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# Exponential Decay Functions

Lesson 6 of 15

## Objective: Students will be able to write functions for and graph exponential decay models.

*51 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 specifically explains this lesson’s Warm up-Exponential Decay which asks students to write a scenario given an exponential function.

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

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This lesson begins with the statement: *"By now, humans are starting to get smart. The military begins a campaign to get rid of the zombies."* We look at a scenario where there are 1,000,000 zombies and one half of them are exterminated each night. This will probably be a struggle for some students. Once some start to get it, I bring the class back together to discuss the model. I ask the students for figure out how many will be left in one week.

They then look at a model where only one fourth of the zombies are exterminated each evening. This may seem like an obvious question, but I ask the students to predict whether the zombies will disappear faster or slower than in the last scenario. Many students will want to make this model: Z(x)=1000000(1/4)^{x}. While this seems to make sense, it is not a correct model. By this model, you start with 1000000 and then go to one quarter of that in the first night. If only one fourth are destroyed, then there should be three quarters left the first night. This would change the model to Z(x)=1000000(3/4)^{x}. I have the students work on this problem individually or with their partner. When we discuss it, I either ask some leading questions or find someone who did the problem correctly and have them present their reasoning.

The general exponential model is now be reintroduced to include decay models.

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Now the students are asked to find the number of remaining zombies from the first three nights of the zombie extermination campaign that kills off half of the remaining zombies. Once these are graphed and the function equation is written or recalled, the students extend the graph into the negative exponents. The domain and range is noted as well as the asymptote. The next task is to perform the same procedure with the scenario where only one quarter of the zombies are exterminated each night. The third task deals with a general exponential decay function. Students will identify the major features of a decay function and compare it with a growth function.

The final goal in this lesson is for students to determine two points that will make a reasonable sketch of an exponential function (**Math Practice 7**). I have the students work with their partner and decide what points are needed. I list several options on the boar, and as a class, we will narrow it down (**Math Practice 3**). My goal is for the students to get down to two points, the intercept and one other point to give the shape. This will vary depending on whether it will be a growth or a decay function. They will then practice this skill with a couple of exponential functions.

#### Resources

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#### Homework

*1 min*

The Home Work asks students to graph exponential functions by choosing two points, has them determine the steepness of exponential functions and has them use the graph of an exponential function to estimate the solution to an exponential equation. This is a skill that has been practiced throughout the year and shouldn't be brand new. The final question is a decay modeling question to make sure they understand how to find the base properly.

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#### Exit Ticket

*5 min*

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

Today's Exit Ticket evaluates student knowledge of the exponential decay by asking them to write a scenario to represent the function: f(x)= 5000(0.4)^{x}.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Rational Exponents
- LESSON 2: Real Number Exponents
- LESSON 3: Exponential Models Day 1 of 2
- LESSON 4: Exponential Models Day 2 of 2
- LESSON 5: Exponential Functions
- LESSON 6: Exponential Decay Functions
- LESSON 7: Simplifying Logarithms
- LESSON 8: Exponential and Logarithmic Equations
- LESSON 9: Logarithmic Functions
- LESSON 10: Exponential Growth and Interest Day 1 of 2
- LESSON 11: Exponential Growth and Interest Day 2 of 2
- LESSON 12: Natural Logarithms
- LESSON 13: Exponential and Logarithmic Functions Review Day 1
- LESSON 14: Exponential and Logarithmic Functions Review Day 2
- LESSON 15: Exponential and Logarithmic Functions Test