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 modeled by the equation f(t) = a(1 + r)^t, where 'a' 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 of reducing an amount by a consistent percentage rate over a period of time. It can be modeled by the equation f(t) = 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 variable 'a' represent in the exponential function f(t) = a(1 + r)^t?

Back

'a' represents the initial amount or starting value before any growth or decay occurs.

Tags

CCSS.HSF-IF.C.8B

Front

What does the variable 'r' represent in the exponential function?

Back

'r' represents the growth or decay rate expressed as a decimal.

Tags

CCSS.HSF-IF.C.8B

Front

How do you determine if a function models growth or decay?

Back

If the growth factor (1 + r) is greater than 1, the function models growth. If the decay factor (1 - r) is less than 1, the function models decay.

Tags

CCSS.HSF-IF.C.8B

Front

What is the formula for compound interest?

Back

The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the number of years.

Front

Calculate the total amount after 2 years for $2750 at 8% interest compounded annually.

Back

Using the formula A = P(1 + r/n)^(nt), we have A = 2750(1 + 0.08/1)^(1*2) = 2750(1.08)^2 = 2750 * 1.1664 = $3207.60.

Front

What is the growth factor in the function f(x) = 10(0.3)^x?

Back

The growth factor is 0.3, indicating a decay since it is less than 1.

Tags

CCSS.HSF-IF.C.8B

Front

What does the y-intercept of an exponential function represent?

Back

The y-intercept represents the initial value of the function when time (t) is zero.

Tags

CCSS.HSF.LE.B.5

Front

If a population of bacteria doubles every hour, what type of growth is this?

Back

This is an example of exponential growth, as the population increases by a constant factor over equal time intervals.

Tags

CCSS.HSF-LE.A.1A