I've never been all that happy with my presentation of exponential functions. I usually slap a couple of functions on the board, plot some points with the class, and then write up a list of these functions' characteristics. Boring.
Thankfully, I recently came across Sam Shah's method of exploring exponential functions using paper folding. I like this idea so much more than plain old point-plotting (though I ask the students to do that as well). Here's the worksheet I had my classes fill out before introducing the formal definition of an exponential function:
Paper Folding Exponential Functions
We spent 15-20 minutes on this worksheet in class before diving into the lecture. That's a lot of time for a college class, and I felt rushed the rest of period, which is never good. So, next semester I may assign it for homework the day before instead to save some time. I would have to give some kind of hint as to how to find f(x) and g(x) though, because the students were a bit unsure of the pattern going on. I don't think anyone got 2^x, but most did see that we were multiplying (or dividing) by two each time.
Once we got into the lecture part, I gave the students a great cheesy word problem:
One of my students noticed that it didn't take much time for the spending to increase significantly, which I was quite happy about. "So the moral of the story," I began to say...
"Is not to be exponential when you're shopping!" she finished for me.