Writing Exponential Functions for Word Problems

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a(b^x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It represents growth or decay processes.

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How do you write an exponential function for a situation where a quantity doubles over time?

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If a quantity doubles every hour, the exponential function can be written as f(t) = a(2^t), where 'a' is the initial amount and 't' is the time in hours.

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What does it mean for a value to depreciate at a certain percentage per year?

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Depreciation at a certain percentage means that the value decreases by that percentage each year. The exponential function for depreciation can be modeled as y = initial_value * (1 - rate)^t.

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How do you calculate the future value of an investment that grows at a constant percentage rate?

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The future value can be calculated using the formula: Future Value = Present Value * (1 + growth_rate)^t, where 't' is the number of time periods.

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What is the formula for calculating population growth?

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The formula for population growth is P(t) = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate, and t is the time in years.

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If a cell phone store sells 417 phones and sales increase by 3.75% per month, how is this modeled?

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The sales can be modeled by the function f(x) = 417(1.0375)^x, where x is the number of months after January.

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What is the difference between growth and decay in exponential functions?

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Growth occurs when the base of the exponential function is greater than 1 (e.g., f(x) = a(b^x) with b > 1), while decay occurs when the base is between 0 and 1 (e.g., f(x) = a(b^x) with 0 < b < 1).

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How do you determine the initial value in an exponential function?

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The initial value is the starting amount before any growth or decay occurs, represented by 'a' in the function f(x) = a(b^x).

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What is the significance of the base in an exponential function?

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The base determines the rate of growth or decay. A base greater than 1 indicates growth, while a base less than 1 indicates decay.

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How can you express a 14% growth rate in an exponential function?

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A 14% growth rate can be expressed as f(x) = initial_value * (1.14)^x, where x is the number of time periods.