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Problem with transformations
Posted:
Nov 16, 2012 7:38 PM


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 selfevident, 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.



