SOLUTION: Consider the parabola through the points (1,5), (2,11), and (3,-3). i) Find the equation for the parabola y = ax² + bx + c. ii) Find the vertex. iii) find the x-intercepts.

Consider the parabola through the points (1,5), (2,11), and (3,-3).
  i) Find the equation for the parabola y = ax² + bx + c.
 ii) Find the vertex.
iii) find the x-intercepts.
 iv) Find the y-intercept.


i)  Substitute (x,y) = (1,5) in y = ax² + bx + c

         5 = a(1)² + b(1) + c
         5 = a + b + c
OR 
a + b + c = 5 

Substitute (x,y) = (2,11) in y = ax² + bx + c

        11 = a(2)² + b(2) + c
        11 = 4a + 2b + c
OR 
4a + 2b + c = 11

Substitute (x,y) = (3,-3) in y = ax² + bx + c

        -3 = a(3)² + b(3) + c
        -3 = 9a + 3b + c
OR 
9a + 3b + c = -3

So now you have a system of three equations in
three unknowns a, b, and c.

 a +  b + c =  5
4a + 2b + c = 11
9a + 3b + c = -3

Can you solve that system?  If not, post again.

Solution to system:  a = -10, b = 36, c = -21

Substituting in 

y = ax² + bx + c

y = -10x² + 36x - 21

---------------------------------

ii)  The x-coordinate h of the vertex is found by the formula:

       h = -b/(2a)
       h = -(36)/(2·-10)
       h = -36/(-20)
       h = 9/5

iii) The y-coordinate k of the vertex is found by substituting
     the x-coordinate in the equation for x and solving for y

y = -10x² + 36x - 21
y = -10(9/5)² + 36(9/5) - 21
y = -10(81/25) + 324/5 - 21
y = -162/5 + 324/5 - 105/5
y = 57/5

So h = 57/5

The vertex is therefore (9/5, 57/5) or in decimals (1.8, 11.4)

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iii) To find the x-intercepts, substitute 0 for y and solve for x

y = -10x² + 36x - 21
0 = -10x² + 36x - 21
    -10x² + 36x - 21 = 0

That does not factor, so we have to use the quadratic formula:
                 ________ 
           -b ± Öb² - 4ac
      x = -----------------
                 2a

                    ___________________ 
           -(36) ± Ö(36)² - 4(-10)(-21)
      x = -------------------------------
                     2(-10)

                  __________ 
           -36 ± Ö1296 - 840
      x = -------------------
                  -20

                  ___ 
           -36 ± Ö456
      x = ------------
              -20

                  _____ 
           -36 ± Ö4·114
      x = --------------
               -20

                   ___ 
           -36 ± 2Ö114
      x = --------------
               -20
     
Make two fractions:
                     ___
           -36     2Ö114
      x = ----- ± ---------
           -20      -20

                  ___
           9     Ö114
      x = --- ± ---------
           5      10

These come out to be approximately

x = .7322921748 and x = 2.867707825

----------------------------      
  
iii) To find the y-intercepts, substitute 0 
     for x and solve for y

y = -10x² + 36x - 21
y = -10(0)² + 36(0) - 21
y = 0 + 0 - 21
y = -21

So the y-intercept is -21.

Edwin