Write the equation for the parabola that has x− intercepts (1.2,0) and (4,0) and y− intercept (0,12).

Equation of a parabola is written in the form of f(x)=ax²+bx+c.
The equation passes through points (4,0), (1.2,0) and (0,12), therefore; 
 replacing the points in the equation y = ax² +bx+c 
we get  0 = a(4)²+b(4) +c   for (4,0)
             0 = a (1.2)²+ b(1.2) +c for (1.2,0)
             12 = a(0)² +b(0) +c   for (0,12)
simplifying the equations we get
16a + 4b + c = 0
1.44a +1.2b + c = 0
+c = 12
thus the first two equations will be
16a + 4b = -12
1.44 a + 1.2b = -12 solving simultaneously 
the value of a = 5/2 and b =-13
Thus, the equation of the parabola will be given by;
 y= 5/2x² - 13x + 12 or y = 2.5x² - 13x + 12

y = 2.5x^2 - 13x + 12
   The equation for a parabola is a quadratic equation of the form
 y = ax^2 + bx + c
   So let's calculate a, b, and c.
First thing to note is that we have the y-intercept, so x is 0. That means that we have
 12 = a*0^2 + b*0 + c
 12 = 0 + 0 + c
 12 = c

   So we now know c which is 12.

   Now with the two intercepts we have
 ax^2 + bx + 12 = 0 where x is 1.2 or 4. So let's set them equal to each other. Then solve for b in terms of a.

   a(1.2)^2 + b(1.2) + 12 = a(4)^2 + b(4) + 12
a(1.44) + b(1.2) = a(16) + b(4)
 1.44a + 1.2b = 16a + 4b
 1.44a = 16a + 2.8b
 -14.56a = 2.8b
 -5.2a = b

   Let's substitute -5.2a for b in the equation we have so far and solve for a.
 y = ax^2 + bx + 12
  y = ax^2 - 5.2ax + 12

    Substitute known x & y value
 0 = a*4^2 - 5.2a*4 + 12
 0 = 16a - 20.8a + 12
 0 = -4.8a + 12
 -12 = -4.8a
 2.5 = a

   We now have both a = 2.5 and c = 12. Let's get b.
 -5.2a = b
 -5.2*2.5 = b
 -13 = b

   So the final equation is: y = 2.5x^2 - 13x + 12

   Let's verify with the given points.
 x = 1.2
 y = 2.5x^2 - 13x + 12
 y = 2.5*1.2^2 - 13*1.2 + 12
 y = 3.6 - 15.6 + 12
 y = 0

   x = 4
 y = 2.5x^2 - 13x + 12
 y = 2.5*4^2 - 13*4 + 12
 y = 2.5*16 - 13*4 + 12
 y = 40 - 52 + 12
 y = 0 x = 0
 y = 2.5x^2 - 13x + 12
 y = 2.5*0^2 - 13*0 + 12
 y = 2.5*0 - 13*0 + 12
 y = 0 - 0 + 12 = 12

   And all three given points confirmed.

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