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Exponential Growth Function

Exponential Growth Function. Exponential Growth. A function of the form y = a • b x is an exponential function. y – int or initial growth. Rate of growth. Lets look at an example of an exponential function:. Make a table of values for the exponential functions y = 2 x

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Exponential Growth Function

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  1. Exponential Growth Function Exponential Growth

  2. A function of the form y = a• b x is an exponential function. y – int or initial growth Rate of growth Lets look at an example of an exponential function: Make a table of values for the exponential functions y = 2x Use x-values of -2, -1, 0, 1, 2, and 3. 1 1 y = 2 –2 = = To get y values substitute the given numbers for x. 4 2 2 x -2 -1 0 1 2 3 y = 2x ¼ ½ 1 2 4 8

  3. Graph the exponential function y = 2x Then evaluate when x = 1.5 7 5 3 1 (3,8) (2,4) (1,2) (-1, ½ ) (0,1) (-2,1/4) -2 -1 1 2 3 4 We can use the table from the last example x -2 -1 0 1 2 3 y = 2x ¼ ½ 1 2 4 8

  4. Exponential Growth A quantity is growing exponentially if it increases by the same percent r in each unit of time t. We model this by the equation: Growth factor C = Initial amount before growth y = C(1 + r) t y = C(1 + r) t time r = growth rate (1 + r) = growth factor Initial amount before growth Growth rate t = time

  5. Example 1: Writing an exponential growth model A newly hatched channel catfish typically weighs about 0.06 grams. During the first six weeks of life, its weight increases by about 10% each day. Write a model for the weight of the catfish during the first six weeks. Use y to represent the catfish weight and t to be the number of days. Solution: The initial weight of the catfish is (C) is 0.06. The growth rate ris 10% or 0.10. Write exponential growth model y = C( 1 + r) t = 0.06(1 + 0.10) t Substitute 0.06 for C and 0.10 for r. = 0.06(1.1) t Add

  6. You try! Write an exponential growth model for the following. A TV station’s local news program has 50,000 viewers. The manager of the station hopes to increase the number of viewers by 2% per month. Write an exponential growth model to represent the number of viewer v in t months.

  7. Example 2: You deposit $500 in an account that pays 8% interest compounded yearly. What will the account balance be after 6 years? C What’s the initial amount? $500 What’s the growth rate? 8% or 0.08 How much time? 6 years r t y = 500 (1 + 0.08)6 = 500 (1.08)6  793 The balance after 6 years will be about $793.

  8. Example 3: An initial population of 20 mice triples each year for 5 years. What is the mice population after 5 years? Solution: We know the population triples every year, this tells us the factor by which the population is growing not the percent of change. This would be therefore be a growth factor(not growth rate) which is 3 Initial population is 20, and the time is 5 years. y = C( 1 + r) t Write the exponential growth model = 20(3)5 Substitute 20 for C, 3 for ( 1 + r), and 5 for t = 4860 There will be 48060 mice after 5 years.

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