03-02 Exponential Models

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What is an exponential model?

Back

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.

Tags

CCSS.HSF-LE.A.1A

Front

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?

Back

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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CCSS.HSF.LE.A.4

Front

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.

Tags

CCSS.HSF-LE.A.1A

Front

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.

Tags

CCSS.HSF-LE.A.1A

Front

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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CCSS.HSF-IF.C.8B

Front

How do you determine the equation of an exponential function given two points?

Back

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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CCSS.HSF.LE.A.2

Front

What is the relationship between exponential functions and logarithms?

Back

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.

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CCSS.HSF.BF.B.5