Exponential Growth and Decay

Front

What is exponential growth?

Back

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to growth that accelerates over time. It can be represented by the equation f(x) = a(1 + r)^x, where 'a' is the initial amount, 'r' is the growth rate, and 'x' is 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, leading to a rapid decline over time. It can be represented by the equation f(x) = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate, and 'x' is time.

Tags

CCSS.HSF-IF.C.8B

Front

What does the decay factor represent in an exponential decay function?

Back

The decay factor is the value by which the quantity decreases in each time period. It is calculated as (1 - decay rate). For example, if the decay rate is 20%, the decay factor is 0.8.

Tags

CCSS.HSF-IF.C.8B

Front

How do you calculate the decay rate from a decay factor?

Back

The decay rate can be calculated using the formula: decay rate = 1 - decay factor. For example, if the decay factor is 0.8, the decay rate is 1 - 0.8 = 0.2 or 20%.

Tags

CCSS.HSF-IF.C.8B

Front

What does the initial value represent in an exponential function?

Back

The initial value represents the starting amount before any growth or decay occurs. In the function f(x) = a(1 + r)^x, 'a' is the initial value.

Tags

CCSS.HSF-IF.C.8B

Front

If a function is modeled as f(x) = 500(0.9)^x, what is the decay rate?

Back

The decay rate is 10%, calculated as 1 - 0.9.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula to find the amount remaining after a certain time in an exponential decay situation?

Back

The formula is f(x) = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate, and 'x' is the time.

Front

In the equation f(x) = 200(1.05)^x, what does 200 represent?

Back

200 represents the initial amount before any growth occurs.

Tags

CCSS.HSF.LE.B.5

Front

How do you find the amount remaining after 3 hours if the initial amount is 80 mg and the decay factor is 0.75?

Back

Use the formula: f(x) = 80(0.75)^x. For x = 3, f(3) = 80(0.75)^3 = 42.1875 mg.

Front

What is the relationship between growth rate and decay rate?

Back

The growth rate is the percentage increase of a quantity, while the decay rate is the percentage decrease. They are complementary; for example, a growth rate of 20% corresponds to a decay factor of 0.8.

Tags

CCSS.HSF-IF.C.8B