How can I determine a parabola given two points?

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To determine a parabola given two points, one must use the general form of a parabola, expressed as y = ax² + bx + c. With only two points, it is impossible to uniquely identify the coefficients a, b, and c due to the infinite number of parabolas that can pass through those points. However, by substituting the coordinates of the two points into the general equation, one can create simultaneous equations. Any solution to these equations will yield a valid parabola that intersects the given points.

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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 [itex]y=ax^2 +bx+c[/itex]

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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