8.1 exponential growth

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8.1 exponential growth

  1. 1. What are Exponential Functions? Exponential functions – functions that include the expression bx where b is a positive # other than 1. b is called the base.
  2. 2. What’s the Shape? Let’s make a table to find the general shape. If we use f(x) = 2x as an example: x f(x) = 2x -3 -2 -1 0 1 2 3
  3. 3. Asymptotes An asymptote is a line that a graph approaches (but does not touch) as you move away from the origin. For example: Our graph has a horizontal asymptote at y = 0.
  4. 4. Graphing y = abx If a > 0 and b > 1, y = abx is an exponential growth function. For all y = abx , b > 1:  Graphs pass through (0, a) (a is the y-int)  x-axis is an asymptote  Domain: all real #s  Range: y > 0 if a > 0 y < 0 if a < 0
  5. 5. To graph: Plot 2 points: (0, a) and (1, __)  Plug in 1 for x to fill the blank Connect with a smooth curve that:  Starts left of the origin, close to the x-axis  Moves up or down quickly to the right
  6. 6. Examples Graph:
  7. 7. Your Turn! Graph
  8. 8. General Exponential Functions General form: As usual: h is horizontal shift k is vertical shift To graph:  Sketch the “parent graph” y = abx  Shift using h and k
  9. 9. Examples Graph and state the domain and range:
  10. 10. Your Turn! Graph and state the domain and range:
  11. 11. Exponential Growth Models When a real-life quantity increases by a fixed % each year, the amount of the quantity after t years can be modeled by: y = a(1 + r)t where a is the initial amount and r is the % increase (as a decimal). (1 + r) is the growth factor.
  12. 12. Example: In January, 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. Write a model giving the number h (in millions) of hosts t years after 1993. How many hosts were there in 1996?
  13. 13. Compound Interest Compound interest is interest paid on the original principal and on previously earned interest. Modeled by an exponential function. If interest is compounded n times per year, the amount A in the account after t years is: where P is the initial principal and r is the annual interest rate.
  14. 14. Example: You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if interest is compounded: A. annually B. quarterly C. daily Which is the best investment?

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