Exponential Functions and Growth

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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. It describes growth or decay processes.

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Define exponential growth.

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Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to a rapid increase over time. It can be modeled by the function f(t) = a * e^(kt), where 'k' is a positive constant.

Tags

CCSS.HSF-LE.A.1A

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What is the formula for calculating future value in exponential growth?

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The formula is: Future Value = Present Value * (1 + r)^t, where 'r' is the growth rate and 't' is the time period.

Tags

CCSS.HSF-IF.C.8B

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What is exponential decay?

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Exponential decay is the process where a quantity decreases at a rate proportional to its current value, often modeled by the function f(t) = a * e^(-kt), where 'k' is a positive constant.

Tags

CCSS.HSF-LE.A.1A

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How do you calculate the remaining amount in exponential decay?

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The remaining amount can be calculated using the formula: Remaining Amount = Initial Amount * (1 - r)^t, where 'r' is the decay rate and 't' is the time period.

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

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The base 'b' in an exponential function determines the rate of growth or decay. If b > 1, the function represents growth; if 0 < b < 1, it represents decay.

Tags

CCSS.HSF-IF.C.8B

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Evaluate the function f(x) = 3(2)^x when x = 3.

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f(3) = 3(2)^3 = 3 * 8 = 24.

Tags

CCSS.HSF-IF.C.8B

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If a population of 500 grows at a rate of 10% per year, what will be the population after 5 years?

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Population = 500 * (1 + 0.10)^5 = 500 * 1.61051 ≈ 805.

Tags

CCSS.HSF-LE.A.1C

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What does it mean if a function is described as exponential growth?

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It means that the function increases rapidly as the input increases, typically doubling at regular intervals.

Tags

CCSS.HSF-IF.C.8B

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What is the half-life in the context of exponential decay?

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Half-life is the time required for a quantity to reduce to half its initial value, commonly used in radioactive decay.