1. PRECALCULUS I EXPONENTIAL FUNCTIONS Dr. Claude S. Moore Danville Community College

2. DEFINITION The exponential function is f(x) = a x where a > 0, a  1, and x is any real number.

3. VALUES OF a INFLUENCE GRAPHS The following are true for f(x) = a x : 1. The graph goes through (0,1). 2. The x-axis is a horizontal asymptote. 3. As a  0, the graph tends to flatten more. 4. If a > 1, the graph of f(x) goes up to the right. 5. If 0 < a < 1, the graph of f(x) goes down to the right.

4. EXAMPLE: y = 2 x This graph of y = f(x) = 2 x was generated with the TI-82. a = 2 > 1, graph goes up to the right. Graph goes through (0,1).

5. GRAPHING f(x) = a -x Before graphing f(x) = a -x, rewrite the function as : f(x) = 1/a x = (1/a) x 1. The graph goes through (0,1). 2. The x-axis is a horizontal asymptote. 3. If (1/a) > 1, the graph of f(x) goes up to the right. 4. If 0 < (1/a) < 1, the graph of f(x) goes down to the right.

6. EXAMPLE: y = 2 -x This graph of y = f(x) = 2 -x = (1/2) x was generated with the TI-82. 0<1/2<1, graph goes down to the right. Graph goes through (0,1).

7. 1. The graph goes through (0,1). 2. The x-axis is a horizontal asymptote. 3. If a > 1, the graph of f(x) goes up to the right. 4. If 0 < a < 1, the graph of f(x) goes down to the right. 1. The graph goes through (0,1). 2. The x-axis is a horizontal asymptote. 3. If a > 1, the graph of f(x) goes down to the right. 4. If 0 < a < 1, the graph of f(x) goes up to the right. f(x) = a x vs. f(x) = a -x

8. EXAMPLE: BACTERIA GROWTH A certain bacteria increases by the model with t in hours. Find P(0), P(5), and P(10). Answers: P(0) = 100P(5) = 299.97 P(10) = 899.8

9. COMPOUND INTEREST Compounded n times per year. A = amount in balance P = principal invested r = annual interest rate t = number of years Compounded continuously.

10. EXAMPLE: COMPOUND INTEREST Find the balance of a $3500 investment compounded monthly at 8% for 5 years. The answer is: A = 3500(1+.08/12) 12(5) = $5214.46

11. EXAMPLE: COMPOUND INTEREST Find the balance of a $3500 investment compounded continuously at 8% for 5 years. The answer is: A = 3500e 0.08(5) = $5221.39. ( $5214.46 compounded monthly)