1. Chapter 7 Exponential and Logarithmic Functions

2. 7-1 Exponential Growth

3. What you’ll learn  Exponential Functions and Their Graphs … and why Exponential functions model many growth patterns: Exponential functions model many growth patterns: human and animal population human and animal population money money

4. vocabulary What is a power? Power: An expression formed by repeated Multiplication of the same factor. ???? ??

5. Exponential Function Looks like a power but the input variable “x” is the exponent instead of the base. “x” is the exponent instead of the base.

6. Your turn: 1. Graph the exponential function on your calculator then copy the graph to your answer sheet.

7. Your turn: 3. f(0) = ? 3. f(0) = ? 2. What is the “base” of the exponential function? 2. What is the “base” of the exponential function?

8. The “Initial” Value The initial value occurs when x = 0. The initial value occurs when x = 0.

9. The “Initial” Value Since “negative time” doesn’t make sense, what is the “domain” of this function? (  what input values are allowed?) If the input variable was time, the previous function If the input variable was time, the previous function would look like: would look like: The initial value occurs when t = 0. The initial value occurs when t = 0. What is the initial value of f(t) ?? What is the initial value of f(t) ??  f(0) = ?  f(0) = ?

10. Vocabulary: The “Initial” Value The initial value of the function is the coefficient The initial value of the function is the coefficient of the power. of the power. What are the initial values of the following functions ? What are the initial values of the following functions ?

11. Your turn: 4. What is the initial value of: 4. What is the initial value of: 5. What is the initial value of: 5. What is the initial value of: 6. The ‘y’ intercept is a point on the y-axis. What is the x-value of every y-intercept ? the x-value of every y-intercept ?

12. 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

13. Exponential Growth Does the output value ever reach ‘0’ ? What do we call the line: y = 0 ? “ Growth” occurs when the growth when the growth factor ‘b’ > 1 factor ‘b’ > 1 Horizontal asymptote

14. Your turn: Graph the function: 7. Where does it cross the y-axis? 9. What is the “intial value of f(t) ? 8. Graph the function: 10. As time passes (input variable ‘t’ increases) what happens to the function value? happens to the function value?

15. 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

16. Negative Coefficient Graph the following: Graph the following: What does a negative coefficient do to the graph of Initial value: reflects across x-axis if negative.

17. Replacing ‘x’ with ‘x – c”: Graph the following: Graph the following: and and Which direction does the graph shift for x – 5 ?

18. Transforming Exponential Functions Describe how to transform the graph of: Into the graph of : 

19. Transforming Exponential Functions Describe how to transform the graph of: Into the graph of :  Why does it cross the y-axis at y = 4? What is the new horizontal asymptote? Graph moved up 4. Horizontal asymptote for parent function is: y = 0 Graph moved up 4. New horizontal asymptote: y = 4 

20. 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” 11. 12. 13.

21. 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)

22. 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?

23. 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? 14. 15. The population of a small town was 1500 in 1989 The population increases by 3% every The population increases by 3% every year. What is the population in 2009? year. What is the population in 2009?

24. Graphing Exponential Growth Use the “power of the calculator” or: 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

25. Your turn: 16. f(0) = ? 17. f(1) = ? 18. Horizontal asymptote = ? 19. Graph the curve. 20. Domain = ? 21. range = ?

26. Putting it all together: If negative: Reflect across x-axis Initial value: Crosses y-axis here Growth factor: Horizontal shift vertical shift

27. 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” d. Any reflections across x-axis? d. Any reflections across x-axis? 22. 23. 24.

28. HOMEWORK  Blog worksheet