Exponential Growth and Decay Functions

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What is an exponential growth function?

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An exponential growth function is a mathematical expression of the form y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time. It describes a situation where the quantity increases at a rate proportional to its current value.

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What is an exponential decay function?

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An exponential decay function is a mathematical expression of the form y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time. It describes a situation where the quantity decreases at a rate proportional to its current value.

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What does the horizontal asymptote of an exponential function represent?

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The horizontal asymptote of an exponential function represents the value that the function approaches as x approaches infinity. For exponential growth, this is typically the x-axis (y=0).

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How do you calculate compound interest?

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Compound interest can be calculated using the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, 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 formula for continuous growth?

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The formula for continuous growth is A = Pe^(rt), where A is the amount after time t, P is the initial amount, r is the growth rate, and e is the base of the natural logarithm.

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What is the significance of the initial amount in exponential functions?

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The initial amount in exponential functions represents the starting value before any growth or decay occurs. It is the value of the function when time (t) is zero.

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How do you determine the growth rate from an exponential function?

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The growth rate can be determined from an exponential function of the form y = a(1 + r)^t by rearranging the equation to solve for r, where r = (y/a)^(1/t) - 1.

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What is the effect of a higher growth rate on an exponential function?

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A higher growth rate in an exponential function results in a steeper curve, meaning the quantity increases more rapidly over time.

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How do you find the value of an asset after a certain period of depreciation?

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To find the value of an asset after depreciation, use the formula V = P(1 - r)^t, where V is the value after time t, P is the initial value, r is the depreciation rate, and t is the time in years.

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What is the relationship between exponential functions and logarithms?

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Exponential functions and logarithms are inverse functions. The logarithm of a number is the exponent to which the base must be raised to produce that number.