Start class by reviewing what rational functions are. Give students the two questions below. After discussing with their tables for about three minutes, have a class discussion about these.
1. What is a rational function?
2. What are some functions that are NOT rational functions?
I chose these two questions because I find that students don't usually have a strong grasp of rational functions like they do for quadratic or exponential functions, for example. Part of the reason for this lack of understanding is because many students often have trouble coming up with a concrete definition for rational functions. For the last two days we have been been working with rational functions without calling them rational functions (besides the second half of yesterday's class), so I want to cement their conceptual understanding of rational functions with precise mathematical language.
The main aspect of rational functions that I want to get out of them is that they involve division of polynomials. The division is vital to these functions and is the reason why rational functions have asymptotes when they are graphed.
The first rational function from the worksheet that we are going to graph is f(x) = x/(x^2-x-2). I want to do this first example as a class; I find that many students will start plugging in random points and I want to establish the structure of how we graph rational functions right away. The ordering of the questions on the worksheet presents the method that I encourage my students to use when graphing a rational function.
We can use our work from the last two days to decide if the rational function is similar to one in the Ultramarathon, Gummy Bear, or Homecoming examples. Students should also be able to identify the asymptotes for this first function. Give them a few minutes to think about letters a) and b) on the worksheet.
After the asymptotes have been added to graph, we want to think about what points are going to be important to the graph. You can ask students what they think the "critical points" of the graph are going to be. They may likely know that the x-intercepts and the points where the function is undefined are really important. Generalize this to show them that the critical points are where either the numerator or denominator are equal to zero.
After adding the intercepts and asymptotes to the graph, ask students how they can figure out the rest of the function. Students will probably say that they should use their graphing calculator or plug in random points. Let them know that these are both good strategies, but really we are looking to see what will happen between each critical point. Remind them that we are not going to plugging in a lot of points to our functions, that would be too time consuming. We just need a few to get a general idea of the function. At this point you can show them how to use a sign diagram (the critical points on a number line) to test every interval of the function. Then, they will have enough to sketch the rest of the graph.
In the video below I comment on the process of taking our rational function in the Launch section and summarizing those steps into an algorithm that would allow us to graph any rational function. The structure is really important as it is something that we can use for any of the rational functions we encounter.
After we summarize to create an algorithm, we can test it with the remaining examples on the worksheet.