Exponential Functions and Their Graphs

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 = Example: Exponential Function

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

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

6 Exponential Function 3 Key Parts 1. Y-intercept 2. Horizontal Asymptote 3. Growth or Decay

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

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

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

10 Calculator Comparison Graph the following on your calculator at the same time and note the trend y 1 = 2 x y 2 = 5 x y 3 = 10 x

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

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

13 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

14 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

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