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Exponential Functions, Growth and Decay Understand how 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: I can graph an exponential function Identify exponential growth and decay Today’s Agenda Do Now Lesson Review Quiz Tomorrow
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Exponential Functions and Their Graphs Section 3-1
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Definition of Exponential Function
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. Definition of Exponential Function
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Example: Exponential Function
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 = The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625 Example: Exponential Function
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Graph of Exponential Function (a > 1)
The graph of f(x) = ax, a > 1 y Exponential Growth Function 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (a > 1)
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Graph of Exponential Function (0 < a < 1)
The graph of f(x) = ax, 0 < a < 1 y Exponential Decay Function 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (0 < a < 1)
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Exponential Function 3 Key Parts 1. Pivot Point (Common Point)
2. Horizontal Asymptote 3. Growth or Decay
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Manual Graphing Lets graph the following together: f(x) = 2x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
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Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x))
y x f(x) (x, f(x)) -2 (-2, ¼) -1 (-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: Graph f(x) = 2x
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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 h(x) = 3x Base is 2. Base is 10. Base is 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
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Calculator Comparison
Graph the following on your calculator at the same time and note the trend y1 = 2x y2= 5x y3 = 10x
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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
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Lesson YOU TRY 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 $ 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
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HW#13 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
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Transformations Involving Exponential Functions
Shifts 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. g(x) = bx+ c Vertical translation Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. g(x) = -bx g(x) = b-x Reflecting 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. g(x) = cbx Vertical stretching or shrinking 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. g(x) = bx+c Horizontal translation Description Equation Transformation
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Example: Translation of Graph
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, ) Example: Translation of Graph
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Example: Reflection of Graph
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, ) Example: Reflection of Graph
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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
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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
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(The Euler #) e is an irrational #, where e 2.718281828…
is used in applications involving growth and decay. The number e
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Graph of Natural Exponential Function f(x) = ex
The graph of f(x) = ex y x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 Natural Exponential Function 6 4 2 x –2 2 Graph of Natural Exponential Function f(x) = ex
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Homework WS 6-1
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