1. Chapter 7 Exponential and Logarithmic Functions

2. 7-1, 7-2, and 7-3 Exponential Growth Exponential Decay The number “e”

3. What you’ll learn Exponential Functions and Their Graphs The Natural Base e Population Models … and why Exponential and logistic functions model many growth patterns: human and animal population money …and patterns of decline radioactive decay dilution of chemical solutions

4. Exponential Growth ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the growth factor ‘b’ > 1 Table of values x f(x) 0 3 1 6 2 12 3 24 448 2 2 2 2 1.5 -20.75 ‘d’ shifts everything up or down

5. Exponential Growth Does the output value ever reach ‘0’ ? What do we call the line: y = 0 ? ‘b’ > 1 Horizontal asymptote

6. Identifying the Parts of the function: ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the growth factor Initial value: 10 f(0) = 10 + 2 = 12 Growth factor: 4 ‘d’ shifts graph up/down and is the horizontal asymptote Horizontal asymptote: 2

7. Your turn: For each of the following what is the: a. “initial value”? a. “initial value”? b. “growth factor”? b. “growth factor”? c. “horizontal asymptote” c. “horizontal asymptote” 1. 2. 3.

8. Graphing Exponential Growth f(1) = ? 3. Horizontal asymptote asymptote 1. f(0) = ? 2. Some other point f(0) = 2 f(1) = 6 y = 0 y = 0 Domain = ? Range = ? All real #’s y > 0

9. What 3 things do you need to graph exponential growth? f(1) = ? f(1) = 6 + 5 = 11 3. Horizontal asymptote 1. f(0) = ? f(0) = 3 + 5 2. Some other point y = 5 y = 5 Domain = ? Range = ? All real #’s y > 5

10. Your turn: 4. f(0) = ? 5. f(1) = ? 6. Horizontal asymptote = ? 7. Graph the curve. 8. Domain = ? 9. range = ?

11. Population Growth Population (as a function of time) function of time) Initial population population Growth rate rate time It’s just a formula!!! The initial population of a colony of bacteria is 1000. The population increases by 50% is 1000. The population increases by 50% every hour. What is the population after 5 hours? every hour. What is the population after 5 hours? Percent rate of change (in decimal form) (in decimal form)

12. Your turn: The population of a small town was 1500 in 1990.The population increases by 3% every year. year. 10. What is the population in 2009?

13. Simple Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”)Interest rate rate time A bank account pays 3.5% interest per year. If you initially invest $200, how much money If you initially invest $200, how much money will you have after 5 years? will you have after 5 years?

14. Your turn: A bank account pays 14% interest per year. If you initially invest $2500, how much money If you initially invest $2500, how much money will you have after 7 years? will you have after 7 years? 11.

15. Exponential Decay ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the decay factor 0 < ‘b’ < 1 Table of values x f(x) 0 4 1 2 2 1 3 0.5 4 0.25 ½ ½ ½ ½ 8 -216 ‘d’ shifts everything up or down

16. Identifying the Parts of the function: ‘a’ is the initial value  f(0) = ‘a’ (plus ‘d’) ‘b’ is called the decay factor Initial value: 10 f(0) = 10 + 2 = 12 Decay factor: 0.4 ‘d’ shifts graph up/down and is the horizontal asymptote Horizontal asymptote: 2

17. Your turn: For each of the following what is the: a. “initial value”? a. “initial value”? b. “decay factor”? b. “decay factor”? c. “horizontal asymptote” c. “horizontal asymptote” 12. 13. 14.

18. Graphing Exponential Decay f(1) = ? 3. Horizontal asymptote asymptote 1. f(0) = ? 2. Some other point f(0) = 6 f(1) = 3 y = 0 y = 0 Domain = ? Range = ? All real #’s y > 0

19. What 3 things do you need to graph exponential growth? 3. Horizontal asymptote 1. f(0) = ? f(0) = 3 + 10 2. Some other point y = 10 y = 10 Domain = ? Range = ? All real #’s y > 10 f(-1) = 7.5 + 10 = 17.5 f(1) = ? f(-1) = 7.5 + 10 = 17.5

20. Exponential Growth and Decay For what range of values of ‘b’ will result in exponential growth ? will result in exponential growth ? For what range of values of ‘b’ will result in exponential decay ? will result in exponential decay ?

21. The slope of the tangent line at any point on the line at any point on the curve is curve is The number ‘e’ (Named after Leonard Euler, a Swiss mathematician) ‘e’ is a very unique number e = 2.71828 18284 59045 23536….

22. The “natural” number ‘e’ works perfectly with natural processes Exponential growth of populations Exponential decay of radioactive material

23. Exponential Functions and the number ‘e’ Any exponential function of the form: Can be written in the form: Exponential Growth: a > 0 and k > 0 Exponential Decay: a > 0 and k 0 and k < 0

24. Exponential Functions and ‘e’

25. HOMEWORK Section 7-1 (page 482) 2, 4, 6, 12, 24, 28 Section 7-2 (page 489) 4, 6, 8, 12, 34c Section 7-3 (page 495) 4, 8, 20, 22, 32, 38, 40, 56 (19 total problems)