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 it mean for a function to exhibit exponential growth?

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Exponential growth occurs when a quantity increases by a consistent percentage over equal time intervals, resulting in a rapid increase. The function has a base greater than 1.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over equal time intervals, resulting in a rapid decrease. The function has a base between 0 and 1.

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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 in the form 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 in the form y = a(b^x) where '0 < b < 1.

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

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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.

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

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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.

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If a population of bacteria doubles every hour, what type of function does this represent?

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This represents an exponential growth function.

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What is the initial value in the function y = 15(1.35)^x?

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The initial value is 15, which represents the starting amount before any growth occurs.

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In the equation A = 1200(0.85)^6, what does the base 0.85 represent?

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The base 0.85 represents a decay factor, indicating that the quantity decreases by 15% each time period.