Exponential Word Problems

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What is an exponential growth function?

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An exponential growth function is a mathematical expression that describes a quantity increasing at a constant percentage rate over time, typically represented as y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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How do you write an exponential growth function for a value that increases by a certain percentage each year?

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To write an exponential growth function, use the formula y = a(1 + r)^t, where 'a' is the initial value, 'r' is the growth rate (as a decimal), and 't' is the number of years.

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What is the formula to calculate the future value of an investment with compound interest?

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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 (the 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.

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If a population increases by 15% each year, how do you express this as an exponential function?

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The exponential function can be expressed as P(t) = P_0(1 + 0.15)^t, where P_0 is the initial population and t is the number of years.

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What does it mean for a function to exhibit exponential growth?

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Exponential growth means that the quantity increases at a rate proportional to its current value, leading to faster growth as time progresses.

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How do you determine the value of an investment after a certain number of years with a fixed annual interest rate?

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

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What is the difference between linear growth and exponential growth?

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Linear growth increases by a constant amount over time, while exponential growth increases by a constant percentage, leading to much larger values over the same period.

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How can you find the population after a certain number of years if it grows exponentially?

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Use the formula P(t) = P_0(1 + r)^t, substituting the initial population for P_0, the growth rate for r, and the number of years for t.

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

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The base of an exponential function indicates the growth factor; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

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How do you calculate the total amount in a savings account after several years with compound interest?

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Use the formula A = P(1 + r/n)^(nt), where A is the total amount, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.