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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What does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when a quantity increases by a consistent percentage over equal time intervals, resulting in a rapid increase.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over equal time intervals, resulting in a rapid decrease.

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How can you identify an exponential decay function from its equation?

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An exponential decay function has a base between 0 and 1 (0 < b < 1) in the equation f(x) = a(b^x).

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What is the formula to calculate the remaining amount after exponential decay?

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The remaining amount can be calculated using the formula: Remaining Amount = Initial Amount * (1 - decay rate)^time.

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If a population of 800 ants decays at a rate of 13% per day, what is the function modeling this decay?

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f(x) = 800(0.87^x), where x is the number of days.

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What is the value of f(20) in the function f(x) = 800(0.87^x)?

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f(20) = 800(0.87^20) ≈ 49 ants.

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

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Future Value = Present Value * (1 + growth rate)^number of years.

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If a car's value increases by 10% each year, how do you express this as an exponential function?

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The function can be expressed as f(t) = 29000(1.10^t), where t is the number of years.

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What is the value of a collectible car worth $29,000 after 5 years with a 10% annual increase?

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The car will be worth approximately $46,704.79 after 5 years.