Transformations of functions are a big foundation of Algebra II, at least in my district. We talk about how to transform nearly every type of function.
Someone at my district wrote up a very thorough activity about how to introduce transformations. It's not a bad lesson. It's a little worksheet/copy heavy and requires a lot of class time. It also requires you to answer the same question about ten times as each kid in turns hits f(-x) and wonders "what the heck?"
So I adapted it for groupwork. The first part of the activity I will not share because I feel like it could be better. In essence, you establish a set of points as "truth" and use transformation rules to use those points to make new graphs that all comply to whatever rule you set like f(x-2). It does work, but the confusion factor is high.
The second phase kicks in when you establish what f(x+2), etc actually do and instead of nit-picking points you're ready to have them perform a transformation whole-sale without the need for a reference set of points.
I give each group this handout:
I leave the points on there to give them something easily identifiable. In a five member group, each person needs to redraw the graph based on two of the ten rules. I give them colored paper and markers and let them socialize. A few still get stuck (especially on f(0.5x)) but you're answering that question 4 times instead of 30. In the end you get some student work you can hang, and you have an ok basis on transformation rules. In three years the kids don't have a real solid feel for the rules until they come up again in quadratics. But I think that's ok. This year I'm going to try and tie the concept of transformations and parent functions together a little better. When it's time to play with things like 2x^2-3x+4 very few of them realize the whole thing is also a transformation of x^2.