1. Warm-up Given the function: Find the inverse function and complete the table below. Given the function: Find the inverse function and complete the table below. x-intercepts y-intercepts vertical asymptotes horizontal asymptotes Sketch the graph of both on the same set of axes

2. 4.3 Exponential Functions

3. Warm-up What are the characteristics of an exponential function, based on the following list of examples. Examples of exponential functions. NOT an Example of exponential function. NOT an Example of exponential function.

4. An exponential function is of the form where: BASE is: a with requirement: a > 0 and a ≠ 1 EXPONENT is: x I. Exponential Function

5. Application Example Example: The function: describes the number of O-Rings expected to fail when the temperature is x (degrees F). The space-shuttle Challenger exploded in 1986 when the temperature was 31 o F. What is the expected number of O-rings to fail at 31 o F ? at 60 o F ?

6. Do you recall these rules? II. Laws of Exponents

7. Example: Plot points and sketch: III. Graphing an Exponential Function Example: Plot points and sketch:

8. III. Properties of the graph of Exponential Function If base is greater than 1 If base is less than 1

9. IV. Transformations of the graph of Exponential Function What do transformations do to the graph of an exponential function? p. 283, # 29-36

10. IV. Transformations of the graph of Exponential Function What do transformations do to the graph of

11. V. Graphing on the Calculator Graph Does your graph demonstrate the correct shape for exponential growth? If not, what happened? Watch your parentheses!!!!! Enter y = 3^(2x) on the calculator

12. VI. A “Natural” Base. The Number e VI. A “Natural” Base. The Number e First approximation for was found in studying continuous compound interest. is the number given by the value of The calculator gives us an approximation to the number. Where does the graph of lie? The calculator gives us an approximation to the number. Where does the graph of lie?

13. A Theorem for Exponential Functions What does the base of an exponential function represent? Rumors: Scenario 1: # people you tell is 2 per day. Scenario 2: # people you tell is 3 per day. What can you say about the ratio of consecutive days ? Theorem: An exponential function satisfies the property:

14. A Theorem for Exponential Functions How can we solve for x using the graphing calculator ? Algebraically ? If an exponential equation can be expressed in the form: (same base on each side of equal sign) Then we are HAPPY, ‘cause it’s pretty easy to solve

15. If a problem is not written in this form, we can TRY to get it that way: where u and v are expressions in x where u and v are expressions in x VII. Solving Exponential Equations If a problem can be not written in this form, we will use logarithms to solve for the unknown

16. Exponential Equations with base e Treat as a number. Solve for x: If we have a problem of the form: