Exponential Functions

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Objective

Students will be able to identify the features of an exponential graph and use transformations to graph them.

Big Idea

We will extend the zombie apocalypse scenario to the graphs of exponential functions.

Warm Up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Exponential Functions which asks students to identify what each portion of an exponential function means in context.

I also use this time to correct and record the previous day's Homework.  

Graphs of Exponential Functions

10 minutes

We begin this lesson by having the students graph Day 0 through Day 4 of our original zombie invasion.  We do a think-pair-share on the meaning of (0, 1) to the invasion.  We then add the negative x values from -1 to -4.  I ask the students whether these make sense to the scenario.  This is a great conversation and really value to their understanding of contextual verse non-contextual functions.

The students then draw the curve through the points.  I ask if rational exponents fit our scenario?  The class then discusses whether the graph will ever cross over the x axis both in terms to the scenario and in terms of the function itself (Math Practice 2).  This leads us to identifying the x-axis as a horizontal asymptote.  The students identify the domain and range of the general function as well as the domain and range of the specific zombie scenario.  

Properties of Exponential Graphs

10 minutes

Next, we look at the graph from their homework assignment for the Exponential Models lesson.  I ask the students to make a list of important attributes (Math Practice 7).  By the end of this activity, the students should have a solid grasp of the important features of exponential graphs. 

Things that are important:

  • (0,1) is a common coordinate to all of them
  • the x axis is the common asymptote
  • as the “b” value increases in each function, the function get taller when x>0 and smaller when x<0
  • Discuss the domain and range of each function as well as the equation of the horizontal asymptote.

I make sure not to downplay any contributions to the list that seem less important.  Sometimes I will randomly call on students rather than asking for volunteers, particularly in an activity like this where there are a lot of possible responses.  Whenever I call on students I try to alternate boy/girl, left side/right side, and any other major difference in the room.  It is hugely important to be mindful of your patterns in calling on students.  

Transformations of Exponential Graphs

30 minutes

Next I ask the students to predict how the graph will change by adding more zombies (three instead of one) at the start.  They either write their predictions down or discuss with their partner.  Then they graph the transformation.  This is a vertical stretch.  My students have been learning about transformations all year so this if familiar to them.  We discuss the domain, range, and horizontal asymptotes of this and each new function during the lesson.  Getting the students away from the charts and into transformations is helpful but not absolutely necessary.  As I tell my students, “When in doubt, make a chart.”   The remainder of the lesson consists of guided practice on graphing vertical and horizontal shifts.  

Homework

1 minutes

The Assignment has students practice both the graphing transformations, and finding the major characteristics of exponential functions.  It also has several extensions.  The students are asked to relate the asymptote to the domain and range as well as explain its location.  They are also asked to graph an exponential decay function and then perform transformations on it.  Finally, students look at an exponential function that has a negative base.   This problem will be discussed the next day.

Exit Ticket

5 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

This Exit Ticket asks the students to graph an exponential function with translations and find the domain and domain.