Exponential Growth & Decay

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What is exponential growth?

Back

Exponential growth occurs when the increase of a quantity is proportional to its current value, leading to growth at an increasing rate over time.

Tags

CCSS.HSF-LE.A.1A

Front

What is exponential decay?

Back

Exponential decay is the decrease of a quantity at a rate proportional to its current value, resulting in a rapid decline that slows over time.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula for exponential growth?

Back

The formula is y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula for exponential decay?

Back

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

Tags

CCSS.HSF-IF.C.8B

Front

What does the term 'initial amount' refer to in exponential functions?

Back

The initial amount is the starting value of the quantity before any growth or decay occurs.

Tags

CCSS.HSF.LE.B.5

Front

How do you calculate the growth factor in exponential growth?

Back

The growth factor is calculated as 1 + r, where r is the growth rate expressed as a decimal.

Tags

CCSS.HSF-IF.C.8B

Front

How do you calculate the decay factor in exponential decay?

Back

The decay factor is calculated as 1 - r, where r is the decay rate expressed as a decimal.

Tags

CCSS.HSF-IF.C.8B

Front

If a population grows from 1000 to 2000 in 5 years, what is the approximate annual growth rate?

Back

The approximate annual growth rate can be calculated using the formula r = (final amount/initial amount)^(1/t) - 1, which gives about 14.87%.

Tags

CCSS.HSF.LE.A.4

Front

What is the significance of the base in exponential functions?

Back

The base indicates the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

Tags

CCSS.HSF-IF.C.8B

Front

What is the relationship between exponential functions and logarithms?

Back

Exponential functions can be inverted using logarithms; for example, if y = a^x, then x = log_a(y).

Tags

CCSS.HSF.BF.B.5