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Have you ever wondered how quickly the money in your bank account will grow? For example, how much money will you have 10 years from now if you put it into a saving account?

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In this lesson you will learn how to create and graph exponential relationships by using a table of values

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Let’s Review Example: You start with 25 dots, and the number of dots increase by 40% in every step. 40% growth

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Let’s Review 40% growth y = a(1+r) x y = 25(1+.4) x y = 25(1.4) x

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Let’s Review Exponential growth Exponential decay

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A Common Mistake If the values are growing, the growth factor is greater than 1 y = 25(1.4) x If it is decaying, the decay factor is less than 1 but more than 0 y = 25(.4) x

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Core Lesson We will investigate the following: Use the following table of values that show your bank account and number of years to create and graph a function relating the time and money in your account. time (years) money ($) 0100 1103 2106.09 3109.27 4112.55 5115.92

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Core Lesson m = a(1+r) t m = (100)*(1+r) t m = (100)*1.03 t Time (years) Money ($) 0100 1103 2106.09 3109.27 4112.55 Each term is 3% larger than the previous term, so r=.03

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Core Lesson time (years) money ($) time (years) money ($) 0100 1103 2106.09 3109.27 4112.55 5115.92 m = (100)*1.03 t 0 20 40 60 80 100 120 0246

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In this lesson you have learned how to create and graph exponential relationships by using a table of values

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Guided Practice We will investigate the following: The following data set shows the amount of caffeine in a person’s bloodstream after a cup of coffee; create and graph the function that describes time and caffeine levels time (hours) caffeine (mg) 035 130.1 225.89 322.26 419.14 516.46

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Guided Practice c = a(1+r) t c = (35)*(1+r) t c = (35)*0.86 t time (hours) caffeine (mg) 035 130.1 225.89 322.26 419.14 516.46 Each term is 14% smaller than the previous term, so r=-.14

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Guided Practice time (hours) caffeine (mg) time (hours)caffeine (mg) 035 130.1 225.89 322.26 419.14 516.46 c = (35)*0.86 t 0 5 10 15 20 25 30 35 40 4.59.5

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Extension Activities 1. Make your own exponential function, create a table of values, and verify that the function you created can be made from the values. 2. Use a computer and explore the “singularity” concept. Investigate how exponential growth is related to “singularity”. 3. Explore “half-life”. How is modeling half- life similar and different to the work you have done with exponential functions?

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Quick Quiz 1. The population of your town increases by 1.4% each year. If the town starts with 65,000 people, create and graph the function that describes time and population size. 2. The number of wolves in the Western US was in serious jeopardy for a while. For some time, the population was decreased by 32% each year. If the original population was 5 million, create and graph the function describing time and wolf population.

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Exponential Functions An exponential function is a function of the form the real constant a is called the base, and the independent variable x may assume.

Exponential Functions An exponential function is a function of the form the real constant a is called the base, and the independent variable x may assume.

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