Date: Nov 16, 2012 7:38 PM
Author: Peter Duveen
Subject: Problem with transformations
The text (Precalculus with limits: a graphing approach Larson, etc.) tells us as follows (p43):
"...you can obtain the graph of g(x) = (x - 2)^2 by shifting the graph of f(x) = x^2 two units to the right, as shown in Figure 1.42 [AN ASSERTION]. In this case, the functions g and f have the following relationship.
g(x) = (x - 2)^2
= f(x - 2) (right shift of two units)[AN ASSERTION]
The following list summarizes vertical and horizontal shifts:" etc. etc.
I feel the assertions are not self-evident, and the treatment is generally confusing.
I would have treated this differently. I would have first attempted to establish a relationship between a function and another function which is the translation of the first so many spaces horizontally.
The relationship is f(x) = g (x + c). That is, the two functions have the same value when the arguments of f and g differ by a particular constant. Assuming we know the form of f(x), what is the form of g(x)?
We introduce the argument f(x - c), and want to see what happens to g, namely, f(x - c) = g[(x - c) + c]
We thus arrive at the expression f(x - c) = g(x). We have now established the form of g(x) in terms of f(x), which we know. It is simply f(x - c), which is not the same as f(x). In other words, we have derived and demonstrated what the textbook merely asserts.