Today we are going to review transformations but the review will be embedded in trying to determine the transformations of an inverse when we know the transformation on the original function.
Students begin class by explaining the transformation to produce the new function when given the parent function. This is the quick review for most of the transformations. Students have worked with transformations in Algebra 1, Geometry and Algebra 2. My students are usually comfortable with all the transformation except for a horizontal stretch and shrink.
After students work for a few minutes the students explain the transformations.
After completing the bell work, students are asked to share the types of transformation they remember. I expect my students to recall both shifts and reflections. Next, I will prompt students with a worksheet about explaining transformations. I let the students work in groups for about 10 minutes. Students may use any resource they have to help answer the questions. Resources include their textbook, the Internet, or any available classroom resources.
As the students research the questions, I move around the room answering questions. One question that occurs frequently is, "Why are the parameters in the book different than the ones that we have used in the past?" I like when students ask a question like this because it provides an opportunity to discuss when and how the parameter name makes a difference. Moreover, this is a good time to remind students to focus on structure, rather than appearance. Students need to see and interpret the structure of the equation, instead of memorizing the letters used in the equation.
As I move around the room I am determining how much more time is needed by the groups. Once most groups are finished I bring the class together for a discussion of the meaning of the parameters.
After the discussion I put a graph on the board. I ask students to draw the graph of g(x) using the graph of f. In this example, some students become confused when required to graph f(2x). I try not to explain, instead asking questions about the changes. Eventually, we will discuss how f(2x) is a horizontal shift. I hope my students will share the observation that the input (x) doubles while the output (y) stays the same. Similarly, if the function is 3*f(x), the output from the function is 3 times as big. Understanding whether the domain or the range is changing helps students in graphing and analyzing tables to determine the transformations.
Since the class just finished 2 lessons on finding inverses, we will now extend this work to find the transformation of an inverse, when the original function is known. I find idea to be an interesting way to look at the structure of a function.
I give each student a copy of the Transformation and Inverses worksheet. As a class we find the answers to the first 3 problems. I give the students about a minute to write the answers and we then check the answers. To make sure the students understand the directions for the activity we do problem 4 together.
I then quickly review the rest of the worksheet. I give students two more problems like Number 4. The students then write a hypothesis about how they can look at the original function to find the inverse's transformation and write the equation. Problem 8 asks students test their hypothesis. Question 9 may be the hardest for many students. This question ask the students to generalize the process.
Students are given 10-15 minutes to work. As students work different students put the inverses for questions 5 and 6 on the board. Putting the inverses on the board allows students to verify their process. If a student is getting the wrong answer for the inverse they will not be able to make the connections wanted on the worksheet.
Students will need to work on the activity some before we discuss the results. Students are told to be prepared to discuss the activity tomorrow.
As class ends I give students 2 tables. I ask students to determine what transformation used to produce the second table. Students put the answer in their notes and show their answer to me.