Our Story So Far: Part I

OK. The snow day probably means postponing the exam again. Meanwhile, here are some remarks while I’m thinking about writing the doggone thing.

All of our work thus far takes place in (various subsets of) \Bbb R (the Real Numbers) or \Bbb R^2 (the xy-plane). We are particularly concerned with real-valued functions of a real variable; typically these are given in the form y = f(x).

The big idea of the course so far is pretty clearly Transformations: when the right-hand side of our typical equation is replaced by f(x) + K or A f(x), one has a vertical transformation (a translation—which for some reason we’ve been calling a shift—or a scaling [i.e., a stretch or a compression]); the corresponding Graphical Transformations can conveniently be expressed as \langle x, y+K \rangle and \langle x, Ay \rangle—”add K to each 2nd co-ordinate” and “multiply each 2nd co-ordinate by 2”. The horizontal transformations associated with y = f(x – H) and y = f(x/W), expressed in the same notation, are then \langle x+H, y\rangle and \langle Wx , y \rangle—note here that the number subtracted from the x variable in the (new) equation is actually added to the x co-ordinate of each ordered pair in “shifting” the (old) graph of f (and, more or less of course, a division “inside the parens” in the functional equation produces a multiplication of first co-ordinates by the value in question [here called W for “wavelength”, by the way … with some loss of accuracy …] in the transformation).

Throw in the reflections \langle x, -y \rangle and \langle -x, y \rangle, and you’ve got what I hope is a pretty good summary of the theory as thus far presented. All of this theory can now be brought to bear on an equation like y = -2(x-1)^2 + 3: beginning with a graph of its “parent function” f(x) = x^2, we can understand this as y = -2f(x-1) + 3 and go on to analyze the corresponding transformation as a reflection in the x axis, followed vertical “stretch” by a factor of 2, followed by a horizontal shift (to the right) by 1 unit and a vertical shift up 3 units (that is, \langle x+1, 2y +3 \rangle; I’ll remark here that even though the angle-bracket notation isn’t an actual course requirement, I’d feel pretty helpless saying most of this without it …).

As to the order in which the “suboperations” forming this transformation are performed, the hard issues are essentially ignored by our text. And, for right now, by me. I’m gettin’ out in the snow and play. As soon as I debug all this TeX-slash-HTML …


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