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Understanding Exponential Functions: Growth, Decay, and Graphing Techniques

This resource provides a comprehensive overview of exponential functions, focusing on their definitions, properties, and graphs. It covers both exponential growth and decay, illustrating how the base of the function affects its graph. Students will learn to identify, graph, and analyze various exponential functions using their calculators. The content includes key aspects like domain, range, horizontal asymptotes, and transformations such as reflections. Complete with practice questions and answers, this guide aims to enhance understanding of exponential concepts.

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Understanding Exponential Functions: Growth, Decay, and Graphing Techniques

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  1. Factoring Practice (5 questions)

  2. Factoring Practice (Answers)

  3. Exponential Functions and Their GraphsSections 7-1,2,3

  4. Objectives • I can recognize growth or decay exponential equations • I can graph exponential growth and decay parent functions

  5. Definition of Exponential Function The exponential function f with base b is defined by f(x) = bx where b > 0, b 1, and x is any real number. For instance, f(x) = 3x and g(x) = 0.5x are exponential parent functions.

  6. The graph off(x) = bx, b > 1 Graph of Exponential Function (a > 1) Exponential Growth y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, )

  7. The graph off(x) = bx, 0 < b < 1 Graph of Exponential Function (0 < a < 1) y Exponential Decay 4 Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x 4 Domain: (–, )

  8. Example: Sketch the graph ofy = 2x. Example: Graph f(x) = 2x y 4 2 x –2 2

  9. Base is 2. Base is 10. Base is 3. Definition of the Exponential Function Put these in your calculator as y1=, y2=, y3= And graph them simultaneously Here are some examples of exponential functions. f (x) = 2xg(x) = 10xh(x) = 3x

  10. When base is a fraction • Graph the following on your calculator at the same time • y1 = (1/2)x • y2= (3/4)x • y3 = (7/8)x

  11. Example: Sketch the graph ofg(x) = 2-x. State the domain and range. Example: Reflection of Graph 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, )

  12. The graph off(x) = ex Graph of Natural Exponential Function f(x) = ex y 6 4 2 x –2 2

  13. The number e The irrational number e, where e 2.718281828… is used in applications involving growth and decay. It is a key on your calculator in the first comlum 2nd LN

  14. What type is it?? • You must know if you are graphing a growth or decay equation? • Let’s look at some examples

  15. Reflection Transformations

  16. Reflections Change Type of Exponential

  17. Growth or Decay??

  18. Growth or Decay??

  19. Growth or Decay??

  20. Growth or Decay??

  21. Growth or Decay??

  22. Growth or Decay??

  23. Growth or Decay??

  24. Homework • Worksheet 11-1

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