Transforming Exponential Functions

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Objective

SWBAT describe how the graph of an exponential function changes based on changes made to the equation.

Big Idea

In this lesson, students will use an interactive calculator to explore transformations of functions.

Launch

10 minutes

The purpose of this launch (transformations_of_exponential) is to help students develop a common language for describing the various transformations that they will be working with in the investigation.  At first, I want students to describe each transformation in either plain or mathematical language (see transformations_movie.mov for view of animations).  The task is to encourage students to listen to each other in order to move towards more sophisticated language.

I will stop after each slide in this slide show.  I want my students to watch the animation and then write down how they would describe what happened using mathematical language when possible (MP3).  I plan to have one or two students share their ideas and determine a common language that can be used as a class.  

As we work, I will keep track of important vocabulary words on the board.  I leave the words posted so that students can use them during their investigation.  Words that students will want to focus on could be: horizontal shift (to the left/right), vertical shift (up/down), stretching, reflection (over the x-axis/y-axis).  

  • Slide 1: This would be described as a vertical shift.
  • Slide 2: This would be described as a horizontal shift.
  • Slide 3: This would be described as an enlargement or dilation.
  • Slide 4: This would be described as a reflection through the y-axis. of the original image.

Investigation

25 minutes

During this section of the lesson, students will use the Desmos graphing calculator to help them explore transformation of exponential functions.  The screenshot at the top of the investigation will help them to set up their calculator appropriately (NOTE: The table of values is included with the first function so that points will be plotted on the graph as a point of reference). I explain to students that they should be as thorough as possible in their explanations.  Also,  I ask students to take their time on the prediction questions.

These questions will serve as immediate feedback.  They can predict how they think the function will look and then graph it using the Desmos calculator.  If the calculator matches their prediction it tells them that they are beginning to grasp that concept.  This metacognitive component to the lesson will help students to build more in-depth meaning around the transformation concept.

Some students may attempt to build sliders (based on previous lessons) for the values that are being varied.  This should be encouraged and can presented to the whole class once students have had a chance to complete the investigation on their own.    

Closure

5 minutes

To complete the lesson, I will write the following functions on the board:

1) f(x) = 2^x

2) g(x) = 2^(x+2) + 4

Then, I will ask my students to describe how the graph of f(x) would be related to the graph of g(x), without graphing the two functions (MP2).  

This task challenges students to visualize how the transformations would effect the original graphs.  Some students may only be able to describe one transformation or the other.  Once students have had a chance to write down their ideas have them do a think-pair-share to discuss their ideas with their partner.  After the partners have come to an agreement, they can check their answer by graphing the functions on the Desmos calculator or on a regular graphing calculator.