### 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) (the **Real Numbers**) or (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 and —”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 and —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** and , 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 : beginning with a graph of its “parent function” , we can understand this as 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, ; 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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[…] case-divisions 01/27/09 Quadratic Formula Lore “complete the square” etcetera 01/28/09 Our Story So Far summarizing […]