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Building Exponential Functions

Building Exponential Functions. A Miscellany of Features of Logarithmic and Exponential Functions. Population Growth / Food Production. A pair of students will model an exponential function and a linear function separately but simultaneously. Population : Food : 10,000,000 15,000,000

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Building Exponential Functions

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  1. Building Exponential Functions A Miscellany of Features of Logarithmic and Exponential Functions

  2. Population Growth / Food Production A pair of students will model anexponential functionand alinear functionseparately but simultaneously. Population:Food: 10,000,00015,000,000 10,000,00015,000,000 ANS*1.02ANS + 500,000 What do the specific numbers represent?

  3. Population Growth / Food Production The food production begins with the ability to feed more than the population. Does that production continue to be able to stay ahead of the population growth? What is the population function? The food function? P(t) =F(t) = 10,000,000*1.02t 15,000,000+500,000*t

  4. Population Growth / Food Production Graph P(t) and F(t) on your calculator. Describe the results. What conclusion can be made about exponential functions and linear functions together? Double the initial amount of food and simulate again. Triple the rate at which food is produced and simulate again. Conclusions?

  5. Building Exponential Functions Given that a generic exponential function is y = a*bx Suppose that the exponential function passes through the two points (0 , 3) and (1 , 6). y = a*bx  3 = a * b0  a = 3 y = 3bx  6 = 3b1  b= 2 y = 3*2x

  6. Building Exponential Functions Build the exponential function which passes through (0 , 7) and (2 , 63)

  7. Building Exponential Functions Build the exponential function which passes through (0 , 7) and (4 , 104)

  8. Building Exponential Functions Build the exponential function which passes through (2 , 7) and (4 , 28)

  9. Building Exponential Functions Build the exponential function which passes through (3 , 1) and (8 , 209)

  10. Matching Graphs to Functions Match each function with a graph above: f(x) = 2*5x g(x) = 9 * 5x h(x) = 2 * 12x j(x) = 2 * (0.5)x

  11. What is Concavity? y = 3 * 4x Find the rate of change from (0 , ) to (1 , ). Find the rate of change from (6 , ) to (7, ). Compare the rates at lower x’s to higher x’s.

  12. What is Concavity? y = 10 * 0.2x Find the rate of change from (0 , ) to (1 , ). Find the rate of change from (10 , ) to (11, ). Compare the rates at lower x’s to higher x’s.

  13. What is Concavity? y = log(x) Find the rate of change from (0.5 , ) to (1 , ). Find the rate of change from (4 , ) to (4.5, ). Compare the rates at lower x’s to higher x’s.

  14. Solving Harder Exponential Equations Solve: 6 * 5x = 73 5x =12.16666 x log 5 = log (12.16666) x = 1.553

  15. Solving Harder Exponential Equations Solve: 8 * 9x = 4 * 20x 1) You can take the log of both sides immediately. ….. Or … 2) You can reduce one of the multipliers before taking logs.

  16. Solving Harder Exponential Equations Solve: 8 * 9x = 4 * 20x log (8 * 9x ) = log (4 * 20x) log 8 + x log 9 = log 4 + x log 20 x(log 9 – log 20) = log 4 – log 8 x = log(4/8) / log (9/20) = 0.868

  17. Solving Harder Exponential Equations Solve: 11 * 6x = 20 * 14x There are no solutions. Why?

  18. Solving Logarithmic Equations Solve: ln (x – 2) + ln (2x – 3) = 2 ln x ln (x-2) (2x - 3) = ln x2 (x – 2) (2x – 3) = x2 2x2 – 7x + 6 = x2 x2 – 7x + 6 = 0 (x – 6) (x – 1) = 0 x = 6 x = 1 only x = 6 is in the domain of the log function.

  19. Solving Logarithmic Equations Solve: log (x) – log (2x – 1) = 0 log (x / 2x – 1) = 0 100 = x / 2x – 1 1 = x / 2x – 1 2x – 1 = x ( cross multiply) x = 1

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