Exponential Functions and Decay

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What is an exponential function?

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An exponential function is a mathematical function of the form y = a(b^x), where 'a' is a constant, 'b' is a positive real number, and 'x' is the exponent. It shows rapid growth or decay.

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What is the difference between exponential growth and exponential decay?

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Exponential growth occurs when a quantity increases at a rate proportional to its current value (b > 1). Exponential decay occurs when a quantity decreases at a rate proportional to its current value (0 < b < 1).

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What is the general form of an exponential decay function?

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The general form of an exponential decay function is y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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How do you identify an exponential growth function from its equation?

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An exponential growth function can be identified by its equation having a base greater than 1, such as y = a(b^x) where b > 1.

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How do you identify an exponential decay function from its equation?

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An exponential decay function can be identified by its equation having a base between 0 and 1, such as y = a(b^x) where 0 < b < 1.

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What does the decay rate represent in an exponential decay function?

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The decay rate represents the percentage decrease of the quantity per time period. For example, a decay rate of 1.5% means the quantity decreases by 1.5% each year.

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What is the formula to calculate the remaining quantity after a certain time in exponential decay?

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The formula is y = a(1 - r)^t, where 'y' is the remaining quantity, 'a' is the initial quantity, 'r' is the decay rate, and 't' is the 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.

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How can you express a percentage decrease as a decimal in an exponential decay function?

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To express a percentage decrease as a decimal, divide the percentage by 100. For example, a 1.5% decrease is expressed as 0.015.

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What is an example of a real-world application of exponential decay?

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An example of exponential decay is radioactive decay, where the quantity of a radioactive substance decreases over time at a rate proportional to its current amount.