Warm Up Determine the better model for each data set Find each function. Round to nearest tenth Calculate f(-12), f(5), and f(21). Round nearest tenth 1 2

Warm Up Answers 1 2 Quadratic Linear

Three Basic Function Models Linear – Steady Increase or Decrease in Values Quadratic – Decreases then Increases OR Increases then Decreases Exponential – Increasing Increase or Decreasing Decrease in Values

Example of Exponential Function Model Suppose the Centers for Disease Control (CDC) has determined that each day that a student with the flu comes to school, they infect two other people. Assume an infected person immediately becomes contagious.

Example of Exponential Function Model Day 0: 1 Person is Infected Day 1: 1+2 New = 3 Infected. (Total Increase: 2) Day 2: 3+6 New = 9 Infected. (Total Increase: 6) Day 3: 9+18 New = 27 Infected. (Total Increase: 18) Day 4: 27+54 New = 81 Infected. (Total Increase: 54)

An Exponential Function can always be written in the following form: In our example, 1 person was infected on Day 0, 3 people were infected by Day 1, 9 people were infected by Day 2, 27 people were infected by Day 3, etc.

This function will model our data perfectly if we set P=1, m=3 and let x represent the day number

If we want to estimate how many people will be infected after 10 days we simply let x=10 As you can see, exponential functions can get very large, quite quickly

Notice what happens to the value of f(x) when m is less than 1.

Exponential Functions are used quite often to model such natural occurrences as population growth and decline.

Example 1: Population Growth In an experiment at the Centers for Disease Control, a specimen of e-coli bacteria is cultured. The population of the bacteria quadruples every hour. If the culture began with 10,000 bacteria, how many bacteria will there be after 24 hours?

Let f(x) represent the population of the bacteria and let x represent the number of hours the culture has been established.

Example 2: Population Decline The bat population in the Northeast United States has been devastated by a phenomena called “white nose syndrome.” Every year, 40% of the bats in one particular cave die from this disease. If the current population of bats in the cave is approximately 2,500, how many living bats would scientists expect to find 5 years from now?

Let f(x) represent the population of the bats and let x represent the number of years from now. Where did the number 0.6 come from?

Exponential Functions – Warm Up The population of a particular beetle is given by the exponential function: where x represents the year. Find the beetle population in 1990, 2000, 2009 and predict the population in 2015.

Finding Exponential Models Enter your data tables as you did with Linear and Quadratic models, then Press STAT Calc 0ExpReg Enter

1. Find the Exponential Model for the Data Table Below

1. Find the Exponential Model for the Data Table Below

2. Find the Exponential Model for the Data Table Below

2. Find the Exponential Model for the Data Table Below

3. Find the Exponential Model for the Data Table Below 3. Find the Exponential Model for the Data Table Below. Round to the nearest thousandth

3. Find the Exponential Model for the Data Table Below

4. Find the Exponential Model for the Data Table Below 4. Find the Exponential Model for the Data Table Below. Round to the nearest thousandth

4. Find the Exponential Model for the Data Table Below

Homework

Homework Answers Pg 452-453 #1-4, 9-20, 37-40 Rises, Falls (0,1) Does Not 243 (1,0) 2 2.56425419972 2.12551979078 1.25056505582 .901620746784 7.41309466897 5.39357064307 .000021096563 .000597989961

Homework Answers Pg 452-453 #1-4, 9-20, 37-40

Homework Answers Pg 452-453 #1-4, 9-20, 37-40 37. a) 0.5°C b) .35°C 37. a) 1.0°C b) 0.4°C 39. a) 1.6°C b) 0.5°C 40. a) 3.0°C b) 0.7°C