Exponential Growth and Decay

Front

What is exponential growth?

Back

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to growth that accelerates over time. It can be modeled by the equation A = A0 * e^(rt), where A0 is the initial amount, r is the growth rate, and t is time.

Tags

CCSS.HSF-LE.A.1A

Front

What is exponential decay?

Back

Exponential decay is the process by which a quantity decreases at a rate proportional to its current value, resulting in a rapid decrease that slows over time. It can be modeled by the equation A = A0 * e^(-rt), where A0 is the initial amount, r is the decay rate, and t is time.

Tags

CCSS.HSF-IF.C.8B

Front

How do you determine the decay factor from an initial value and a value after a certain time?

Back

The decay factor can be determined by dividing the value after a certain time by the initial value. For example, if a car is worth $20,000 initially and $15,000 after 1 year, the decay factor is 15,000 / 20,000 = 0.75.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula for exponential decay when given a percentage decay rate?

Back

The formula for exponential decay when given a percentage decay rate is A = A0 * (1 - r)^t, where A0 is the initial amount, r is the decay rate (as a decimal), and t is time.

Tags

CCSS.HSF-IF.C.8B

Front

What does the base of an exponential function represent in growth and decay models?

Back

In growth models, the base is greater than 1, indicating an increase over time. In decay models, the base is between 0 and 1, indicating a decrease over time.

Tags

CCSS.HSF-IF.C.8B

Front

How can you identify whether a model represents exponential growth or decay?

Back

You can identify a model as exponential growth if the base of the exponential function is greater than 1, and as exponential decay if the base is between 0 and 1.

Tags

CCSS.HSF-IF.C.8B

Front

What is the significance of the decay rate in an exponential decay model?

Back

The decay rate indicates the percentage by which the quantity decreases over a specific time period. A higher decay rate results in a faster decrease.

Tags

CCSS.HSF-IF.C.8B

Front

How do you calculate the remaining amount after multiple periods of decay?

Back

To calculate the remaining amount after multiple periods of decay, use the formula A = A0 * (1 - r)^t, where A0 is the initial amount, r is the decay rate, and t is the number of periods.

Tags

CCSS.HSF.BF.A.2

Front

What is the relationship between half-life and exponential decay?

Back

Half-life is the time required for a quantity to reduce to half its initial value. In exponential decay, the half-life is constant and can be calculated using the decay rate.

Front

How do you derive the exponential decay formula from a percentage decay rate?

Back

The exponential decay formula can be derived by recognizing that if a quantity decreases by a percentage r, the remaining amount after one time period is A0 * (1 - r). Repeating this process leads to A = A0 * (1 - r)^t.

Tags

CCSS.HSF-IF.C.8B