# 7.7 EXPONENTIAL GROWTH AND DECAY:

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2.5 Linear Equations and Formulas:

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.GOAL:

Definition:An EXPONENTIAL FUNCTION is a function of the form:BaseExponentWhere a 0, b > o, b 1, and x is a real number. ConstantGRAPHING: To provide the graph of the equation we can go back to basics and create a table.Ex: What is the graph of y = 32x?4GRAPHING:Xy = 32xy-232(-2) 32(-1) -1032(0) 3 = 31 132(1) 6 = 32 232(2) 12 = 34 5GRAPHING:Xy-2-103 16 212

This graph grows fast = Exponential Growth6YOU TRY IT:7GRAPHING:-2126-103 = 31 1 2=3(2)2 =3(2)1 8GRAPHING:Xy-2-103 16 212

This graph goes down = Exponential Decay9YOU TRY IT:10y = 32x Base = 2 Exponential growth y- intercept (x=0) = 3

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12MODELING: 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?13EVALUATING: To provide the solution we must know the following formula:y = abxy = totala = initial amount b = growth factor (1 + rate)x = time in years. 14SOLUTION: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 = abxy = 36(1.07)10y = 36(1.967)y = 70.8 b.15BANKING: We also use the concept of exponential growth in banking:A = total balanceP = Principal (initial) amount n = # of times compound interestt = time in years. r = interest rate in decimal form 16MODELING 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 annuallyb) Compounded quarterlyc) Compounded monthly17COMPOUNDED ANNUALLY:A = ?P = $6000 n = 1t = 4 yrs r = 0.07 A = 6000(1.07)4A = 6000(1.3107)A = $7864.7718COMPOUNDED QUARTERLY:A = ?P = $6000 n = 4 timest = 4 yrs r = 0.07 A = 6000(1.0175)16A = 6000(1.3199)A = $7919.5819COMPOUNDED MONTHLY:A = ?P = $6000 n = 12 timest = 4 yrs r = 0.07 A = 6000(1.0058)48A = 6000(1.3221)A = $7932.3220MODELING 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?:

21DECAY: To provide the solution we g back to the following formula:y = abxy = totala = initial amount b = decay factor (1 - rate)x = time in years. 22SOLUTION: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?:12Initial: Growth: 1- 1/2 Time (x): 16/8 = 2 unknown y = abxy = 12(1/2)2y = 12(.25)y = 3 23VIDEOS:ExponentialFunctionsGrowthhttps://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/exponential-growth-functions

Graphinghttps://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphing-exponential-functions

VIDEOS:ExponentialFunctionsDecayhttps://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/word-problem-solving--exponential-growth-and-decay

CLASSWORK:

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Problems: As many as needed to master the concept.