Exponential Growth/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 at an increasing rate. Example: A population of bacteria doubles every hour.

Front

What is Exponential Decay?

Back

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. Example: A radioactive substance decays by 5% each year.

Front

What is the formula for compound interest?

Back

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

Front

How do you calculate the future value of an investment compounded annually?

Back

Use the formula A = P(1 + r)^t, where A is the future value, P is the principal, r is the annual interest rate, and t is the number of years.

Front

What is a common ratio in exponential functions?

Back

The common ratio is the factor by which a quantity increases or decreases in exponential growth or decay. Example: In the function y = a * r^x, r is the common ratio.

Front

How do you find the value of x in an exponential equation?

Back

To find x in an equation like h(x) = a^x, take the logarithm of both sides. Example: If h(x) = 27 and a = 3, then x = log_3(27) = 3.

Front

What is the relationship between exponential functions and logarithms?

Back

Exponential functions and logarithms are inverse operations. If y = a^x, then x = log_a(y).

Front

What is the formula for exponential decay in population?

Back

The formula is P(t) = P_0 * e^(-kt), where P(t) is the population at time t, P_0 is the initial population, k is the decay constant, and e is Euler's number.

Front

How do you model a population that grows by a percentage?

Back

Use the formula P(t) = P_0 * (1 + r)^t, where P(t) is the population at time t, P_0 is the initial population, r is the growth rate, and t is the time.

Front

What is the significance of the base in an exponential function?

Back

The base of an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.