03-02 Exponential Models

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What is an exponential model?

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An exponential model is a mathematical representation of a situation where a quantity grows or decays at a rate proportional to its current value, typically expressed in the form y = ab^x, where a is the initial amount, b is the growth (b > 1) or decay (0 < b < 1) factor, and x is the time.

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What is the half-life of a substance?

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The half-life of a substance is the time required for half of the initial amount of the substance to decay or be transformed into another substance.

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How do you calculate the remaining amount of a substance after a certain time using the half-life?

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To calculate the remaining amount of a substance after a certain time, use the formula: Remaining Amount = Initial Amount * (1/2)^(Time / Half-life).

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What is the formula for exponential decay?

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The formula for exponential decay is y = a * e^(-kt), where y is the remaining quantity, a is the initial quantity, k is the decay constant, and t is time.

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How do you find the decay constant from a percentage decay rate?

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To find the decay constant (k) from a percentage decay rate (r), use the formula: k = -ln(1 - r).

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What is the formula for exponential growth?

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The formula for exponential growth is y = a * e^(kt), where y is the amount after time t, a is the initial amount, k is the growth constant, and t is time.

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How do you convert a percentage growth rate to a growth constant?

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To convert a percentage growth rate (r) to a growth constant (k), use the formula: k = ln(1 + r).

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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 the equation of an exponential function given two points?

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To determine the equation of an exponential function given two points (x1, y1) and (x2, y2), use the form y = ab^x. Solve for a and b using the coordinates of the points.

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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.