find the general equation of the parabola that passes through the three points (1,3), (-1,9) and (2,6)

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

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A parabola is an equation of the form `y=ax^2+bx+c`

(where `a != 0`  )

If we had a,b, and c, we would have the equation.


We know the parabola passes through (1,3), which means we can plug in 1 for x and 3 for y, and the equation should be satisfied.  So:


Similarly for (-1,9) and (2,6)


So we have:




Thus:` `




We want to solve for a,b,c.  Subtract the first equation from the second equation:

`6=-2b`      so   `b=-3`

Plug this into the equations to get:





The first two equations essentially say the same thing.  Simplifying, we have:




Subtract the first from the second, to get:

`6=3a` so `a=2`   

Plug this into either equation:


We must have `c=4`





The equation must be `y=2x^2-3x+4`

(If you plot that graph, you can see that it really does go through the points)



The parabola that passes through the points is:

`y=2x^2-3x+4 `