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

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GOAL:

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

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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 = 32 x ?

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GRAPHING: Xy = 32 x y -232 (-2) 32 (-1) 0 32 (0) 3 = 31 1 32 (1) 6 = 32 2 32 (2) 12 = 34

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

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YOU TRY IT:

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GRAPHING: Xy -212 6 0 3 = 31 1 2 =3(2) 2 =3(2) 1

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

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YOU TRY IT:

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y = 32 x Base = 2 Exponential growth y- intercept (x=0) = 3

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MODELING: We use the concept of exponential growth in the real world: 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?

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

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

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

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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

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

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

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

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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?:

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

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SOLUTION: Y= total: 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?: 12 Initial: Growth: 1- 1/2 Time (x): 16/8 = 2 unknown y = ab x y = 12(1/2) 2 y = 12(.25) y = 3

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VIDEOS: Exponential Functions Growth https://www.khanacademy.org/math/trigonometry/expon ential_and_logarithmic_func/exp_growth_decay/v/expone ntial-growth-functions Graphing https://www.khanacademy.org/math/trigonometry/expon ential_and_logarithmic_func/exp_growth_decay/v/graphi ng-exponential-functions

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

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