Question 619129
  i have been stuck on this problem for days can someone please help me.use 
completing the square to describe the graph f(x)=40-12x-x^2.support your answer graphically.

   First  review  what  a  perfect square is.  It  is  a  trinomial  which  can  be  obtained  by  squaring  a  binomial(  multiplying  a  binomial  by  itself.

   For  example  a  binomial  has  2  terms.  2x^2   +  2x
    or  4x  + 5
    When  you have  a  trinomial  with  an  x^2  term   and  
  one  x  term
  and  one  constant(number)
  then  the  kind of  binomial  we
  are  looking  to  square  is  of  the  
   form  bx  +  c.

  Let's  square  some  binomials.

    a) (x + 2)  ^2   is  (x + 2)(x + 2)  =  x ^2   +   4x   +  4

    b)   (x +   3 )  ^2  =  (x + 3)(x + 3)  =   x ^2  + 6x  +  9
    c)   (x + 4) ^2 =  (x + 4)(x +4)  =    x^2   +   8x    +16
    d)   AND   (x + 5) ^ 2  =  (x+5)(x+5)  =   x^2  +10x  + 25

     I'm   sensing  a  pattern.

     a)     2(2)  =4,    2^ 2  =4
     b)     3(2) = 6,    3^2  =9
     c)     4(2) =8 ,    4^2  =8
     d)     5(2 = 10,    5^2  =25

    Now practice  squaring  binomials  without writing each one twice as aabove.

      (x + 6)^2  =   X^2   +  ___x    +  ___
            
For  the  x  coefficient,  double   the  6.         
For  the  constant  term,  square  the  6

     so  (x + 6) ^ 2 =    x^2   + 12x  +  36

       x^2   +12x   +   ?   to  make  a  perfect  square?
        Take  1/2  of  12  and  square it .  6^2  =  36.
           Make  this binomial  into  a  perfect  square. 
     x^2+  16x    +  ?      ( 1/2  of  16  =  8.   8^2  =64)
      x^2  +16x + 64

   We  cannot  have  a   -  in front of  the  x,  because  squaring a number
    never  produces  a  negative.

     Write     40 -12x  - x^2  in  standard  form  as
   A.    -x^2 -12x  + 40  =  y(or  f(x))
    Next we  replace   y  with  0.
     Why?   Because  points  with  y coordinate =  0 
     are  on  the  x-axis., giving us the
     x-intercepts.
     Once we know  the position  of the intercepts  we
     find the line of symmetry  that is 
     halfway  between  these  points.
  A.   -x^2 -12x  +40  =  0
       -x^2  -12x   =  -40  (Multiply  by  -1.
     B.  x^2  + 12x  =  +40
       Now  what do we add to  left 
     side  to  complete  the  square?
     We  add  1/2(12)   and  square  it,  getting  36.   Add   to  BOTH   sides.
   B.   x^2   +12x     +36   =  40 +36
            (x + 6)^2  =   76.  taking  sq.  roots:
        x  +  6   =  sq. root  of  76
         x + 6 =   8.7    0r    x +  6 =  -8.7 
          x  =  2.7        and    x  = -14.7
     Intercepts  are   (2.7, 0)   and  ( -14.7, 0)
       Line  of  symmetry  is  x  =  -6
       The vertex  is  on  this  line .  
   The x-coordinate  is  -6,   and    y  coordinate  is f(-6)