##
* *Reflection: Grappling with Complexity
Shifting Exponential Functions - Section 2: Exploration: Shifting Exponentials

Is it a horizontal shift or a vertical stretch? Well, it's both!

In today's lesson, there was some confusion as to which transformation the students were seeing (and rightly so!): Shifting Exponential Functions, video reflection, stretching and shifitng.

So although a vertical stretch is not wrong at all, guide your students into seeing these transformations in questions 3-4 as a horizontal shift. The questions are worded "How did the graph SHIFT?" The questions are not how did the graph TRANSFORM. It is important that students establish this left/right shifting and how this affects the parent equation today. We will explore more about how horizontal shifting and vertical stretching are the same in the lesson Stretching Exponential Functions (And your mind!).

*Grappling with Complexity: Is it a horizontal shift or a vertical stretch?*

# Shifting Exponential Functions

Lesson 4 of 14

## Objective: SWBAT use horizontal and vertical shifts to sketch the graphs of exponential functions and identify key features of the graphs and how transformations change these key features.

## Big Idea: Building on yesterday’s idea, key features of exponential graphs, students see how parameter changes of the function can change these features.

*41 minutes*

#### Warm-up

*5 min*

To warm-up for today’s lesson I want students to review the important aspects of basic exponential functions by doing a * ***Think-Pair-Share** over the prompts on page 2 of the flipchart. I plan to give students time to discuss the prompts in their team and then randomly call on individuals to share with the class.

I predict students will struggle with this question due to the abstract nature of the problem. Students worked with domain and range yesterday, but are used to having a specific equation. Right now, my students are very calculator reliant to visualize the graph and then identify the domain and range. I chose this problem specifically so that students couldn't graph it in their calculator.

My main goal here is for students to reflect on the general properties of an exponential function that has not been transformed in any way. Students should reflect on the domain and range of exponential functions and how the ‘b’ value affects our functions. In particular, that the domain has no restrictions, but that the range is restricted because the function values can never reach zero. Students should also know that exponential functions grows quickly when b>1 and decreases quickly when 0<b<1.

*expand content*

Students should now work through the Student Worksheet Exploring shifting. This worksheet could be completed by students individually or in their teams. I am going to allow students to use a graphing calculator to speed up the activity. My goal today is not for them to be able to graph (that was yesterday!) but for them to quickly identify the parameter changes and make conclusions based on these changes.

Today’s lesson is a great opportunity to reinforce **Mathematical Practice 7: look for and make use of structure**. Are students picking up on the fact that exponential functions behave (in regards to transformations) the same as other functions we’ve studied? Are they able to generalize these past findings and the structure of an exponential function to quickly identify the shifts of these functions without a calculator? Students should be encouraged to sketch the graph of the function before graphing on their calculator to reinforce what they already know, or think they know.

I also think that depending on how students are required to work thought this lesson there could be great opportunities for students to practice **Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. **If students are working though this independently they should be given the opportunity to discuss some of their findings with their teammates. I would like my students to discuss the ‘reflect’ questions in their teams and come to an agreed upon conclusion. If students are working independently, I will require them to team up and review these in the last 5 minutes of this section.

Toward the end of this section, I plan to present page 3 of the flipchart to summarize the findings. I don’t expect students to copy this, just read it and make sure they agree. They will be encouraged to copy it (or snap a quick photo with their phone!) if they think they need it.

*expand content*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cat Island (Day 1 of 2): Cats can’t add but they do multiply!
- LESSON 2: Cat Island (Day 2 of 2): Cats can’t add but they do multiply!
- LESSON 3: Graphing Exponential Functions
- LESSON 4: Shifting Exponential Functions
- LESSON 5: Counting the Change: Linear, quadratic, or exponential? (Day 1 of 2)
- LESSON 6: Counting the Change: Linear, quadratic, or exponential? (Day 2 of 2)
- LESSON 7: Sorting out the Change
- LESSON 8: Stretching Exponential Functions (and your mind)
- LESSON 9: Credit Card Investigation: What is interest? (Day 1 of 4)
- LESSON 10: Credit Card Investigation: What is interest? (Day 2 of 4)
- LESSON 11: Credit Cards Investigation: How is interest really calculated? (Day 3 of 4)
- LESSON 12: Credit Card Investigation: A Dastardly Scheme (Day 4 of 4)
- LESSON 13: Exponential Functions Test Review
- LESSON 14: Exponentials Functions Unit Test