At the start of this lesson I'll want to call to mind the Factor Theorem. First, I'll point out that we've learned quite a bit about cubic functions over the past few lessons such as seen the variety of graphical forms and we've made use of the structure of the equation in both its expanded and its factored forms.
When students enter the room, I'll have the equation f(x) = (x - 1)(x + 2)(x - 3) on one side of the board and a sketch of a (different) cubic function crossing the x-axis at three distinct points, such as x = -5, -3, and 2 on the other side of the board. I'll ask the students to take a minute to think-pair-share what they can tell me the two functions.
Of course, they should reply that the graph that goes with the given equation will have its roots at x = 1, -2, and 3. They should also tell me that the equation that goes with the given graph must have the factors (x + 5), (x + 3), and (x - 2).
Great! With that, we're ready to get started.
As I hand out Graphs of Cubics, I will explain that today's task is to create an equation to match the given graph. First, they are asked to simply list the given roots. Second, to write the linear factors that are implied by those roots. Third, to create the final equation to match the given graph.
During these first 10 minutes, I expect everyone to work individually with the aim of completing the first page of the handout. I'll move around the room checking in with students one-on-one to ensure that they've made sense of the problem.
I expect all students to quickly identify the three linear factors, and many will simply multiply these together for their final equation. When I see a student that believes she's found the final equation, I'll be sure to ask how she knows that it's correct. She'll point out the three linear factors and explain that they guarantee the graph of her equation will have the correct roots. So far, so good; but what about the y-intercept?
The y-intercept is the thing that makes this first equation challenging, and I do not tell my students how to fix it. (MP 1) They realize that they cannot change the three linear factors without changing the roots of the graph; they also realize they can't insert another linear factor without creating an additional root. The key is to insert a constant factor. You can guide them carefully to this insight, but be sure to challenge them to explain why this will not alter the roots, but will alter the y-intercept. (MP 3)
After the previous 10 minutes, some students will have found the correct equation, but others will still be trying to make sense of it. (Here's my answer key.)
At this point I'll sketch the given graph on the board and then ask the class, "Would someone like to come up to explain his or her solution to this problem?" From the volunteers I'll pick someone that I know has the correct solution, but I'll give preference to any student who doesn't typically put himself forward.
In front of the class, the student should first explain how he decided on the three linear factors. At this point, I might ask a question like, "Why did you choose the factor (x + 2) rather than (x - 2)?" Next, the student should point out that these three factors do not yield the correct y-intercept and then go on to explain how he determined which constant factor was necessary. When he's done, I'll ask the class if anyone has any questions or if anyone had a different way of thinking about the problem that might be helpful for the class. (MP 3)
Finally, I'll summarize with the observation that while the three roots determine the three linear factors, there may be any number of cubic equations having those three roots. (An image like this is very helpful if students need a visual example.) Our job is to use the fourth given point (the y-intercept, in this case) to determine which of these cubic equations we're looking at.
With about 15 minutes remaining in the lesson, I'll now let the students know that they may begin working together on the remaining two problems. Having seen this first problem worked out in detail, they should be better prepared to move forward.
Since the next two problem become progressively more challenging, however, I do not expect anyone to complete both of them in class. I'll move around the room to answer questions, offer encouragement, and keep everyone on task.
Tonight's homework is to spend about 15 minutes persevering to solve these problems. Come to class tomorrow prepared to share your insights or struggles.