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Section 4.2 Exponential Functions and Graphs Exponential Function The exponential function is very important in math because it is used to model many real life situations. For example: population growth and decay, compound interest, economics, and much more. Exponent Remember x 1 0 Exponential Function The function f(x) = ax, where x is a real number, a > 0 and a 1, is called the exponential function, base a. The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers. Examples: f ( x) 3 x 1 f ( x) 3 x f ( x) (4.23) x Graphing Exponential Functions To graph an exponential function, follow the steps listed: 1. Compute some function values and list the results in a table. 2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises. Example Graph the exponential function y = f(x) = 3x. x y = f(x) = 3x (x, y) 0 1 (0, 1) 1 3 (1, 3) 2 9 (2, 9) 3 27 (3, 27) 1 1/3 (1, 1/3) 2 1/9 (2, 1/9) 3 1/27 (3, 1/27) Example Graph y = 3x + 2. The graph is the graph of y = 3x shifted _____ 2 units. x y 3 1/3 2 1 1 3 0 9 1 27 2 81 3 243 Example 1 Graph the exponential function y f ( x) 3 1 x x y f ( x) 3 (x, y) 0 1 (0, 1) 1 3 (1, 3) 2 9 (2, 9) 3 27 (3, 27) 1 1/3 (1, 1/3) 2 1/9 (2, 1/9) 3 1/27 (3, 1/27) x Example Graph y = 4 3x The graph is a reflection of the graph of y = 3x across the _______, followed by a reflection across the _______ and then a shift _______ of 4 units. x 3 2 1 0 1 2 3 y 23 5 1 3 3.67 3.88 3.96 Observing Relationships Connecting the Concepts The Number e e is known as the natural base (Most important base for exponential functions.) e is an irrational number (can’t write its exact value) We approximate e e 2.718281 The Number e Find each value of ex, to four decimal places, using the ex key on a calculator. a) e4 b) e0.25 c) e2 d) e1 Natural Exponential Function f(x) e x Remember e is a number e lies between 2 and 3 Graphs of Exponential Functions, Base e Graph f(x) = ex. x f(x) 2 0.135 1 0.368 0 1 1 2.718 2 7.389 Example Graph f(x) = ex+2. x f(x) 4 0.135 3 0.368 2 1 1 2.718 0 7.389 1 20.086 Example Graph f(x) = 2 e3x. x f(x) 2 401.43 1 18.09 0 1 1 1.95 2 1.99 Compound Interest Formula r A P 1 n nt A amount accumulated after t years P principal dollar s in vested r int erest rate (decimal form) n number of times money is compounded per year t number of years money is compounded Example A father sets up a savings account for his daughter. He puts $1000 in an account that is compounded quarterly at an annual interest rate of 8%. How much money will be in the account at the end of 10 years? (Assume no other deposits were made after the original one.)