E.g. to find the extremum of we can notice that . This operation maps every value x to through the axis of symmetry and vice-versa (e.g. 0 is mapped to 6 and 6 mapped to 0). This operation preserves symmetry for the axis of symmetry and hence the extremum of the function.
What about the function ? We know that the minimum of this function is at but . Also this operation maps e.g 4 to 10, but 10 to 16! Hence, this approach fails with this example (symmetry is not preserved).
Why is that so? Why is this approach not working for all the quadratic functions?