Writing 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 f(x) = a(b)^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

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How do you model depreciation using an exponential function?

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To model depreciation, use the formula V(t) = V_0(1 - r)^t, where V_0 is the initial value, r is the depreciation rate, and t is the time in years.

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What does the base of an exponential function represent in a growth scenario?

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In a growth scenario, the base represents the growth factor. For example, a base of 1.06 indicates a 6% growth.

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What does the base of an exponential function represent in a decay scenario?

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In a decay scenario, the base represents the decay factor. For example, a base of 0.85 indicates a 15% decay.

Front

Write the exponential function for a car worth $20,000 depreciating at 10% per year.

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V(t) = 20000(0.90)^t.

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What is the formula for exponential growth?

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The formula for exponential growth is f(t) = 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 the formula for exponential decay?

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The formula for exponential decay is f(t) = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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If a population of bacteria triples every hour, what is the exponential function that models this growth?

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P(t) = P_0(3)^t, where P_0 is the initial population.

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What does the term 'initial value' mean in the context of exponential functions?

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The initial value is the starting amount before any growth or decay occurs, represented by 'a' in the function f(x) = a(b)^x.

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

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The decay rate can be found by subtracting the base from 1. For example, in f(x) = 800(0.85)^x, the decay rate is 1 - 0.85 = 0.15 or 15%.