How can I determine a parabola given two points?

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To determine a parabola given two points, it's essential to recognize that two points alone are insufficient to uniquely define the coefficients a, b, and c in the general form y = ax^2 + bx + c. By substituting the coordinates of the two points into the equation, simultaneous equations can be formed, but these will not yield a unique solution due to the existence of multiple parabolas passing through the same points. Additional information about the parabola, such as its vertex or direction, is necessary to narrow down the possibilities. If no extra conditions are provided, any solution to the equations can represent a valid parabola. Thus, without further constraints, one can only find a family of parabolas rather than a single explicit equation.
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If given two arbitrary points on a parabola, is there anyway to determine an explicit equation for that parabola. I know there are multiple possible parabolas, but how can I find just one of them?
 
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the general form of a parabola is y=ax^2 +bx+c

so given two points on the parabola, that would be insufficient to find a,b and c. If there is something else given about those 2 points then you may be able to find it.
 
If given two points, and you want the family of parabolas to go through them, just substitute those points into the general form of a parabola and you should get some simultaneous equations. You won't able able to get unique solutions for them, as you know, because there are many parabolas that go through those 2 points. If you want just any, choose any solution to the equations.
 

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