Exponential Functions

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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.

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What does the base of an exponential function represent?

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The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

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How do you model a situation where a quantity decreases by a certain percentage each year?

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You can model this situation using the formula f(t) = a * (1 - r)^t, where a is the initial amount, r is the decay rate (as a decimal), and t is the time in years.

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What is the formula for compound interest?

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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 deposit), 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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How do you determine the time it takes for an investment to grow to a certain amount with compound interest?

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You can use the formula A = P(1 + r/n)^(nt) and solve for t, often requiring logarithmic functions to isolate t.

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What is the growth rate in an exponential growth model?

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The growth rate in an exponential growth model is represented by the base of the exponential function. For example, in the function f(t) = a * b^t, the growth rate is (b - 1) * 100%.

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What is the decay rate in an exponential decay model?

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The decay rate in an exponential decay model is represented by the base of the exponential function. For example, in the function f(t) = a * (1 - r)^t, the decay rate is r * 100%.

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How can you express a percentage increase in an exponential function?

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A percentage increase can be expressed in an exponential function as f(t) = a * (1 + r)^t, where r is the growth rate as a decimal.

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What is the significance of the initial value in an exponential function?

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The initial value in an exponential function represents the starting amount before any growth or decay occurs, denoted by 'a' in the function f(x) = a * b^x.

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How do you find the number of periods required for an investment to reach a certain value?

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You can find the number of periods by rearranging the compound interest formula and using logarithms to solve for t.