1. DAY 5 – EXPONENTIAL GROWTH AND DECAY

2. ZOMBIES! A rabid pack of zombies is growing exponentially! After an hour, the original zombie infected 5 people. Now those 5 zombies went on to infect 5 more people each! After a zombie bite, it takes an hour to become infected. Develop a plan to determine how many newly infected zombies will be created after 4 hours. If possible, draw a diagram, create a table, a graph, and an equation. WARM UP

3. HOMEWORK CHECK

4. QUIZ

5. How can exponential functions model real- world (or sci-fi) problems and solutions? ESSENTIAL QUESTIONS

6.  Exponential Functions: Functions in which the variable (x) appears in the exponent.  f(x) = ab x  Initial Value: The first term in a sequence. Represented by “a” in the function or the y-intercept on a graph, occurs when x = 0.  Growth/Decay Factor: The rate at which the values increase or decrease, represented by “b” in the function.  If b > 1, then the function is growing.  If 0 < b < 1, then the function is decaying. REVIEW OF VOCABULARY

7. BACK TO ZOMBIE PROBLEM  What was the initial value, a, from the warm up question?  a=1 zombie  What was the growth/decay factor, b?  b=5 because the number of zombies increased by a factor of 5 each time.  What function represents this model?  f(x) = 15 x, where x is hours since first zombie

8. KNIGHTDALE APOCALYPSE!  Use your graphing calculator to determine the time when all of Knightdale has been infected. That is, when 12,724 people are infected. 12,724 = 1(5) x x= 5.87 hours…..scary….

9.  Common Ratio: The ratio of one term in a sequence and the previous term.  Common Difference: The difference between one term in a sequence and the previous term.  Recursive function: a function that can be used to find any term in a sequence if you have the previous term (i.e. NOW-NEXT)  Explicit function: a function that can be used to find any term in the sequence without having to know the previous term. (i.e. y = or f(x) = ) REVIEW OF VOCABULARY

10.  Domain: the set of all input values for a function  Theoretical Domain: the set of all input values for a function without consideration for context  Practical Domain: the set of all input values for a function that are reasonable within context REVIEW OF VOCABULARY

11. LET’S CHECK OUT AN EXAMPLE

12.  Everyone needs to stand so that the recorder can count everyone and record the number of people standing.  Use your random number generator to “roll the dice”  If you roll a 1, sit down. Otherwise remain standing so the recorder can count the number of people standing.  We will continue until less than 3 people are standing. STANDING – SITTING INVESTIGATION

13. Investigation #1 1. What is your initial value for this set of data? What does it represent in the investigation? 2. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 3. Based on your answer to Question 2, find the r OR d for the data you collected. Show the process you used to do so.

14. 4.Could you estimate your answer to Question 3 without conducting the exploration? If so, how? 5. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 6. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the stage of the investigation and let f(x) equal the number of people standing in that stage.

15. EXAMPLE 2  In a laboratory, one strain of bacteria can double in number every 15 minutes.  Suppose a culture starts with 60 cells. Use your graphing calculator or a table of values to show the sample’s growth after 2 hours.  This can be modeled by the equation y = 60(2) x, where x is sets of 15 minutes.  How many sets of 15 minutes has happened in 2 hours?  8 sets of 15 minutes  Plug in x and solve y = 60(2) 8  15,360

16. EXAMPLE 3  You are investing $10,000 at 6% interest, compounded annually. Use y = 10,000(1.06) t.  How long will it take for there to be $25,000 in the account? Round to nearest year.

17.  You decide to invest $1000 in a savings account that earns 2.5% interest annually. How much money will you have at the end of 10 years? EXAMPLE 4

18. Summary Exponential Growth This is because you begin with 100% (1 when written as a decimal) and add the same percentage each time. Exponential Decay This is because you begin with 100% (1 when written as a decimal) and subtract the same percentage each time. Recursive: NEXT = NOW ∙ b Explicit: Starting at a, y = ab x

19. At 9:00 am, the official count of the zombie infestation was 16384. Every hour the number of zombies quadruples. Around what time did the first zombie roll into town? CLOSURE

20.  Independent practice HOMEWORK