Writing 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 y = a(b)^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It describes growth or decay processes.

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What does the base of an exponential function represent?

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The base of an exponential function indicates 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 write an exponential function for a situation where a quantity triples every day?

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The function can be written as y = a(3)^x, where 'a' is the initial amount and 'x' is the number of days.

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What is the general form of an exponential growth function?

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The general form of an exponential growth function is y = a(1 + r)^t, where 'r' is the growth rate and 't' is time.

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What is the general form of an exponential decay function?

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The general form of an exponential decay function is y = a(1 - r)^t, where 'r' is the decay rate and 't' is time.

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How do you determine the initial value in an exponential function?

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The initial value in an exponential function is represented by 'a' in the equation y = a(b)^x. It is the value of y when x = 0.

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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 intervals that have passed, affecting the growth or decay of the quantity.

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How can you model a situation where a quantity decreases by half each round?

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You can model this situation with the function y = a(0.5)^x, where 'a' is the initial quantity and 'x' is the number of rounds.

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What is the formula to calculate compound interest using exponential functions?

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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, '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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How do you find the value of an exponential function at a specific point?

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To find the value of an exponential function at a specific point, substitute the value of 'x' into the function and calculate the result.