Question 157345

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)}}}.

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

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

{{{x=(-2)/(-4)}}} Multiply 2 and {{{-2}}} to get {{{-4}}}.

{{{x=1/2}}} Reduce.

So the x-coordinate of the vertex is {{{x=1/2}}}. Note: this means that the axis of symmetry is also {{{x=1/2}}}.

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

{{{y=-2(1/2)^2+2(1/2)+9}}} Plug in {{{x=1/2}}}.

{{{y=-2(1/4)+2(1/2)+9}}} Square {{{1/2}}} to get {{{1/4}}}.

{{{y=-1/2+2(1/2)+9}}} Multiply {{{-2}}} and {{{1/4}}} to get {{{-1/2}}}.

{{{y=-1/2+1+9}}} Multiply {{{2}}} and {{{1/2}}} to get {{{1}}}.

{{{y=19/2}}} Combine like terms.

So the y-coordinate of the vertex is {{{y=19/2}}}.

So the vertex is *[Tex \LARGE \left(\frac{1}{2},\frac{19}{2}\right)].

Also, since the leading coefficient is negative, this means that the vertex is at a maximum.

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So using this info we can answer the following:

The x coordinate of the vertex is {{{1/2}}}

The y coordinate of the vertex is {{{19/2}}}

The equation of the line of symmetry is {{{x=1/2}}}

The maximum or minimum of f(x) is {{{f(x)=19/2}}}

The value of f(x)=19/2 is a maximum