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Exponential Functions

Exponential Functions. Exponential Functions and Their Graphs. 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 . Example 1:.

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Exponential Functions

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  1. Exponential Functions

  2. Exponential Functions and Their Graphs

  3. Irrational Exponents If b is a positive number and x is a real number, the expression bx 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

  5. Example 1: Use properties of exponents to simplify

  6. Example 1: Use properties of exponents to simplify

  7. Example 1: Use properties of exponents to simplify

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

  9. Graphing Exponential Functions

  10. Graphing Exponential Functions

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

  12. Example 2:

  13. Example 2:

  14. Properties: Exponential Functions

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

  16. Example 3: • 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:

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

  27. Translations For k>0 • 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

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

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

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

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