1. Pre-Algebra 12-6 Exponential Functions 12-6 Exponential Functions Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Pre-Algebra 12-6 Exponential Functions Warm Up Write the rule for each linear function. 1. 2. f(x) = -5x - 2 f(x) = 2x + 6 Pre-Algebra 12-6 Exponential Functions

3. Pre-Algebra 12-6 Exponential Functions 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. Pre-Algebra 12-6 Exponential Functions Learn to identify and graph exponential functions.

5. Pre-Algebra 12-6 Exponential Functions Vocabulary exponential function exponential growth exponential decay

6. Pre-Algebra 12-6 Exponential Functions An exponential function has the form f(x) = p  a x, where a > 0 and a ≠ 1. If the input values are the set of whole numbers, the output values form a geometric sequence. The y-intercept is f(0) = p. The expression a x is defined for all values of x, so the domain of f(x) = p  a x is all real numbers.

7. Pre-Algebra 12-6 Exponential Functions Create a table for the exponential function, and use it to graph the function. A. f(x) = 3  2 x Additional Example 1A: Graphing an Exponential Function 3  2 0 = 3  1 3  2 1 = 3  2 3  2 2 = 3  4 3  2 -2 = 3  1 4 3  2 -1 = 3  1 2 xy –2 –1 0 1 2 3434 3232 3 6 12

8. Pre-Algebra 12-6 Exponential Functions Additional Example 1B: Graphing an Exponential Function Create a table for the exponential function, and use it to graph the function. B. f(x) = 4  1212 x xy –2 –1 0 1 2 4 2 1 16 8 4  = 4  4 1212 –2 4  = 4  2 1212 –1 4  = 4  1 1212 0 4  = 4  1212 1 1212 1212 2 1414

9. Pre-Algebra 12-6 Exponential Functions Create a table for the exponential function, and use it to graph the function. A. f(x) = 2 x Try This: Example 1A 2020 2121 2 -2 2 -1 xy –2 –1 0 1 2 1414 1212 1 2 4 2

10. Pre-Algebra 12-6 Exponential Functions Create a table for the exponential function, and use it to graph the function. B. f(x) = 2 x + 1 Try This: Example 1B 2 0 + 1 2 1 + 1 2 -2 + 1 xy –2 –1 0 1 2 5454 3232 2 3 5 2 2 + 1 2 -1 + 1

11. Pre-Algebra 12-6 Exponential Functions If a > 1, the output f(x) gets larger as the input x gets larger. In this case, f is called an exponential growth function.

12. Pre-Algebra 12-6 Exponential Functions 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? Day MonTueWedThu Number of days x 0123 Number of bacteria f(x) 500010,00020,00040,000

13. Pre-Algebra 12-6 Exponential Functions Additional Example 2 Continued f(x) = 5000  a x f(x) = 5000  2 x A week is 7 days so let x = 7. f(7) = 5000  2 7 = 640,000 If the number of bacteria doubles each day, there will be 640,000 bacteria in the culture after a week. f(0) = p f(1) = 5000  a 1 = 10,000, so a = 2. f(x) = p  a x Substitute 7 for x.

14. Pre-Algebra 12-6 Exponential Functions Try This: 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? Year 2003200720112015 Every 4 years x 0123 Account balance f(x) 30060012002400

15. Pre-Algebra 12-6 Exponential Functions Try This: Example 2 Continued f(x) = 300  a x f(x) = 300  2 x 20 years will be x = 5. f(5) = 300  2 5 = 9600 In 20 years, Robin will have a balance of $9600. f(0) = p f(1) = 300  a 1 = 300 so a = 2. f(x) = p  a x Substitute 5 for x.

16. Pre-Algebra 12-6 Exponential Functions In the exponential function f(x) = p  a x, if a < 1, the output gets smaller as x gets larger. In this case, f is called an exponential decay function.

17. Pre-Algebra 12-6 Exponential Functions 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. Seconds 0153045 Number of Half-lives x 0123 Bohrium-267 f(x) (mg) 16842

18. Pre-Algebra 12-6 Exponential Functions Additional Example 3 Continued f(x) = 16  a x Since 2 minutes = 120 seconds, divide 120 seconds by 15 seconds to find the number of half-lives: x = 8. There is 0.0625 mg of Bohrium-267 left after 2 minutes. f(0) = p f(x) = p  a x Substitute 8 for x. f(x) = 16  1212 x f(1) = 16  a 1 = 8 so a =. 1212 f(8) = 16  1212 8

19. Pre-Algebra 12-6 Exponential Functions Try This: 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. Seconds 0255075 Number of Half-lives x 0123 Element (mg) 8421

20. Pre-Algebra 12-6 Exponential Functions Try This: Example 3 Continued f(x) = 8  a x Since 3 minutes = 180 seconds, divide 180 seconds by 25 seconds to find the number of half-lives: x = 7.2. There is approximately 0.054 mg of the element left after 3 minutes. f(0) = p f(x) = p  a x Substitute 7.2 for x. f(x) = 8  1212 x f(1) = 8  a 1 = 4 so a =. 1212 f(7.2) = 8  1212 7.2

21. Pre-Algebra 12-6 Exponential Functions Lesson Quiz: Part 1 1. Create a table for the exponential function f(x) =, and use it to graph the function. 3  1212 x xy –212 –16 03 1 2 3434 3232

22. Pre-Algebra 12-6 Exponential Functions Lesson Quiz: Part 2 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  2 x, where x is the number of 3- year periods; $3200.