}}} to get {{{9/4}}}.



{{{f(3/2)=-27/4+9(3/2)+2}}} Multiply {{{-3}}} and {{{9/4}}} to get {{{-27/4}}}.



{{{f(3/2)=-27/4+27/2+2}}} Multiply {{{9}}} and {{{3/2}}} to get {{{27/2}}}.



{{{f(3/2)=35/4}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=35/4}}}.



So the vertex is *[Tex \LARGE \left(\frac{3}{2},\frac{35}{4}\right)].



Since the leading coefficient (which is -3) is negative, this means that the y coordinate of the vertex is the max of f(x). 


So the max of f(x) is {{{35/4}}}