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Page 1 of 2 Math 112 Section 3.1 Exponential Functions and Graphs Exponential Function An exponential function with base a is defined as f ( x) a x , where a > 0 and a 0. x Example 1: f ( x) 2 1 x Example 2: g ( x) 2 2 x x f x = 2x 8 f(x) g(x) f x = 1 2x 8 0 6 1 6 4 2 4 -1 2 2 -2 -5 -5 a > 1 (increasing) 5 0 < a < 1 (decreasing) Compound Amount If P dollars is invested at a yearly rate of interest r per year, compounded m times per year for t years, the compound amount is r A P 1 m tm dollars. Example 3: If $82,000 is invested at 4.5% compounded quarterly, find the sum in 5 years. Example 4: Interest is compounded semiannually. Find the amount in the account and the interest earned if the principal is $3,000, the rate of interest is 7%, and the time is 3 years. r Definition of e: As m becomes larger and larger, 1 number e, whose approximate value is 2.7182818. Example 5: Calculate e21, e-5 m m becomes closer and closer to the Page 2 of 2 Shifting exponential functions: Example 6: f ( x) 2 x Example 7: f ( x) 2 x 2 hx = 2x+2 8 f x = 2x 8 6 6 4 4 2 2 -5 -5 Example 8: f ( x) 2 2 x Example 9: f ( x) 2 x rx = 2x+2 s x = -2x 8 5 6 -2 4 -4 2 -6 -8 -5 Example 10: f ( x) e x 5 Example 11: f ( x) 2 x4 1