Exponential Functions and Their Graphs The exponential function

Exponential Functions and Their Graphs

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) = 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) = bx, b > 1 Exponential Growth Function y 4 Range: (0, ) (0, 1) x 4 Domain: (– , ) Horizontal Asymptote y=0 4

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

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

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

Example: Sketch the graph of f(x) = 2 x. x -2 -1 0 1 2 f(x) (x, f(x)) ¼ (-2, ¼) ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) y 4 2 x – 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 8

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) = 2 x g(x) = 10 x Base is 2. Base is 10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. h(x) = 3 x Base is 3. 9

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 10

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 11

Transformations Involving Exponential Functions Transformation Equation Description Horizontal translation g(x) = bx+c • Shifts Vertical stretching or shrinking g(x) = cbx 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. Reflecting g(x) = -bx g(x) = b-x • Reflects Vertical translation g(x) = bx+ c • 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. the graph of f (x) = bx about the x-axis. • Reflects the graph of f (x) = bx about the y-axis. 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. 12

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 13

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 14

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 15