Translation
You'd like to translate a parabola y = a x 2 + b x + c horizontally, for a vector t = ( t , 0 ) , t ∈ R . New parabola is y t = a 2 x 2 + b 2 x + c 2 . What is c 2 in terms of a , b , c ?
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1 solution
Better alternative is to notice that c 2 = y ( − t ) .
Translating a curve in a fixed reference system is equivalent to translating the reference system in the opposite direction keeping the curve fixed in position . In the transformed reference system, the coordinates of a point are : x t = x + t and y t = y . So the equation of the curve y = f ( x ) in the original system should be y = f ( x + t ) in the transformed system. Hence c 2 in this case should be a t 2 + b t + c , isn't it?
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Yes, that's why added this solution as "a better alternative" in the comment.
substituting x − t for x performs the translation: y t = a ( x − t ) 2 + b ( x − t ) + c = a x 2 + ( b − 2 a t ) x + a t 2 − b t + c
so we get a 2 = a , b 2 = b − 2 a t , and c 2 = a t 2 − b t + c .
We'll take a 2 = a for granted. The symmetry axis x = 2 a − b will translate to x t = 2 a − b + t = 2 a 2 − b 2 so we obtain b 2 = b − 2 a t . Discriminant won't change because distance between solutions don't change with horizontal translation (equivalent condition would emerge by taking the minimal or maximal value as invariant under horizontal translation). Either way, b 2 − 4 a c = b 2 2 − 4 a c 2 . Inserting b 2 = b − 2 a t yields c 2 = c − b t + a t 2 .