Exponential growth and decay1

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What is Exponential Growth?

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Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to a rapid increase over time. It can be modeled by the equation: A = A₀(1 + r)^t, where A₀ is the initial amount, r is the growth rate, and t is time.

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What is Exponential Decay?

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Exponential decay is the process of reducing an amount by a consistent percentage rate over a period of time. It can be modeled by the equation: A = A₀(1 - r)^t, where A₀ is the initial amount, r is the decay rate, and t is time.

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How do you identify Exponential Growth vs. Decay in an equation?

Back

In an exponential growth equation, the base of the exponent is greater than 1 (e.g., A = A₀(1 + r)^t). In decay, the base is between 0 and 1 (e.g., A = A₀(1 - r)^t).

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What does the equation A = 1200(0.85)^6 represent?

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This equation represents exponential decay, as the base (0.85) is less than 1, indicating a decrease in value over time.

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What is the decay rate in the equation A = 60(0.8)^x?

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The decay rate is 20% per hour, as the base (0.8) indicates that 80% of the original amount remains after each hour.

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What does the function f(x) = 354(1.15)^x represent?

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This function represents exponential growth, with an initial value of 354 and a growth rate of 15% per day.

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How do you write an exponential decay function for an initial value of 1450 and a decay of 18%?

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The function is written as y = 1450(0.82)^x, where 0.82 represents the remaining amount after an 18% decay.

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What is the formula for exponential growth?

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The formula for exponential growth is A = A₀(1 + r)^t, where A₀ is the initial amount, r is the growth rate, and t is time.

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What is the formula for exponential decay?

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The formula for exponential decay is A = A₀(1 - r)^t, where A₀ is the initial amount, r is the decay rate, and t is time.

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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 between 0 and 1 indicates decay.