Exponential Functions and Their Graphs The exponential function

Exponential Functions and Their Graphs

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) = 3 x and g(x) = 0. 5 x are exponential functions. 2

The value of f(x) = 3 x when x = 2 is f(2) = 32 = 9 The value of f(x) = 3 x when x = – 2 is f(– 2) = 3– 2 = The value of g(x) = 0. 5 x when x = 4 is g(4) = 0. 54 = 0. 0625 3

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

The graph of f(x) = abx, 0 < b < 1 y 4 Since a< 1; a is decay factor. Range: (0, ) Horizontal Asymptote y=0 (0, 1) x 4 Domain: (– , ) 5

Example: Sketch the graph of f(x) = 2 x. x y f(x) -2 2 -2 = 1/4 -1 2 -1 = 1/2 0 20 = 1 1 21 = 2 2 22 = 4 4 2 x – 2 2 6

Example: Sketch the graph of g(x) = 2 x – 1. State the domain and range. The graph of this function is a vertical translation of the graph of f(x) = 2 x down one unit. y f(x) = 2 x 4 2 Domain: (– , ) x Range: (– 1, ) y = – 1 7

Example: Sketch the graph of g(x) = 2 -x. State the domain and range. y The graph of this function is a reflection the graph of f(x) = 2 x in the yaxis. f(x) = 2 x 4 Domain: (– , ) Range: (0, ) x – 2 2 8

Example: Sketch the graph of g(x) = 4 x-3 + 3. State the domain and range. y Make a table. x y 1 3. 0625 2 3. 25 3 4 4 4 7 5 19 x – 2 2 Domain: (– , ) Range: (3, ) or y > 3 9

Write an exponential function y = abx for the graph that includes the given points. (2, 2) and (3, 4) 2 y = abx 2 2 = a b 2 Substitute in (2, 2) for x and y. Solve for a Now substitute (3, 4) in for x and y into y = abx 4 = ab 3 Set them equal to each other: Now solve for b to get b=2 2 = 4=a b 3 b 2 4=a You must solve for a: 2/2^2 = (1/2) = a b 3 Subst you’re a = (1/2) & b = 2 into either one of your 2 equations: y = abx Y = (1/2)(2)^x 10

Write an exponential function y = abx for the graph that includes the given points. (2, 4) and (3, 16) y = abx 11

The Natural Base e The irrational number e, where e 2. 71828… is used in applications involving growth and decay. Using techniques of calculus, it can be shown that 12

The graph of f(x) = ex y x -2 -1 0 1 2 6 4 2 f(x) 0. 14 0. 38 1 2. 72 7. 39 x – 2 2 13

Example: Sketch the graph of g(x) = ex-5 + 2. State the domain and range. y Make a table. x y 7 9. 39 6 4. 72 5 3 4 2. 36 3 2. 14 4 x – 2 2 Domain: (– , ) Range: (2, ) or y > 2 14

One to One Property If the bases are the same, then set the exponents equal. Example— 3 x = 32 x = ___ Example— 3 x+ 1 = 9 Example— 2 x - 1 = 1/32 Example— e 3 x + 2 = e 3 x = ___ Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

Interest Applications Formulas for Compound Interest— 1. ) compound per year -- A = P 1 + r n Balance in account nt Principal ($ you invest) r is the rate n is the number times you compound your money per year t is time. 2. Compounded continuously– A = Pert 16

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 17