Section Summary: 1.3

1. Definitions
   • translations - rigid shifts of a graph, either horizontally or vertically.
   • stretching - the graph is distorted in height or width, but the general shape remains the same.
   • reflections - the graph is reflected about either of the two axes.
   • function composition - a process whereby functions are applied successively to elements of the domain. For example, the composite function is defined by

     

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.
   • One reason why these transformations and compositions occur in the same section is that the transformations are examples of compositions: given the two functions defined by their formulas f(x) and g(x)=mx+b, the horizontal stretches and shifts are given by , whereas the vertical stretches and shifts are given by .

     These transformations are called ``linear transformations'' for a reason! All of them fit in the forms of either f(mx+b) (horizontal shifts or stretches) or mf(x)+b (vertical shifts or stretches). Given c > 0,

     

     Given c > 1,

     

   • Function algebra:

     

4. Summary

   This section concerns creating ``new functions from old'', by function composition, by linear transformations (a form of composition), and by ``function algebra'' (i.e. forming arithmetic combinations of functions, such as their sums, products, etc.). Of particular importance is the computation of the domains of such new functions.

5. Problems we might do together:

   pp. 46-49, #1, 3, 5, 6, 20, 27, 30, 54