Transformations of y=1/x

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Objective

SWBAT graph rational functions that have been transformed in one way from the original function. Students will be able to explain how each of four transformations affects the graph of the original function.

Big Idea

Apply previous understanding of function transformations to a new function family to create a deeper generalization about transformations.

Warm-Up

30 minutes

There are two key parts to this Warm-Up. The first part is a follow-up on yesterday’s lesson and an attempt to make students think more about the graph of 1/x. It is important that students can connect their answers to the questions to the graph and explain how the graph relates to the function. Also, ideally a student would be able to use the concept of asymptotes when answering these first four questions. If a student had solid answers to those questions, I would ask them to connect each answer to either the vertical or horizontal asymptote.

The second problem might take students forever. In terms of calculator use, this is where I differentiate. I ask all students to fill out at least some of the data table without a calculator, because I want to make sure that they have mastered the skill. Then, if I know that they are able to do this, I allow them to use a calculator. For some of my students who have weaker math backgrounds, I ask them to show me every single calculation without a calculator. It might take them a while, but these are the kinds of skills they need in order to avoid having to take remedial math in college, so I like to use this opportunity to push them on this. Finally, if students are taking longer than 20-30 minutes on the warm-up, I assign them partners and have them collaborate to finish problem #2.

Note that problem #2 is an example of a function with many transformations. Because we have not talked about transformations yet, I don’t expect them to address this, but if students produce a high quality graph with time to spare, I would take a minute to ask them how the changes to the original affected the graph. This is a good way to differentiate for advanced students because they will be able to anticipate the lesson better after thinking about this.

As always, the back page is not required for all students, but pushes advanced students to figure some stuff out in advance. 

 

Investigation and New Learning

30 minutes

Closing

10 minutes

The matching part of the Exit Ticket seems way too easy, but it is really helpful for students who might not have used the key terms correctly because they can see the explanations and it is also helpful for students who may not have figured out all 4 transformations, because they will see all 4 laid out.

The much deeper piece of this Exit Ticket is attempting to explain why the transformations have this effect. Honestly, this question goes beyond what I was taught in my math classes, and my answers below are my attempt at explaining them. I ask students to attempt to explain these, but I am not looking for one particular answer. The Sample Explanations provided below are an example of what they might try to explain.