Modeling Exponential Functions (Percent Growth Decay)

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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 models growth or decay processes.

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What does it mean for a population to decrease by a certain percentage?

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When a population decreases by a certain percentage, it means that a portion of the population is lost over a specific time period. For example, a 15% decrease means that 15% of the population is removed each time period.

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How do you model exponential decay?

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Exponential decay can be modeled using the function f(x) = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate (as a decimal), and 'x' is the time period.

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What is the formula for exponential growth?

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The formula for exponential growth is f(x) = a(1 + r)^x, where 'a' is the initial amount, 'r' is the growth rate (as a decimal), and 'x' is the time period.

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How do you convert a percentage to a decimal for calculations?

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To convert a percentage to a decimal, divide the percentage by 100. For example, 15% becomes 0.15.

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

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The base in an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

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If a population of 10,000 decreases by 20% each year, what is the function that models this?

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f(x) = 10000(0.80)^x, where 0.80 represents the remaining 80% after a 20% decrease.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over time, while exponential growth increases by a constant percentage, leading to faster growth as time progresses.

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How do you find the remaining population after a certain number of years with exponential decay?

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Use the formula f(t) = a(1 - r)^t, where 'a' is the initial population, 'r' is the decay rate, and 't' is the number of years.

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What does 'compounded annually' mean in terms of interest?

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Compounded annually means that the interest earned is added to the principal once a year, and the next year's interest is calculated on the new total.