## Sinusoidal Definitions - Section 3: Connecting Definitions to Parameters

*Sinusoidal Definitions*

*Sinusoidal Definitions*

# Trigonometric Graphs (Day 2 of 3)

Lesson 2 of 13

## Objective: SWBAT determine how parameters change the graphs of trigonometric functions.

## Big Idea: Through the use of technology students discover how transformations of trigonometric functions are like the transformations of previously studied functions.

*30 minutes*

#### Bell work

*5 min*

Today, we begin by quickly reviewing transformations of algebraic functions. Students are given a couple of minutes to discuss the bell problem. Afterward, students will share out the solution.

After discussing the bell work I will use one of the possible questions from today's main activity to start my students' thinking about the transformations of graphs of trigonometric functions. I plan to let the students talk about this for about a minute, as a form of brainstorming, then we will list their ideas on the board. The list will remain posted throughout the lesson. We'll come back to this question at the end of the class in Closure.

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#### Exploring Transformations

*15 min*

After considering an example (or two) I will give students the Transformation and the Trigonometric Functions activity. As in yesterday's lesson, I will have my students will use Desmos to explore the transformations.

During this exploration I am not worried about the mathematical terminology such as amplitude or period. I want students to understand how the graph changes and write this in their own words. Some students will know the terminology from other courses but most will not. I will allow students to work in pairs so that they can discuss the ideas and exchange observations.

As I move around the room during the activity, I will ask questions to help students see how transformations of different graphs are similar. Here are some of the questions plan to ask:

- How does parameter "a" change the graph graph? If you have the graph y=3x^2 what does the 3 do to the graph? Is this similar?
- Why does the shape of the sine graph repeat?
- What did the parameter "b" do to the graph? If I have sqrt(3x) what does that do to the graph?
- How does the parameter "c" affect the graph? Is that similar the 4 does in y=ln(x-4)?
- Did you expect the parameter "d" to move the graph up or down? Why?

As students work on the Activity, they often understand how the shifts and the vertical stretch (change in amplitude) work. The period, however, is harder to understand. For this reason, today's work focuses on how the parameter "b" changes the graph, but I only have students predict the shifts and vertical stretch in Problems 5 and 6. I will further develop the change in period in the next lesson.

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After my students have completed today's exploration, I will give them the Sinusoidal Definitions worksheet. This worksheet contains textbook definitions for key features of the graphs we have been working with over the last two days. I ask my students to read the definitions and then try to connect the definitions to the parameters they analyzed in today's transformation activity. After a few minutes, I plan to tell my students that they should finish this task for homework to prepare for a discussion at the start of tomorrow's lesson.

#### Resources

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#### Closure

*5 min*

As class ends comes to a close I return to the brainstorming list from the start of class. I ask the students to look at what they predicted. Then, I ask, "Do you want to change anything? How would you explain the transformations to someone else?"

I encourage students to revise phrase and identify misconceptions they might have had. We will also discuss anything that surprised them. I expect students will ask about the parameter, **b**, which allows me the chance to foreshadow what we will be doing tomorrow.

#### 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: Trigonometric Graphs (Day 1 of 3)
- LESSON 2: Trigonometric Graphs (Day 2 of 3)
- LESSON 3: Trigonometric Graphs (Day 3 of 3)
- LESSON 4: Frequency versus Period
- LESSON 5: Graphing Sine and Cosine Functions (Day 1 of 2)
- LESSON 6: Graphing Sine and Cosine Functions (Day 2 of 2)
- LESSON 7: Graphing the nonsinusoidal trigonometric functions Day 1 of 2
- LESSON 8: Graphing the nonsinusoidal trigonometric functions Day 2 of 2
- LESSON 9: Writing Sinusoidal Models
- LESSON 10: Sinusoidal Functions and Climate Changes
- LESSON 11: Sinusodial Project Day 1 of 3
- LESSON 12: Sinusodial Project Day 2 of 3
- LESSON 13: Sinusoidal Project Day 3 or 3