First of all, when the coefficient of the highest term is a letter, it should be classified and discussed;

Secondly, the positional relationship between the symmetry axis of quadratic function and the interval is investigated: on the left or right side of the midpoint of the interval;

Finally, in order to make it easier to understand, I divide the discussion about the classification of quadratic functions into four categories:

The symmetry axis is on the left side of the interval, passing through the interval (left side of the midpoint, right side of the midpoint) and on the right side of the interval;

Details are as follows:

Note: The above results can be combined appropriately and written in the form of piecewise function.