Exponential Growth, Decay, and Compound Interest

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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 the value increasing rapidly over time. It can be modeled by the equation y = a(1 + r)^t, where 'a' 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 y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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What is compound interest?

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Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It can be calculated using the formula A = P(1 + r/n)^(nt), where 'A' is the amount of money accumulated after n years, 'P' is the principal amount, 'r' is the annual interest rate, 'n' is the number of times that interest is compounded per year, and 't' is the number of years.

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How do you calculate the time required for an investment to reach a certain amount with compound interest?

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To calculate the time required, you can rearrange the compound interest formula to solve for 't': t = (log(A/P)) / (n * log(1 + r/n)).

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What is the formula for calculating the future value of an investment with annual compounding?

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The future value (FV) can be calculated using the formula FV = P(1 + r)^t, where 'P' is the principal, 'r' is the annual interest rate, and 't' is the number of years.

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What is the formula for calculating the future value of an investment with quarterly compounding?

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The future value (FV) can be calculated using the formula FV = P(1 + r/n)^(nt), where 'P' is the principal, 'r' is the annual interest rate, 'n' is the number of compounding periods per year, and 't' is the number of years.

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

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When a value decreases at a certain percentage per year, it means that each year the value is reduced by that percentage of its current value. This is modeled by exponential decay.

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How do you determine if a function represents growth or decay?

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A function represents growth if the base of the exponent is greater than 1 (e.g., y = a(1 + r)^t). It represents decay if the base is between 0 and 1 (e.g., y = a(1 - r)^t).

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What is the significance of the initial value in exponential functions?

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The initial value in exponential functions represents the starting amount before any growth or decay occurs. It is the value of 'a' in the equations y = a(1 + r)^t or y = a(1 - r)^t.

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How can you find the value of an asset after a certain number of years if it depreciates at a constant rate?

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You can find the value of an asset after a certain number of years using the formula V = P(1 - r)^t, where 'V' is the future value, 'P' is the initial value, 'r' is the depreciation rate, and 't' is the number of years.