We've just finished a unit on transforming trig function graphs:
y = a sin b(x + c) + d.
We did various things and practice, but I think I found my new BFF towards the end of the unit once we were reviewing for the test.
A student was having a hard time looking at a graph of a transformed function and getting the equation. I had some old transparencies that had the coordinate plane on it with radians on the x axis. I also had half sheets of cut up transparencies and a ton of markers. I laid the 1/2 sheet over the graph and had the student draw the parent function of the sine graph (for example). Then she was able to slide the 1/2 transparency any place she wanted to see what the transformed graph would look like (barring amplitude and period changes ...... though I guess we could have used another 1/2 transparency overlayed on that one).
This seemed to help gel in her mind the steps she had to take. I think next year I may give all students such transparencies just to keep and play with for the duration of the unit. .... My 2nd thought was to have pipe cleaners they could manipulate into the shapes and then move those around on a graph.
Also, I loved this particular review question I gave. There was a picture of a (say a potential sine graph) that has been transformed in all sorts of ways. The question was: find 2 possible equations for this graph. I guess a variation question could be: find a sine and a cosine equation for this graph ... or find 2 sine equations and 2 cosine equations for this graph.