Exponential growth and decay1

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What is Exponential Growth?

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Exponential Growth occurs when a quantity increases by a constant percentage over a period of time. The general formula is A = P(1 + r)^t, where A is the amount after time t, P 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 occurs when a quantity decreases by a constant percentage over a period of time. The general formula is A = P(1 - r)^t, where A is the amount after time t, P 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?

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

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What does the decay rate of 20% mean in terms of an equation?

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A decay rate of 20% means that 80% of the quantity remains after each time period. The equation would be y = initial amount * (0.8)^x.

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What is the growth rate in the function f(x) = 354(1.15)^x?

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The growth rate is 15%, as indicated by the (1 + r) term in the equation, where r = 0.15.

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What is the initial value in the equation A = 1200(0.85)^6?

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The initial value is 1200, which represents the starting amount before any decay occurs.

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How do you write an exponential decay function for a quantity that decreases by 18%?

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The function would be written as y = initial amount * (0.82)^x, where 0.82 represents the remaining 82% after an 18% decay.

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

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The exponent represents the number of time periods that have passed, affecting how much the quantity has grown or decayed.

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What does it mean if a function is classified as Exponential Decay?

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It means that the quantity decreases over time at a rate proportional to its current value, leading to a rapid decrease initially that slows over time.

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How can you determine the amount remaining after a certain time in an exponential decay model?

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You can determine the remaining amount by substituting the time into the decay equation, A = P(1 - r)^t.