Exponential Functions

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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 the base (a positive real number), and 'x' is the exponent. It shows rapid growth or decay.

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What does the base of an exponential function represent?

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The base of an exponential function indicates the growth (if greater than 1) or decay (if between 0 and 1) rate of the function.

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How do you identify exponential growth?

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Exponential growth occurs when the base of the exponential function is greater than 1 (e.g., y = a(1 + r)^x, where r is the growth rate).

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How do you identify exponential decay?

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Exponential decay occurs when the base of the exponential function is between 0 and 1 (e.g., y = a(1 - r)^x, where r is the decay rate).

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What is the formula for calculating the amount after 't' years in exponential growth?

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The 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 the time in years.

Front

What is the formula for calculating the amount after 't' years in exponential decay?

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The 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 the time in years.

Front

If a quantity doubles every year, what is the growth factor?

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The growth factor is 2, meaning the function can be modeled as y = a(2^x).

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What does the initial amount in an exponential function represent?

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The initial amount represents the value of the function when x = 0, indicating the starting point of the growth or decay.

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How can you determine the percentage increase from an exponential function?

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The percentage increase can be found by subtracting 1 from the base of the function and multiplying by 100. For example, if the function is y = a(1 + r)^x, then the percentage increase is r * 100.

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

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The exponent indicates the number of time periods that have passed, affecting the growth or decay of the function.