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Topic: Problem with transformations
Replies: 5   Last Post: Nov 17, 2012 1:20 PM

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Peter Duveen

Posts: 163
From: New York
Registered: 4/11/12
Problem with transformations
Posted: Nov 16, 2012 7:38 PM
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The text (Precalculus with limits: a graphing approach Larson, etc.) tells us as follows (p43):
" 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.

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