Exponential Growth and Decay_use after 1st day_identify

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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 growth that accelerates over time. It 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 at a rate proportional to its current value, leading to a rapid decrease initially that slows over time. It 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 decay from a function?

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You can identify exponential decay if the base of the exponent is between 0 and 1 (e.g., f(x) = a(0.5)^x).

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What does the base of an exponential function indicate?

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The base of an exponential function indicates the growth or decay factor. If the base is greater than 1, it indicates growth; if the base is between 0 and 1, it indicates decay.

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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, and x is the exponent.

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What is the difference between linear and exponential functions?

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Linear functions have a constant rate of change, represented by a straight line, while exponential functions have a variable rate of change that increases or decreases rapidly, represented by a curve.

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

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The initial value (a) in an exponential function represents the starting amount before any growth or decay occurs.

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How do you determine if a function is linear or exponential from a graph?

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A linear function will show a straight line on a graph, while an exponential function will show a curve that either rises or falls steeply.

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What is a real-world example of exponential growth?

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A real-world example of exponential growth is population growth, where the number of individuals increases rapidly as the population grows.

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What is a real-world example of exponential decay?

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A real-world example of exponential decay is radioactive decay, where the quantity of a radioactive substance decreases over time.