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7.7 EXPONENTIAL GROWTH AND DECAY:

Exponential Growth: An equation that increases. 7.7 EXPONENTIAL GROWTH AND DECAY:. Exponential Decay: An equation that decreases. Growth Factor: 1 plus t he percent rate of change which is expressed as a decimal. Decay Factor: 1 minus t he percent rate of change expressed as a decimal.

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7.7 EXPONENTIAL GROWTH AND DECAY:

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  1. Exponential Growth: An equation that increases. 7.7 EXPONENTIAL GROWTH AND DECAY: Exponential Decay: An equation that decreases. Growth Factor: 1 plusthe percent rate of change which is expressed as a decimal. Decay Factor: 1minusthe percent rate of change expressed as a decimal.

  2. GOAL:

  3. Definition: An EXPONENTIAL FUNCTION is a function of the form: Constant Base Exponent Where a ≠ 0, b > o, b ≠ 1, and x is a real number.

  4. GRAPHING: To provide the graph of the equation we can go back to basics and create a table. Ex: What is the graph of y = 3∙2x?

  5. GRAPHING: = -2 3∙2(-2) = 3∙2(-1) -1 = 3∙1 3 3∙2(0) 0 = 3∙2 6 3∙2(1) 1 = 3∙4 12 3∙2(2) 2

  6. GRAPHING: -2 -1 3 0 6 1 12 2 This graph grows fast = Exponential Growth

  7. YOU TRY IT: Ex: What is the graph of y = 3∙x?

  8. GRAPHING: 3∙(-2) 12 -2 =3∙(2)2 3∙(-1) -1 =3∙(2)1 6 3∙(0) 0 3 = 3∙1 3∙(1) =3∙ 1 3∙(2) 2 =3∙

  9. GRAPHING: -2 12 -1 6 3 0 1 2 This graph goes down = Exponential Decay

  10. YOU TRY IT: Ex: What are the differences and similarities between: y = 3∙2x and y = 3∙x?

  11. y = 3∙2x Base = 2  Exponential growth  y- intercept (x=0) = 3

  12. y = 3∙x  Base =  Exponential Decay  y- intercept (x=0) = 3

  13. We use the concept of exponential growth in the real world: MODELING: Ex:Since 2005, the amount of money spent at restaurants in the U.S. has increased 7% each year. In 2005, about 36 billion was spend at restaurants. If the trend continues, about how much will be spent in 2015?

  14. To provide the solution we must know the following formula: EVALUATING: y = a∙bx y = total a = initial amount b = growth factor (1 + rate) x = time in years.

  15. SOLUTION: Since 2005, … has increased 7% each year. In 2005, about 36 billion was spend at restaurants…. about how much will be spent in 2015? Y= total: unknown y = a∙bx $36 billion Initial: y = 36∙(1.07)10 1 + 0.07 Growth: y = 36∙(1.967) 10 years(2005-2015) Time (x): y= 70.8 b.

  16. We also use the concept of exponential growth in banking: BANKING: A = P(1+)nt A = total balance P = Principal (initial) amount r = interest rate in decimal form n = # of times compound interest t = time in years.

  17. MODELING GROWTH: Ex:You are given $6,000 at the beginning of your freshman year. You go to a bank and they offer you 7% interest. How much money will you have after graduation if the money is:a) Compounded annually b) Compounded quarterly c) Compounded monthly

  18. COMPOUNDED ANNUALLY: A = P(1+)nt A = ? A = 6000(1+)1(4) P = $6000 A = 6000(1.07)4 r = 0.07 n = 1 A = 6000(1.3107) t = 4 yrs A = $7864.77

  19. COMPOUNDED QUARTERLY: A = P(1+)nt A = ? A = 6000(1+)4(4) P = $6000 A = 6000(1.0175)16 r = 0.07 n = 4 times A = 6000(1.3199) t = 4 yrs A = $7919.58

  20. COMPOUNDED MONTHLY: A = P(1+)nt A = ? A = 6000(1+)12(4) P = $6000 A = 6000(1.0058)48 r = 0.07 n = 12 times A = 6000(1.3221) t = 4 yrs A = $7932.32

  21. MODELING DECAY: Ex:Doctors can use radioactive iodine to treat some forms of cancer. The half-life of iodine-131 is 8 days. A patient receives a treatment of 12 millicuries (a unit of radioactivity) of iodine-131. How much iodine-131 remains in the patient after 16 days?:

  22. To provide the solution we g back tothe following formula: DECAY: y = a∙bx y = total a = initial amount b = decay factor (1 - rate) x = time in years.

  23. SOLUTION: The half-life of iodine-131 is 8 days. A patient receives a treatment of 12 millicuries(a unit of radioactivity) of iodine-131. How much iodine-131 remains in the patient 16 days later?: Y= total: unknown y = a∙bx 12 Initial: y = 12∙(1/2)2 1- 1/2 Growth: y = 12∙(.25) 16/8 = 2 Time (x): y = 3

  24. VIDEOS: Exponential Functions Growth https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/exponential-growth-functions Graphing https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphing-exponential-functions

  25. VIDEOS: Exponential Functions Decay https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/word-problem-solving--exponential-growth-and-decay

  26. CLASSWORK:Page 450-452: Problems: As many as needed to master the concept.

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