Rate of growth or decay (rate of change) Modeling with exponential functions (PART 2) The future value function For exponential functions 𝑦=𝑎∙ 𝑏 𝑥 the value of b represents growth or decay factor. 𝑏=1+𝑟, 𝑟= 𝑦 2 − 𝑦 1 𝑦 1 You can model exponential growth or decay with Future Value Function: Initial amount Number of time periods 𝐹 𝑡 =𝑃∙ (1+𝑟) 𝑡 Amount after t periods Rate of growth or decay (rate of change)

Classify each function as linear, quadratic or exponential. Explain why.

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U.S. Population (in millions) We do The following table contains U.S. population data for the two most recent census years, 2000 and 2010. If the U.S. population growths exponentially, Census Year U.S. Population (in millions) 𝟐𝟎𝟎𝟎 𝟐𝟖𝟏.𝟒 𝟐𝟎𝟏𝟎 𝟑𝟎𝟖.𝟕 Write the exponential function 𝐸, to estimate the population 𝑡 years since the year 2000. What will be the U.S. population in 2020? How long will it take to the country to reach the billion?

We do Census Year U.S. Population (in millions) 𝟏𝟗𝟎𝟎 𝟕𝟔.𝟐 𝟏𝟗𝟐𝟎 𝟏𝟎𝟔.𝟎 𝟏𝟗𝟒𝟎 𝟏𝟑𝟐.𝟐 𝟏𝟗𝟔𝟎 𝟏𝟕𝟗.𝟑 𝟏𝟗𝟖𝟎 𝟐𝟐𝟔.𝟓 𝟐𝟎𝟎𝟎 𝟐𝟖𝟏.𝟒 The following table contains additional U.S. census population data. Find the growth factor for each 20-year period and record it in the table below. What do you observe about these growth factors? Census Year U.S. Population (in millions) Growth Factor (2𝟎-year period) 𝟏𝟗𝟎𝟎 𝟕𝟔.𝟐 𝟏𝟗𝟐𝟎 𝟏𝟎𝟔.𝟎 𝟏𝟗𝟒𝟎 𝟏𝟑𝟐.𝟐 𝟏𝟗𝟔𝟎 𝟏𝟕𝟗.𝟑 𝟏𝟗𝟖𝟎 𝟐𝟐𝟔.𝟓 𝟐𝟎𝟎𝟎 𝟐𝟖𝟏.𝟒 Find an average 20-year growth factor for the population data in the table. What does that number represent? Use the average growth factor to find an exponential function 𝑔 that can model this data.

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