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If you want the parameters a, b and c then you must know the values of x and y. Even in this case you will not be able to calculate unique values for a,b and c but rather get a set of answers which describes the relationship necessary between a,b and c that will satisfy your values for x and y.

Here is how to do that:

Start with the usual parabolic equation: y = ax^2 + bx + c

There is a standard formula to solve this equation for x when you want y to be zero (also called the 'roots' of the equation). That formula is x = (-b +/- sqrt ( 4ac - b^2 ) ) / 2a).

The sqrt means take the square root of what in parentheses. and +/- means the formula must be calculated twice, once using a plus in place of +/- and once using a minus.

Instead of y being zero we want to be able to make it any value we like but our formula only works for y being zero. So we need to subtract the y value from the original parabolic equation to make one side zero, that give us 0 = ax^2+bx + (c-y).

Now we can use the standard formula again , but this time with c-y in place of just c.

x = (-b +/- sqrt( 4a(c-y) - b^2 ) / 2a

Using this new formula, if you know x and y you can calculate a relationship between a,b and c that will always give you that value of x and y in a parabolic equation.

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Q: How do you find a b and c in a parabolic equation of y equals a times x to the power of 2 plus b times x plus c?
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