Exponential Functions and Their Graphs Presented by the best teacher in the world: Mr. Peter Richard

The Exponential Function y = abx b is the base: GROWTH It must be greater than 0 It cannot equal 1. x can be any real number .

Identify Exponential Functions Which of the following are exponential functions? y = 3x y = x3 yes no y = 2(7)x y = 2(-7)x yes no

Exponential Factors If the factor b is greater than 1, then we call the relationship exponential growth. If the factor b is less than 1, we call the relationship exponential decay.

Determine Whether Growth or Decay State whether the function is an exponential growth or exponential decay function. (Look at the “b” Value. Is it more or less than one?) decay growth decay decay growth

GRAPHING EXPONENTIAL DECAY MODELS EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL y = C (1 + r)t y = C (1 – r)t An exponential model y = a • b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. 1 + r > 1 0 < 1 – r < 1

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved

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, ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved

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, ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved

The irrational number e, where e  2.718281828… is used in applications involving growth and decay. Using techniques of calculus, it can be shown that Copyright © by Houghton Mifflin Company, Inc. All rights reserved

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved

Quiz and Homework Quiz: Page 435 # 16, 18, 20, 24, 26 Homework # 17, 19, 21, 27, 29 Copyright © by Houghton Mifflin Company, Inc. All rights reserved