Exponential Functions Growth and Decay Growth how and

Exponential Functions, Growth and Decay Growth, how and Decay Understand to write and evaluate exponential expressions to model growth and decay situations. Do Now: - What is the Domain and Range. Is there a x and y intercept? If so where? Success Criteria: q I can graph an exponential function q Identify exponential growth and decay ü ü Today’s Agenda Do Now Lesson Review Quiz Tomorrow

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) = 3 x and g(x) = 0. 5 x are exponential functions. 3

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 4

The graph of f(x) = ax, a > 1 Exponential Growth Function y 4 Range: (0, ) (0, 1) x 4 Domain: (– , ) Horizontal Asymptote y=0 5

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 6

Exponential Function • • 3 Key Parts 1. Pivot Point (Common Point) 2. Horizontal Asymptote 3. Growth or Decay 7

Manual Graphing • Lets graph the following together: • f(x) = 2 x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

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 9

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. 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

Exponential Functions, Lesson YOU TRY Growth, and Decay In 2000, the world population was 6. 08 billion and was increasing at a rate 1. 21% each year. 1. Write a function for world population. Does the function represent growth or decay? P(t) = 6. 08(1. 0121)t 2. Use a calculator to predict the population in 2020. ≈ 7. 73 billion The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? V(t)≈ 3000(0. 7)t 4. Use a calculator to predict the value in 4 years. ≈ $720. 30 Gabby purchased a car for $5000. A year later the car was valued at $4500. 5. Write a function that represents the value of the car. C(t)≈ 5000(1 -0. 1)t 6. At this rate of depreciation, how many years until her car is worth $1000? ≈ ? ? years

HW#13 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14

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. 15

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 16

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 17

Discuss these transformations • • • y = 2(x+1) Left 1 unit y = 2 x + 2 Up 2 units y = 2 -x – 2 Ry, then down 2 units 18

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 19

(The Euler #) e is an irrational #, where e 2. 71828… is used in applications involving growth and decay. 20

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 21

Homework • WS 6 -1 22