Exponential Growth vs Decay

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What is exponential growth?

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Exponential growth occurs when a quantity increases by a consistent percentage over a period of time, resulting in a rapid increase that can be represented by the function f(x) = a(1 + r)^x, where 'a' is the initial amount and 'r' is the growth rate.

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What is exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over a period of time, leading to a rapid decrease that can be represented by the function f(x) = a(1 - r)^x, where 'a' is the initial amount and 'r' is the decay rate.

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How can you identify exponential growth on a graph?

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Exponential growth is identified on a graph as a curve that rises steeply to the right, indicating that the quantity increases rapidly as the independent variable increases.

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How can you identify exponential decay on a graph?

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Exponential decay is identified on a graph as a curve that falls steeply to the right, indicating that the quantity decreases rapidly as the independent variable increases.

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What is the general form of an exponential function?

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The general form of an exponential function is f(x) = a * b^x, where 'a' is a constant, 'b' is the base (b > 0), and 'x' is the exponent.

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What is the difference between linear and exponential functions?

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Linear functions increase or decrease by a constant amount, resulting in a straight line, while exponential functions increase or decrease by a constant percentage, resulting in a curve.

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What does the base 'b' represent in an exponential function?

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In an exponential function f(x) = a * b^x, the base 'b' represents the growth (b > 1) or decay (0 < b < 1) factor of the function.

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What is an example of an exponential growth function?

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An example of an exponential growth function is f(x) = 2 * 3^x, where the quantity triples for each unit increase in 'x'.

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What is an example of an exponential decay function?

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An example of an exponential decay function is f(x) = 5 * (0.5)^x, where the quantity halves for each unit increase in 'x'.

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What is the significance of the initial value 'a' in an exponential function?

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The initial value 'a' in an exponential function represents the starting amount or value of the quantity at x = 0.