Exponential Growth and Decay

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What is exponential growth?

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Exponential growth occurs when a quantity increases by a consistent percentage over a period of time, resulting in a rapid increase that can be modeled by the function f(x) = a(1 + r)^x, where 'a' is the initial amount and 'r' is the growth rate.

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What is exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over time, resulting in a rapid decrease that can be modeled by the function f(x) = a(1 - r)^x, where 'a' is the initial amount and 'r' is the decay rate.

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How can you identify exponential decay in a function?

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You can identify exponential decay in a function if the base of the exponent is between 0 and 1 (e.g., f(x) = a(0.20)^x).

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What does the initial value represent in an exponential function?

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The initial value in an exponential function represents the starting amount before any growth or decay occurs, typically denoted as 'a' in the function f(x) = a(1 + r)^x or f(x) = a(1 - r)^x.

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What is the difference between linear and exponential functions?

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Linear functions have a constant rate of change and produce straight-line graphs, while exponential functions have a variable rate of change that increases or decreases rapidly, producing curved graphs.

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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 and 'r' is the growth rate.

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What is the formula for exponential decay?

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The formula for exponential decay is f(x) = a(1 - r)^x, where 'a' is the initial amount and 'r' is the decay rate.

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How do you determine the growth or decay rate from a function?

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The growth or decay rate can be determined from the function by examining the base of the exponent: if the base is greater than 1, it indicates growth; if the base is between 0 and 1, it indicates decay.

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What does a graph of an exponential growth function look like?

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A graph of an exponential growth function starts at the initial value and rises steeply as x increases, showing a rapid increase.

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What does a graph of an exponential decay function look like?

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A graph of an exponential decay function starts at the initial value and decreases rapidly, approaching zero but never actually reaching it.