Exponential Functions and Their Graphs Digital Lesson

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

3 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) = = The value of f(x) = 3 x when x = –2 is 9 f(–2) = 3 –2 =

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

5 The graph of f(x) = a x, 0 < a < 1 y x (0, 1) Domain : (– ,  ) Range : (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (0 < a < 1) 4 4 Represents Exponential Decay

6 Example: Sketch the graph of f(x) = 2 x. x xf(x)(x, f(x)) -2¼(-2, ¼) ½(-1, ½) 01(0, 1) 12(1, 2) 24(2, 4) y 2–2 2 4 Example: Graph f(x) = 2 x Domain = (- ,  ) Range = (0,  )

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

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

9 Example: Sketch the graph of g(x) = 4 x State the domain and range. x y Make a table. Domain : (– ,  ) Range : (3,  ) or y > 3 2 –2 4 Example: Reflection of Graph x y

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

11 The graph of f(x) = e x y x 2 – xf(x)f(x) Graph of Natural Exponential Function f(x) = e x

12 Example: Sketch the graph of g(x) = e x State the domain and range. x y Make a table. Domain: (– ,  ) Range: (2,  ) or y > 2 2 –2 4 x y

13 Formulas for Compound Interest — 1.) compound per year : A = P 1 + r nt n Interest Applications Balance in account 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 = Pe rt

14 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