Modeling Exponential Functions (Percent Growth Decay)

Front

What is an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It represents growth or decay processes.

Front

What does it mean for a quantity to grow by a percentage?

Back

When a quantity grows by a percentage, it increases by that percentage of its current value. For example, a 10% growth means the new value is the original value plus 10% of the original value.

Front

What does it mean for a quantity to decay by a percentage?

Back

When a quantity decays by a percentage, it decreases by that percentage of its current value. For example, a 10% decay means the new value is the original value minus 10% of the original value.

Front

How do you write a function for exponential decay?

Back

A function for exponential decay can be written as V(t) = V0 * (1 - r)^t, where V0 is the initial value, r is the decay rate (as a decimal), and t is time.

Front

How do you write a function for exponential growth?

Back

A function for exponential growth can be written as V(t) = V0 * (1 + r)^t, where V0 is the initial value, r is the growth rate (as a decimal), and t is time.

Front

What is the formula for calculating the value of an investment after t years with compound interest?

Back

The formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount (initial investment), 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

What is the difference between simple interest and compound interest?

Back

Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal and also on the accumulated interest from previous periods.

Front

How do you convert a percentage to a decimal?

Back

To convert a percentage to a decimal, divide the percentage by 100. For example, 25% becomes 0.25.

Front

What is the initial value in an exponential function?

Back

The initial value in an exponential function is the value of the function when time (t) is zero. It represents the starting amount before any growth or decay occurs.

Front

What does it mean for a population to decrease by a certain percentage each year?

Back

It means that each year, the population is reduced by that percentage of its current size. For example, a 5% decrease means the population is 95% of its size from the previous year.