When we left off on yesterday's task (Copy of Task - Distance between Functions), we alluded to the fact that there had to be a more efficient way to find the maximum vertical distance between the two functions. Many students used guess-and-check. In class we looked at how they arrived at their solution. The big idea at the end of yesterday's class was that we subtracted the y-values to find the vertical distance. We can use that to come up with our function that represents the vertical distance at any point x, mainly d(x) = g(x) - f(x).
Once we come up with this function describing the vertical distance between g and f, we can consider the problem as finding the maximum value of this function. Perhaps students figured out how to find the maximum while finishing the problem for homework. We will see.
At the start of class, I ask students to show the class an algebraic approach (using x = -b/(2a) to find the vertex), and, a graphical approach using technology (using the maximum feature on the graphing calculator). If one solution strategy is missing, ask them how they could find the maximum if they did not have a graphing calculator, for example.
Then, I extend the problem. Today I have decided to give the students an exponential function and a cubic function. I pose the question: What function describes the vertical distance between the graphs at any value of x? Hopefully, it will be crystal clear to students that subtracting any two functions will give us the vertical distance between them.
After we have closure with this problem, it is a good idea to ask students about what Mathematical Practices we used during this task. The task was challenging, rich, and we spent a lot of time on it, so I want to recognize some of the great practices that I noticed over the past two days. Show the list to students and see if they can recognize how the strategies they used make them great mathematicians. Comment on their perseverance and let them know that it is imperative to their success!
Since we always feel the crunch of trying to squeeze in all of our content throughout the year, it can be difficult to choose tasks to spend extended amounts of time on; I really believe that this task is worth it (Perseverence)!
After the yesterday's task has been completed, it is a good time to get into the nitty-gritty of function combinations and their notations. Ask students to summarize - how did they find the distance between the two functions for all of these problems? Once they reiterate that d(x) = g(x) - f(x), make a big deal about the distance between the two functions just being their difference. Then, tell them that instead of writing it as g(x) - f(x), we can write it as (g - f)(x) and it will mean the exact same thing. You might want to have a quick discussion here about what happens if you calculated f(x) - g(x) for the task instead of g(x) - f(x). What would f(x) - g(x) tell us?
At this point you can also go over the notations for addition, multiplication, and division. Then, give them two functions, f(x) and g(x), and ask them to calculate (f + g)(4) or (f/g)(x), for instance. This discussion will take the focus of the lesson from the single task to something more universal. Students got to see an example where subtracting two functions would be appropriate, challenge them to think of a context where addition of functions might be necessary.
Finally, an assignment (Homework - Combinations of Functions) is attached to get them thinking further about this concept and some of its applications. This assignment can be started in class and finished for homework.