Review of exponential functions & Sequences

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What is an exponential function?

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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.

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What does the base 'b' in an exponential function represent?

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The base 'b' in an exponential function represents the growth (if b > 1) or decay (if 0 < b < 1) factor of the function.

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How do you calculate compound interest?

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Compound interest can be calculated using the formula 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.

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What is the formula for exponential growth?

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The formula for exponential growth is f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is the formula for exponential decay?

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The formula for exponential decay is f(t) = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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What is the starting term in an exponential function?

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The starting term in an exponential function f(x) = a(b^x) is the value of 'a', which represents the initial quantity before any growth or decay occurs.

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How do you determine the growth rate from an exponential function?

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The growth rate can be determined from the base 'b' in the function f(x) = a(b^x) using the formula r = b - 1.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over equal intervals, while exponential growth increases by a constant percentage over equal intervals, leading to much larger values over time.

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How do you model a situation with exponential growth?

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To model a situation with exponential growth, identify the initial amount, determine the growth rate, and use the formula f(t) = a(1 + r)^t.

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

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The exponent in an exponential function indicates the number of time periods that have passed, affecting the total amount based on the growth or decay rate.