Exponential Functions and Their Graphs Section 3-1

Definition of Exponential Function 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. Definition of Exponential Function

Example: Exponential Function The value of f(x) = 3x when x = 2 is f(2) = 32 = 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 Example: Exponential Function

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

Exponential Function 3 Key Parts 1. Pivot Point (Common Point) 2. Horizontal Asymptote 3. Growth or Decay

Manual Graphing Lets graph the following together: f(x) = 2x Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) -2 ¼ (-2, ¼) 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph f(x) = 2x

Definition of the Exponential Function The exponential function f with base b is defined by f (x) = bx or y = bx Where b is a positive constant other than and x is any real number. Here are some examples of exponential functions. f (x) = 2x g(x) = 10x h(x) = 3x Base is 2. Base is 10. Base is 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Calculator Comparison Graph the following on your calculator at the same time and note the trend y1 = 2x y2= 5x y3 = 10x

When base is a fraction Graph the following on your calculator at the same time and note the trend y1 = (1/2)x y2= (3/4)x y3 = (7/8)x

Transformations Involving Exponential Functions Shifts the graph of f (x) = bx upward c units if c > 0. Shifts the graph of f (x) = bx downward c units if c < 0. g(x) = bx+ c Vertical translation Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. g(x) = -bx g(x) = b-x Reflecting Multiplying y-coordintates of f (x) = bx by c, Stretches the graph of f (x) = bx if c > 1. Shrinks the graph of f (x) = bx if 0 < c < 1. g(x) = cbx Vertical stretching or shrinking Shifts the graph of f (x) = bx to the left c units if c > 0. Shifts the graph of f (x) = bx to the right c units if c < 0. g(x) = bx+c Horizontal translation Description Equation Transformation

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

Discuss these transformations y = 2(x+1) Left 1 unit y = 2x + 2 Up 2 units y = 2-x – 2 Ry, then down 2 units

Special Symbols Math uses special symbols at times to represent special numbers used in calculations. The symbol  (pi) represents 3.14….. The symbol “i” represents

(The Euler #) e is an irrational #, where e  2.718281828… is used in applications involving growth and decay. 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 Natural Exponential Function 6 4 2 x –2 2 Graph of Natural Exponential Function f(x) = ex

Homework WS 6-1