Is there a formula to discovering if a given table of values is linear or quadric? :

One method I use is to look for the differences between the values.

In a linear table of values, the difference between each and every value should be the same.

In a quadric table of values, the difference will vary.


Also is there a formula to find the transformation that applies between the parent & the transformed?:

There is no such formula but one can plot the graph and possibly through intelligent guessing, figure out the transformation.