3.1 Exponential Functions and Their Graphs CW: Exponential Growth and Decay

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

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

The Graph of f(x) = ax, a > 1 y Range: (0, ) (0, 1) x Horizontal Asymptote y = 0 Domain: (–, )

The Graph of f(x) = ax, 0 < a <1 y Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x Domain: (–, )

Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) y x f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) 4 2 x –2 2

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: Sketch the graph of g(x) = 2x+1 + 2 Example: Sketch the graph of g(x) = 2x+1 + 2. 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 left one unit and up two units. 4 2 y = 2 x Domain: (–, ) Range: (2, )

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, )

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

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

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