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Objectives: 1. Understand the exponential growth/decay function family. 2. Graph exponential growth/decay function using y = ab x – h + k 3. Use exponential function to models in real life. Vocabulary: Euler’s number, natural base 5.3 Exponential Growth/Decay

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Question 1: Interest of a bank Account You deposit P dollars in the bank and receive an interest rate of r compounded annually for t year. Initial AssetA 0 = P The end of 1 st year: A 1 = P + P·r = P(1 + r) The end of 2 nd year: A 2 = A 1 + A 1 ·r = A 1 (1 + r) = P(1 + r)(1 + r) = P(1 + r) 2 The end of 3 rd year: A 3 = A 2 + A 2 ·r = A 2 (1 + r) = P(1 + r) 2 (1 + r) = P(1 + r) 3

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: t-th year: A t = P(1 + r) t Since r > 0, then 1 + r > 1. Denote b = 1 + r, then b > 1. So the above model can be written as: A t = P ·b t (b > 1)

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Question 2: Cell split You have C cells and each cell will split to 2 new cells. After n splits how many cells you will have? Initial Cell:C 0 = C = C ·2 0 1 st split: C 1 = C ·2 = C ·2 1 2 nd split: C 2 = C 1 ·2 = C ·2 2 3 rd split:C 3 = C 2 ·2 = C ·2 3 : n-th split: C n = C ·2 n C n = C ·b n (b = 2 > 1)

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The observation we had from the above 2 Questions attracts us to have a deeper study to some function like: f (x) = a b x where a ≠ 0, b > 0 and b ≠ 1 Definition Exponential Function The function is of the form: f (x) = a b x, where a ≠ 0, b > 0 and b ≠ 1, x R.

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Definition Exponential Growth Function The function is of the form: f (x) = a b x, where a > 0, b > 1, x R. Definition Exponential Decay Function The function is of the form: f (x) = a b x, where a > 0, 0 < b < 1, x R.

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Summary f (x) = a b x a ≠ 0, b > 0, b ≠ 1 x R Exponential Function a > 0, b > 0, b ≠ 1 Exponential Growth ( b > 1) Exponential Decay ( 0 < b < 1) f (x) = a b x a > 0, b > 0, b ≠ 1 x R Domain x R Range y > 0 a a

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Challenge Question 1: In the definition, why b > 0 and b ≠ 1? Function Family -- functions whose graphs are identical if the graphs are shifted some units horizontally and/or vertically. Activity Using the graphing calculator to graph: a) y = 2 · 3 x b) y = 2 · 3 x+1 - 5

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Graph b) is -- shifted 1 unit to the left of a) (horizontally) -- shifted 5 unit down of a) (vertically) -- “new” y-intercept is at (-1, -3) -- “new” asymptotes are line y = - 5 and line x = -1. New center is at (-1, - 5) Procedures 1) Find the new center (h, k) 2) Draw 2 perpendicular dash lines to represent new x-axis and new y-axis 3) Label a + k as “new” y-intercept 4) Treat horizontal and vertical dash line as “new” x-axis and “new” y-axis.

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c)Example 1 Graph y = 2 x y = (1/2) x y = -3 · 2 x d)y = -2 (1/3) x Conclusions: 1) Graph a) is an exponential growth function graph and y – intercept is 1. 2) Graph b) is an exponential decay function graph and y – intercept is 1. 3) Graphs c) and d) are not an exponential growth or decay function.

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Challenge Question 2: The product of two exponential growth(or decay) functions is still an exponential growth(or decay) function? Justify your answer. Suppose that y 1 = a 1 b 1 x and y 2 = a 2 b 2 x are exponential growth function, where a 1, a 2 R +, b 1, b 2 R +, b 1 > 1, b 2 > 1 then since b 1 b 2 > 1 y 1 y 2 = a 1 a 2 b 1 x b 2 x = a 1 a 2 (b 1 b 2 ) x is also an exponential growth function.

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Challenge Question 3: Given an exponential growth(or decay) function y = a b x, can you construct an exponential decay(or growth) function? Given that y 1 = a 1 b 1 x is an exponential growth function, take a 2 = a 1 R +, b 2 = 1/ b 1 R +, and then 0 < b 2 < 1 Therefore, y 2 = a 2 b 2 x is an exponential decay function.

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Example 2 Graph y = -3 · 2 x Example 3 Graph y = 3 · 4 x-1 Example 4 Graph y = 4 · 3 x-2 + 1

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Example 5 Graph y = 3 · (1/2) x Example 6 Graph y = -3 · (1/4) x-1 Example 7 Graph y = 4 · (1/2) x-3 + 2

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The variations of exponential growth/decay function model is very practical in a wide variety of application problems. Growth:y = a (1 + r ) t Decay:y = a (1 – r ) t Note: the rate r is measured the same time period as t, where a stands for the initial amount.

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Example 8 You purchase a baseball card for $54. If it increases each year by 5%, write an exponential growth model.

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Example 9 In 1980 wind turbines in Europe generated about 5 gigawatt-hours of energy. Over the next 15 years, the amount of energy increased by about 5.9% per year. a) Write a model giving the amount E (gigawatt-hours) of energy and t years after 1980. About how much wind energy was generated in 1984? b) Graph the model. c) Estimate the year when 8.0 gigawatt-hours of energy were generated?

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Example 10 You drink a beverage with 120 milligrams of caffeine. Each hour, the amount c of caffeine in your system decreases by about 12%. a) Write an exponential decay model. b) How much caffeine remains in our system after 3 hours? c) After how many hours, the amount caffeine in your system is 50 milligrams of caffeine?

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Example 11 You have a new computer for $2100. The value of the computer decreases by about 30% annually. a) Write an exponential decay model for the value of the computer. Use the model to estimate the value after 2 years. b) Graph the model. c) Estimate when the computer will be worth $500?

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Assignment: 5.1 P469 #14-40 even 5.1 P469 #43-48, 56-66 5.2 P 477 #11-52 5.1/5.2 Exponential Growth/Decay

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