I begin this class with this question posted on my board:
"What can you say about the appearance of different quadratic function graphs - how do you know they're quadratics?"
As my students discuss this question I call on them to share what they're saying with the class, with a focus on key features and particularly on end behavior. (MP7) Most students say they recognize the arc of a parabola even if it's upside down, which leads to a further discussion of the end behavior of quadratic functions. I ask for an explanation of what end behavior is, playing the new student and continuing to ask questions of my students until the explain it satisfactorily. If you've never tried this, it's kind of fun - just keep asking student-like questions and rephrasing their answers as though you don't understand until they make it crystal clear.
I tell my students that they will now have the opportunity to explore and discover, by graphing an assortment of functions, their own understanding end behavior for several common polynomial functions.
I distribute the problem set, ask if there are any questions, then say they have about 20 minutes to complete the assignment. (MP1, MP7) While they're working I walk around offering encouragement and assistance as needed. There are always a few students who still struggle with graphing. For these students I'll suggest extra practice time and help them with one or two of the more complicated problems to get them rolling.
After about 20 minutes or when everyone is done, I have students work with their right-shoulder partner for the next activity. I explain that I want each team to write a brief "guide" that summarizes their observations about end behavior. (MP3, MP7) I give the example of a field guide to birds, that allows the user to quickly identify a bird by its description and/or picture. I say that they should include the kinds of polynomials they just practiced on and may include additional polynomials if they choose. I explain this further in my video.
I close this lesson by projecting the graph of (f(x)=(x^2-3)^2) from my calculator onto the board and say that I used a standard viewing window and functions they are familiar with. I then challenge my students to duplicate the graph, telling them I need to see that actual equation they use, not just a graph. (MP1, MP2, MP7)
This activity usually generates a lot of interesting discussion as they work to figure it out. I've heard comments like "This must be a function with an exponent of at least four because there are four zeros and maybe even more roots!" and "I tried x^4 but I didn't get anything like this."