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.

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.

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.

Front

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

Back

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

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.

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.

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.

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.