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13-5

Exponential Functions. 13-5. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Exponential Functions. 13-5. Course 3. Warm Up Write the rule for each linear function. 1. 2. f ( x ) = -5 x - 2. f ( x ) = 2 x + 6. Problem of the Day

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13-5

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  1. Exponential Functions 13-5 Course 3 Warm Up Problem of the Day Lesson Presentation

  2. Exponential Functions 13-5 Course 3 Warm Up Write the rule for each linear function. 1. 2. f(x) = -5x - 2 f(x) = 2x + 6

  3. Problem of the Day One point on the graph of the mystery linear function is (4, 4). No value of x gives a y-value of 3. What is the mystery function? y = 4

  4. Learn to identify and graph exponential functions.

  5. Vocabulary exponential function exponential growth exponential decay

  6. A function rule that describes the pattern is f(x) = 15(4)x, where 15 is a1, the starting number, and 4 is r the common ratio. This type of function is an exponential function.

  7. In an exponential function, the y-intercept is f(0) = a1. The expression rx is defined for all values of x, so the domain of f(x)= a1 rx is all real numbers.

  8. 3 4 3 2 3  2-2 = 3  1 1 4 2 3  2-1 = 3  Additional Example 1A: Graphing Exponential Functions Create a table for the exponential function, and use it to graph the function. f(x) = 3  2x 3 3  20 = 3  1 6 3  21 = 3  2 3  22 = 3  4 12

  9. 2 3 Additional Example 1B: Graphing Exponential Functions Create a table for the exponential function, and use it to graph the function. f(x) = x 2.25 1.5 1 0.67 0.44…

  10. 1 4 1 2 Check It Out: Example 1A Create a table for the exponential function, and use it to graph the function. f(x) = 2x 2-2 2-1 1 20 2 21 22 4

  11. 5 4 3 2 Check It Out: Example 1B Create a table for the exponential function, and use it to graph the function. f(x) = 2x+ 1 2-2 + 1 2-1 + 1 2 20 + 1 3 21 + 1 22 + 1 5

  12. In the exponential function f(x) = a1 rx if r > 1, the output gets larger as the input gets larger. In this case, f is called an exponential growth function.

  13. Additional Example 2: Using an Exponential Growth Function A bacterial culture contains 5000 bacteria, and the number of bacteria doubles each day. How many bacteria will be in the culture after a week?

  14. Additional Example 2 Continued f(x) = a1 rx Write the function. f(x) = 5000  rx f(0) = a1 The common ratio is 2. f(x) = 5000  2x A week is 7 days so let x = 7. f(7) = 5000  27 = 640,000 Substitute 7 for x. If the number of bacteria doubles each day, there will be 640,000 bacteria in the culture after a week.

  15. Check It Out: Example 2 Robin invested $300 in an account that will double her balance every 4 years. Write an exponential function to calculate her account balance. What will her account balance be in 20 years?

  16. Check It Out: Example 2 Continued f(x) = a1 rx Write the function. f(x) = 300 rx f(0) = a1 The common ratio is 2. f(x) = 300 2x 20 years will be x = 5. f(5) = 300 25 = 9600 Substitute 5 for x. In 20 years, Robin will have a balance of $9600.

  17. In the exponential function f(x) = a1 rx, if r < 1, the output gets smaller as x gets larger. In this case, f is called an exponential decay function.

  18. Additional Example 3: Using an Exponential Decay Function Bohrium-267 has a half-life of 15 seconds. Find the amount that remains from a 16 mg sample of this substance after 2 minutes.

  19. The common ratio is . x 8 f(x) = 16  1 2 1 2 1 2 f(8) = 16  Additional Example 3 Continued f(x) = a1 rx Write the function. f(x) = 16 rx f(0) = a1 Since 2 minutes = 120 seconds, divide 120 seconds by 15 seconds to find the number of half-lives: x = 8. Substitute 8 for x. There is 0.0625 mg of Bohrium-267 left after 2 minutes.

  20. Check It Out: Example 3 If an element has a half-life of 25 seconds. Find the amount that remains from a 8 mg sample of this substance after 3 minutes.

  21. The common ratio is . x f(x) = 8  1 2 1 2 1 2 7.2 f(7.2) = 8  Check It Out: Example 3 Continued f(x) = a1 rx Write the function. f(x) = 8 rx f(0) = p Since 3 minutes = 180 seconds, divide 180 seconds by 25 seconds to find the number of half-lives: x = 7.2. Substitute 7.2 for x. There is approximately 0.054 mg of the element left after 3 minutes.

  22. 1. Create a table for the exponential function f(x) = , and use it to graph the function. x 3  1 2 3 4 3 2 Lesson Quiz: Part I

  23. Lesson Quiz: Part II 2. Linda invested $200 in an account that will double her balance every 3 years. Write an exponential function to calculate her account balance. What will her balance be in 12 years? f(x) = 200 2x, where x is the number of 3-year periods; $3200.

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