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3.1 Exponential Functions and Their Graphs

Learn about exponential functions with various bases, shifts on their graphs, and formulas for compound interest calculations. Explore the concept of Euler's number (e), compound interest with different compounding frequencies, and solve practical examples.

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3.1 Exponential Functions and Their Graphs

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  1. 3.1 Exponential Functions and Their Graphs The exponential function f with base a is denoted by f(x) = ax and x is any real number.

  2. Graph y = 2x x y -2 -1 0 1 2 2-2 = 2-1 = 1 2 4

  3. y = 2-x x y -2 -1 0 1 2 4 2 1 ½ ¼

  4. y = 2x + 1 x y -2 -1 0 1 2 5/4 3/2 2 3 5 Graph shifted up one.

  5. y = 2x+1 x y -2 -1 0 1 2 ½ 1 2 4 8 -3 ¼ Graph shifted left one.

  6. y = 2x y = -2x x y -2 -1 0 1 2 -1/4 -1/2 -1 -2 -4 Reflects about x-axis

  7. The number e e = 2.71828… y = ex x y -2 -1 0 1 2 1/e2 1/e 1 e e2 = 7.389

  8. Using the One-to-One Property Ex.

  9. Formulas for Compound Interest For n compoundings per year: yearly 1 monthly 12 semi-ann 2 daily 365 For continuous compounding:

  10. Ex. A total of $12,000 is invested at an annual percentage rate of 9%. Find the balance after 5 years if it is compounded a. quarterly b. continuously A = 12,000e.09(5) A = $18,819.75 A = $18,726.11

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