Question 312383
Please help! I am lost in how to figure this out:

{{{"f(x)"= -2x^2+2x+8}}}
<pre><b>
That's the same as

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

Plot some points:

(-2,-4), (-1,4), (0,8), (1,8), (2,4), (3,-4)

and get this graph:

{{{graph(200,400,-3,4,-5,9, -2x^2+2x+8)}}}
</pre></b>
This one has several parts..

The x coordinate of the vertex?
<pre><b>
Learn the formula:

The x-coordinate of {{{f(x)=Ax^2+Bx+C}}} is {{{-B/(2A)}}}

Compare
{{{"f(x)"= -2x^2+2x+8}}}
to
{{{"f(x)"= Ax^2+Bx+C}}}

and observe that {{{A=-2}}}, {{{B=2}}}, {{{C=8}}}

Then substitute and get

{{{-B/(2A)=-2/(2*(-2))=-2/(-4)=1/2)}}}
</pre></b>
The y coordinate of the vertex?
<pre><b>
Learn the rule: To get the y-coordinate of the vertex,
after finding the x-coordinate of the vertex, substitute
the value of the x-coordinate of the vertex x in the original 
equation:

{{{"f(x)"= -2x^2+2x+8}}}
{{{f(1/2)= -2(1/2)^2+2(1/2)+8}}}
{{{f(1/2)= -2(1/4)+1+8}}}
{{{f(1/2)= -1/2+9}}}
{{{f(1/2)=-1/2+18/2=17/2=8&1/2}}}

So the y coordinate of the vertex is {{{17/2}}} or {{{8&1/2}}}

That makes the vertes ({{{1/2}}},{{{8&1/2}}}), or (0.5,8.5)

and we see that looks right according to the graph:

{{{drawing(200,400,-3,4,-5,9, 
graph(200,400,-3,4,-5,9, -2x^2+2x+8),
locate(.3,9,"(0.5,8.5)")
)}}}

</pre></b>
The equation of the line of symmetry?
<pre><b>
Learn the rule that the equation of the line of symmetry
is always the vertical line whose equation is

x = the x-coordinate of the vertex

So the axis of symmetry has the equation 

{{{x = 1/2}}} or {{{x=0.5}}}

which is a vertical line {{{1/2}}} a unit right of the y-axis
and parallel to it (vertical), and when we graph it (in green,

{{{drawing(200,400,-3,4,-5,9, 
graph(200,400,-3,4,-5,9, -2x^2+2x+8),
locate(.3,9,"(0.5,8.5)"), green(line(1/2,-6,1/2,10))
)}}}

and we see that this vertical line whose equation is {{{x=1/2}}},
bisects the parabola. 
</pre></b>
The maximum/minimum of f(x)?
<pre><b>
Learn that the maximum or minimum value is the y-coordinate of
the vertex.

The maximum or minimum point is the same as the vertex.

To determine whether or not it is a maximum or a minimum
we need to know whether the parabola opens upward or downward.
To determine this we look at the coefficient of {{{x^2}}}.
If it is positive then the parabola opens upward and the
vertex is a minimum point. If it is negative, the parabola 
opens downward and the vertex is a maximum.  

{{{"f(x)"= -2x^2+2x+8}}}

Here we see that the coefficient of {{{x^2}}} is negative and
that the parabola opens downward, so the vertex is a maximum.
We knew it opened downward because we graphed it, but we
would have known it opened downward even if we had not
graphed it because the coefficient of {{{x^2}}} is negative. 

</pre></b>
Is the value of {{{f(1/2)=17/2}}} maximum or minimum?
<pre><b>
It is a maximum as we have just seen.

Thanks!