How do you motivate vertex form for a quadratic? Do you just drop it on them? That's pretty much what I used to do in Algebra 1. Hey, kids, you want to model this u-shaped path, like you get when you toss a basketball. I'm just going to tell you to start with y = a(x - h)2 + k. Start messing with a, h, and k, and see what happens. Let's see what we can say about how they each affect the graph.
There's an awful lot of "building functions" in the common core, and an awful lot of modeling, and I think it's great. The whole F-BF header should be a playground. I'm just not clear on how you take a class there.
You can look at sequences of patterns easily enough that result in y = ax2, and I suppose patterns that result in y = (x - h)2 and y = x2 + k. Do those arise naturally anywhere? Or do you choose carefully something from Visual Patterns?
What else? "Fit a quadratic function to a photograph" seems to be a favorite of presenters at conferences who want to browbeat teachers for not making class real-world enough. But how do students develop those functions in the first place, in an authentic way? I feel a little awkward for asking, because I feel like I should already know this. But I also suspect that not that many people have a great answer.