1. Exponential Functions

2. Exponential Functions and Their Graphs

3. Irrational Exponents Irrational Exponents If b is a positive number and x is a real number, the expression b x always represents a positive number. It is also true that the familiar properties of exponents hold for irrational exponents.

4. Example 1: Use properties of exponents to simplify Use properties of exponents to simplify

5. Example 1: Use properties of exponents to simplify Use properties of exponents to simplify

6. Example 1: Use properties of exponents to simplify Use properties of exponents to simplify

7. Example 1: Use properties of exponents to simplify Use properties of exponents to simplify

8. Exponential Functions An exponential function with base b is defined by the equation An exponential function with base b is defined by the equation x is a real number. x is a real number. The domain of any exponential function is the interval The domain of any exponential function is the interval The range is the interval The range is the interval

9. Graphing Exponential Functions

11. Example 2: Let’s make a table and plot points to graph.

12. Example 2:

14. Properties: Exponential Functions

15. Example 3: Given a graph, find the value of b: Given a graph, find the value of b:

16. Example 3: Given a graph, find the value of b: Given a graph, find the value of b:

17. Increasing and Decreasing Functions

18. One-to-One Exponential Functions

19. Compound Interest

20. Example 4: The parents of a newborn child invest $8,000 in a plan that earns 9% interest, compounded quarterly. If the money is left untouched, how much will the child have in the account in 55 years?

21. Example 4 Solution: Using the compound interest formula: Future value of account in 55 years

22. Base e Exponential Functions Sometimes called the natural base, often appears as the base of an exponential functions. It is the base of the continuous compound interest formula:

23. Example 5: If the parents of the newborn child in Example 4 had invested $8,000 at an annual rate of 9%, compounded continuously, how much would the child have in the account in 55 years?

24. Example 5 Solution: Future value of account in 55 years

25. Graphing Make a table and plot points: Make a table and plot points:

26. Exponential Functions Horizontal asymptote Horizontal asymptote Function increases Function increases y-intercept (0,1) y-intercept (0,1) Domain all real numbers Domain all real numbers Range: y > 0 Range: y > 0

27. Translations For k>0 y = f(x) + k y = f(x) + k y = f(x) – k y = f(x) – k y = f(x - k) y = f(x - k) y = f(x + k) y = f(x + k) Up k units Down k units Right k units Left k units

28. Example 6: On one set of axes, graph On one set of axes, graph

29. Example 6: On one set of axes, graph On one set of axes, graph Up 3

30. Example 7: On one set of axes, graph On one set of axes, graph Right 3

31. Non-Rigid Transformations Exponential Functions with the form f(x)=kb x and f(x)=b kx Exponential Functions with the form f(x)=kb x and f(x)=b kx are vertical and horizontal stretchings of the graph f(x)=b x. Use a graphing calculator to graph these functions.