Question 83113
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a. y=4x^2+8x-4        answer: y = 4(x+1)²-8 
b. y=-3x^2-4x-1       answer: y = -3{{{(x-2/3)^2}}} + {{{1/3}}}
c. y=-5x^2+10x+1      answer: y = 5(x-1)²+6  
d. y=-2x^2+10x-11     answer: y = -2(x-5/2)²+3/2
e. y=3x^2+9x+6        answer: y = -3{{{(x+3/2)^2}}} - {{{3/4}}}

I'll do (b) only.  The others are the same.

 y = -3x² - 4x - 1

Factor the coefficient of x² out of the first two terms.
(Do not factor out x with it)

y = -3(x² + {{{4/3}}}x) - 1

To the side multiply {{{4/3}}} by {{{1/2}}}, getting {{{2/3}}}
Square {{{2/3}}} getting {{{4/9}}}

Add and then subtract {{{4/9}}} inside the parentheses:

y = -3(x² + {{{4/3}}}x + {{{4/9}}} - {{{4/9}}}) - 1

Change the parentheses to brackets so parentheses may be
inserted:

y = -3[x² + {{{4/3}}}x + {{{4/9}}} - {{{4/9}}}] - 1

Factor the first three terms in the brackets as a perfect
square:

y = -3[{{{(x+2/3)^2}}} - {{{4/9}}}] - 1

Remove the bracket by distributing the -3, remembering
to leave the {{{(x+2/3)^2}}} intact:

y = -3{{{(x+2/3)^2}}} + {{{4/3}}} - 1

Write the 1 as {{{3/3}}}

y = -3{{{(x+2/3)^2}}} + {{{4/3}}} - {{{3/3}}}

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

Now plot the vertex ({{{-2/3}}},{{{1/3}}})

{{{drawing(400,400,-2,2,-2,1.5, locate(-.7,.415,o),
     graph(400,400,-2,2,-2,1.5) )}}}

Find the x-intercepts by setting y = 0

              y = -3x² - 4x - 1 

              0 = -3x² - 4x - 1

   3x² + 4x + 1 = 0

(3x + 1)(x + 1) = 0

3x + 1 = 0                x + 1 = 0
    3x = -1                   x = -1
     x = {{{-1/3}}}
    
So the x intercepts are ({{{-1/3}}},0) and (-1,0)
Plot them.

{{{drawing(400,400,-2,2,-2,1.5, locate(-.7,.415,o),
     locate(-.37,.083,o), locate(-1.035,.083,o),  
     graph(400,400,-2,2,-2,1.5)) }}}
           
Find the y-intercept by letting x = 0 in the original

 y = -3x² - 4x - 1
 y = -3(0)² -4(0) - 1
 y = 0 - 4 - 1
 y = -1

So plot the y-intercept (0.-1) too

{{{drawing(400,400,-2,2,-2,1.5, locate(-.7,.415,o),
     locate(-.37,.083,o), locate(-1.035,.083,o), locate(-.05, -.93,o),
     graph(400,400,-2,2,-2,1.5) )}}}

Now draw a smooth parabola through those 4 points:

{{{drawing(400,400,-2,2,-2,1.5, locate(-.7,.415,o),
     locate(-.37,.083,o), locate(-1.035,.083,o), locate(-.05, -.93,o),
     graph(400,400,-2,2,-2,1.5,-3x^2-4x-1) )}}}


Edwin</pre>