Concepts of Geometric Transformations (operations) on functions are a difficult one for students. Getting students to grasp these abstract concepts prove even more challenging for the tutor. To do so one must convert these concepts into a concrete visualization and at the same time connect these operations performed on the function with the result. Such topics are best taught with visualization tools such as Geometer's Sketchpad.
This past semester I had the opportunity working with a high school student in this area. While there are many types of transformations such as shears and rotations, we only covered the basic transformations; dilation/shrinking, reflections and shifts. Honing in on my skills, I was able to successfully teach the concepts very quickly. Using Sketchpad we were able to cover this topic in two lessons and effectively solidify his understanding of Transformations. Sketchpad enabled us to focus on manipulating the functions to see the transformation and eliminating tedious sketching by hand.
All shapes and images around us from mountains and trees to snowflakes, no matter how complex, can be approximated geometrically by functions.
For anyone learning Geometry, particularly Euclidean Geometry or any advanced subject area involving analysis of functions up to calculus, Geometer's Sketchpad is an invaluable tool. The link below showcase a sample of the work of one student in the area of Function Transformations. The image created is a simplified graphical view of a clown face using some basic functions and transformations. No drawing required, just math. This is merely a small taste of the power of geometry!
To view this file, copy and paste the link in your browser. http://dl.dropbox.com/u/17545844/Transformations%20for%20a%20Face_1%20%281%29.pdf