Question 371261
<pre>
{{{y=8-3x^2-2x}}}

Arrange the right side in descending order:

{{{y=-3x^2-2x+8}}}

Factor -3 out of the first two terms on the right:

{{{y=-3(x^2+expr(2/3)x)+8}}}

[Note that I got the {{{2/3}}}
 by dividing -2 by -3]


To the side multiply the coefficient of x, which is {{{2/3}}} by {{{1/2}}},
which gives {{{1/3}}} then square it, getting {{{1/9}}}, then add and
subtract {{{1/9}}} after the {{{expr(2/3)x}}}

{{{y=-3(x^2+expr(2/3)x+1/9-1/9)+8}}}

Notice that the first three terms inside the parentheses, 
{{{x^2+expr(2/3)x+1/9}}} can be factored as {{{(x+1/3)(x+1/3)}}} and
then as {{{(x+1/3)^2}}}.  So replace the first three terms by this:

{{{y=-3((x+1/3)^2-1/9)+8}}}


Remove the larger outside parentheses leaving the smaller parentheses
intact:

{{{y=-3(x+1/3)^2+3/9+8}}}

Reduce {{{3/9}}} to {{{1/3}}}

{{{y=-3(x+1/3)^2+1/3+8}}}

To add the last two terms:  {{{1/3+8 = 1/3+24/3=25/3}}}, so
the final answer is:

{{{y = -3(x+1/3)^2+25/3}}}

Compare that to

{{{y = a(x-h)^2+k}}}

and {{{a=-3}}}, {{{h=-1/3}}}, {{{k=25/3}}}

so the vertex is the point {{{V(h,k)=V(-1/3,25/3)}}}

the line of symmetry has the equation {{{x=h}}} or {{{x=-1/3}}}.

We know it opens downward since "a" is a negative number.

We find the y-intercept by substituting x = 0 in the original

equation:

{{{y=8-3x^2-2x}}}
{{{y=8-3(0)^2-2(0)}}}
{{{y=8-0-0}}}
{{{y=8}}}

So the y-intecept is (0,8), 

We find the x-intercepts from the original equation,
setting y=0

{{{y=8-3x^2-2x}}}

{{{0=8-3x^2-2x}}}

Rewrite as

{{{-3x^2-2x+8=0}}}

Multiply through by -1

{{{3x^2-2x+8=0)}}}

{{{(3x-4)(x+2)=0}}}

{{{3x-4=0}}} gives {{{x=4/3}}}

{{{x+2=0}}} gives {{{x=-2}}}

So the x-intercepts are the points ({{{4/3}}},0) and (-2,0)


so we plot the vertex, the y-intercept, the x-intercepts,
and the vertical line of symmetry:

{{{drawing(1000/3,800,-3,2,-2,10, graph(1000/3,800,-3,2,-2,10),
circle(4/3,0,.05), circle(-2,0,.05),
green(line(-1/3,-3,-1/3,11)), circle(0,8,.05), circle(-1/3,25/3,.05)  )}}}

and sketch in the parabola graph, through those points,
symmetrical about the green line of symmetry:

{{{drawing(1000/3,800,-3,2,-2,10, graph(1000/3,800,-3,2,-2,10,8-3x^2-2x ),
circle(4/3,0,.05), circle(-2,0,.05),
green(line(-1/3,-3,-1/3,11)), circle(0,8,.05), circle(-1/3,25/3,.05)  )}}}

Edwin</pre>