## Transformation and Inverses - Section 3: Using algebra to prove the transformations of inverses

# Transforming Functions Day 2 of 2

Lesson 13 of 15

## Objective: SWBAT use transformations to write the inverse of a function when given the original function.

*45 minutes*

#### Bell work

*10 min*

The students ended class yesterday working on Transformation and Inverses worksheet. I begin class today by asking students to use their hypothesis from the worksheet to find the inverse of a function. I give my students sticky notes so that they can post their answers on the board. This gives students a chance to share their response somewhat anonymously.

After 3 to 4 minutes I will review the posted responses. I will start by grouping the responses so that similar answers are together. Then, I will share the different answers and ask students to explain why an answer is correct or incorrect. Through a process of elimination, the class works together to determine the correct answer. The class will also discuss how a vertical shift of 1 makes the inverse have a horizontal shift to the left.

This class discussion time allows me to assess students understanding of how transformations on a function can be used to determine an inverse. The discussion helps students who are not understanding. As students share an idea I have another student restate the idea or I say "so you are saying when the original function shifts up 2 the inverse shifts 2 to the left?"

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After completing the bell work, the students need to process what was discussed. I plan to ask students questions comparing the transformations of a function and its inverse. I want the students to discuss the answer for a minute in small groups before discussing the answer as a class. I will ask different groups to share their answer to start each review. I will scribe the conversations on the board as the initial answer is validated or adjusted.

As the class works to find inverses I will provide a series of problems to help students deepen their understanding. Once the class gets into a flow, I will have students put answers on individual white boards so I can quickly verify their responses. I have different students show their answer to the class. If students do not agree a discussion on what needs correct occurs.

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The complete process of verifying the relationship between a function and its inverse can be confusing. Some confusion involves understanding what the parameters of the expression are saying. Most students are not able to show an argument for the transformations of the inverse until they have seen an example.

I begin the discussion by stating stating f(x) is a parent function like problems 1-3 Transformation and Inverses worksheet. "What does f^-1(x) represent?" Students realize this is the inverse of the parent function. This idea is important as we move to verifying the transformations.

I begin the with a function that has a vertical shift. This is probably the simplest to verify but gives students a process to follow. The class works through the proof and then use the proof to find an inverse for a function.

After seeing one proof, groups are given different transformations (y=a f(x);y=f(x-c); y=(bx); y=f(bx-c)) to determine the inverse. Groups share their work use a document camera, or sharing their white board work. The groups are given the challenge of using the proofs just shown to find the general inverse equation if a function is y=a f(bx-c)+d. As groups work I am listen to discussions to determine which students are struggling. If a group is not sure what to do I will sit with the group and ask questions.

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#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Interval Notation
- LESSON 2: Evaluating Piecewise Defined Functions
- LESSON 3: Writing Piecewise Functions
- LESSON 4: Graphing Piecewise Defined Functions
- LESSON 5: Function Notation
- LESSON 6: Operations of Functions
- LESSON 7: Presentation on Functions Operations
- LESSON 8: Composition of Functions, Day 1 of 2
- LESSON 9: Composition of Functions, Day 2 of 2
- LESSON 10: Finding the Inverse of a Function Day 1 of 2
- LESSON 11: Finding the Inverse of a Function Day 2 of 2
- LESSON 12: Transforming Functions Day 1 of 2
- LESSON 13: Transforming Functions Day 2 of 2
- LESSON 14: Review for Assessment
- LESSON 15: Assessment