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.

5. Problems we might do together:

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