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SWBAT determine the "k-onstant" given transformed graphs of functions.

Big Idea

Working backwards from the graphs to determine shifts and changes gives students good real-world experience.

Set the Stage

15 minutes

For this lesson I begin with the following set of data on the board: (0,4) (1,5) (2,7) (3,11).  I challenge my students to graph these points on graph paper and fit a curve to them. (MP2) I explain in my video why my students are using graph paper instead of calculators at this point.  As they're working I walk around and collect a couple to share with the class via the document camera.  I look for graphs that are labeled, neatly and accurately marked with different curves to fit the points.  I can usually find at least one student who thinks a straight line works well and also one who tries to connect the dots rather than making a smooth curve.  By collecting an projecting a collection of graphs I don't put any one student on the spot, but allow the class to discuss and critique each graph independent of its creator. (MP3) I've included a sample graph with a smooth curve.  If none of the graphs fit the points exactly I encourage my students to try again.  Usually someone will see that it's a variation of the 2^x function and then it doesn't take long for them to come up with f(x)= (2^x)+3. (MP7) If they're struggling to find the parent function I ask questions like "Which function does this look like?" and "What functions that we've been studying have f(x) values that increase exponentially?"  Once we've identified the equation I ask for a volunteer to describe the parent function and how it has been changed. 

Put It Into Action

30 minutes

I tell my students that they will be working independently to identify functions from graphs with points given.  This is a non-calculator activity which can be intimidating to some students, so I give a quick review of the parent functions we've been studying by sketching graphs on the board.  I check for understanding then distribute today's handout.  After giving my students time to read the directions I ask if there are any questions then tell them they have about 20 minutes to complete the handout.  (MP1)  The most frequently asked question at this point is how they can tell if they correct.  I rarely answer that directly but instead might respond with "What could you do to check how well your equation fits?"

When 20 minutes are over or everyone is done I ask them to exchange papers with their right-shoulder partner.   I read through the equations used to generate the graphs while my students make corrections on their classmates papers. They know they earn credit for these corrections which gives added incentive to pay attention and get them right. (MP6)  I have them return the corrected papers to their owners and give everyone an opportunity to look over their own work and ask questions, then I collect the papers so I can look them over.

Wrap It Up

5 minutes

To close this lesson I give each student a notecard and ask them to briefly summarize (using complete sentences) the process they use to try to figure out a parent function from a given graph. (MP2, MP7) If this is too general for your students you might ask specific questions like:

  1. What do you look at first when you're trying to figure out the parent function for a graph?
  2. Which features of a graph are most important to you for identifying the parent function?
  3. What strategies would you suggest to a classmate who is absent today and trying to complete this lesson?