Exponential Growth and Decay

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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 y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is exponential decay?

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Exponential decay is the decrease in a quantity at a rate proportional to its current value, resulting in a rapid decline that slows over time. It can be represented by the function 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 linear growth?

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Linear growth occurs when a quantity increases by a constant amount over equal intervals of time. It can be represented by the function y = mx + b, where 'm' is the slope (rate of change) and 'b' is the y-intercept.

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What is the formula for calculating compound interest?

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The formula for calculating compound interest is A = P(1 + r/n)^(nt), where 'A' is the amount of money accumulated after n years, including interest, 'P' is the principal amount (initial investment), 'r' is the annual interest rate (decimal), 'n' is the number of times that interest is compounded per year, and 't' is the number of years.

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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 and produce a straight line when graphed, while exponential functions have a variable rate of change that increases or decreases rapidly, producing a curve.

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How do you calculate the future value of an investment with exponential growth?

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To calculate the future value of an investment with exponential growth, use the formula A = P(1 + r)^t, where 'A' is the future value, 'P' is the principal amount, 'r' is the growth rate, and 't' is the time in years.

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

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The base in an exponential function 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 do you determine if a function is exponential?

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A function is exponential if it can be expressed in the form y = ab^x, where 'a' is a non-zero constant, 'b' is a positive constant, and 'x' is the exponent. The variable is in the exponent.

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What is the formula for exponential decay?

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The formula for exponential decay 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 time.

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How do you calculate the value of an asset after a certain number of years with a constant percentage increase?

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To calculate the value of an asset after a certain number of years with a constant percentage increase, use the formula V = P(1 + r)^t, where 'V' is the future value, 'P' is the present value, 'r' is the rate of increase, and 't' is the number of years.