Exponential Functions and Their Graphs

Definition of Exponential Function The exponential function f with base b is defined by f(x) = abx where b > 0, b  1, and x is any real number. ** when b> 1; b is considered a growth factor. For instance, f(x) = 3x and g(x) = 0.5x are exponential functions. Definition of Exponential Function

The value of f(x) = 3x when x = 2 is 9 The value of f(x) = 3x when x = –2 is f(–2) = 3–2 = The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625

Graph of Exponential Function (a > 1) The graph of f(x) = abx, b > 1 y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (a > 1)

Graph of Exponential Function (0 < a < 1) The graph of f(x) = abx, 0 < b < 1 Since a< 1; a is decay factor. y 4 Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x 4 Domain: (–, ) Graph of Exponential Function (0 < a < 1)

Example: Sketch the graph of f(x) = 2x. y x f(x) -2 -1 1 2 2-2 = 1/4 4 2-1 = 1/2 2 20 = 1 21 = 2 x 22 = 4 –2 2 Example: Graph f(x) = 2x

Example: Translation of Graph Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y f(x) = 2x The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2 Domain: (–, ) x y = –1 Range: (–1, ) Example: Translation of Graph

Example: Reflection of Graph Example: Sketch the graph of g(x) = 2-x. State the domain and range. y f(x) = 2x The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4 Domain: (–, ) x –2 2 Range: (0, ) Example: Reflection of Graph

Example: Reflection of Graph Example: Sketch the graph of g(x) = 4x-3 + 3. State the domain and range. y Make a table. x y 4 1 3.0625 2 3.25 3 4 x 4 7 –2 2 5 19 Domain: (–, ) Range: (3, ) or y > 3 Example: Reflection of Graph

Write an exponential function y = abx for the graph that includes the given points. (2, 2) and (3, 4) 2 = ab2 y = abx Substitute in (2,2) for x and y. Solve for a 2 = a b2 Now substitute (3,4) in for x and y and 2/b2 in for a 4 = (2)b3 b2 4 = (2)b 4 = b y = ½ (4)x

Write an exponential function y = abx for the graph that includes the given points. (2, 4) and (3, 16) y = abx

The Natural Base e The irrational number e, where e  2.718281828… is used in applications involving growth and decay. Using techniques of calculus, it can be shown that The number e

Graph of Natural Exponential Function f(x) = ex The graph of f(x) = ex y x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 6 4 2 x –2 2 Graph of Natural Exponential Function f(x) = ex

Example: Sketch the graph of g(x) = ex-5 + 2. State the domain and range. y Make a table. x y 4 7 9.39 6 4.72 5 3 x 4 2.36 –2 2 3 2.14 Domain: (–, ) Range: (2, ) or y > 2

One to One Property If the bases are the same, then set the exponents equal. Example— 3x = 32 x = ___ Example— 3x+ 1 = 9 x = ___ Example— 2x - 1 = 1/32 x = ___ Example— e3x + 2 = e3 x = ___ Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Interest Applications Formulas for Compound Interest— 1.) compound per year -- A = P 1 + r nt n r is the rate n is the number times you compound your money per year t is time. Balance in account Principal ($ you invest) 2. Compounded continuously– A = Pert

A total of $12000 is invested at an annual interest rate of 9% A total of $12000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly b. monthly c. continuously