Date: Nov 17, 2012 9:39 AM
Author: Peter Duveen
Subject: Re: Problem with transformations
Undoubtedly true, Clyde, that it will help for students to be familiar with various curves of the form (x-h)^2. etc.
But in addition to this, a kind of definition derived from the above experience should be formulated about exactly what a translation is. The vertical translation is easier to grasp, I think, but the horizontal one requires a bit more careful analysis.
A function is a horizontal translation of another function when, for every value of x, or at least for a range of x, f(x) = g(x + c). Such a definition must proceed determining the form of g, if the form of f is known. To merely assert the result, that g(x) = f(x - c) is the main point I am addressing. This conclusion should be derived from our definition, which ought to agree with our experience. If the conclusion is merely asserted, the students 1. may not really understand what is going on, and 2. will have trouble applying the formula with confidence.