Exponential Functions and Their Graphs Section 3-1 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. 2 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 1 = 9 The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625 3 The graph of f(x) = ax, a > 1 Exponential Growth Function y 4 Range: (0, ) (0, 1) x 4 Domain: (–, ) Horizontal Asymptote y=0 4 The graph of f(x) = ax, 0 < a < 1 y Exponential Decay Function 4 Range: (0, ) (0, 1) x 4 Domain: (–, ) Horizontal Asymptote y=0 5 Exponential Function • • • • 3 Key Parts 1. Pivot Point (Common Point) 2. Horizontal Asymptote 3. Growth or Decay 6 Manual Graphing • Lets graph the following together: • f(x) = 2x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Sketch the graph of f(x) = 2x. x y f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 0 1 2 ½ 1 2 4 (-1, ½) (0, 1) (1, 2) (2, 4) 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) = 2x g(x) = 10x Base is 2. Base is 10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. h(x) = 3x Base is 3. 9 Calculator Comparison • Graph the following on your calculator at the same time and note the trend • y1 = 2 x • y 2= 5 x • y3 = 10x 10 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 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) = 2x – 1. State the domain and range. The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . y f(x) = 2x 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) = 2x in the yaxis. f(x) = 2x 4 Domain: (–, ) Range: (0, ) x –2 2 14 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 15 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 1 16 (The Euler #) e is an irrational #, where e 2.718281828… is used in applications involving growth and decay. 17 The graph of f(x) = ex y Natural Exponential Function x -2 -1 0 1 2 6 4 2 f(x) 0.14 0.38 1 2.72 7.39 x –2 2 18 Homework • WS 6-1 19