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 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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How do you write an exponential function for a situation involving growth?

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To write an exponential function for growth, use the formula C(t) = C_0(1 + r)^t, where C_0 is the initial amount, r is the growth rate (as a decimal), and t is time.

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How do you write an exponential function for a situation involving decay?

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To write an exponential function for decay, use the formula C(t) = C_0(1 - r)^t, where C_0 is the initial amount, r is the decay rate (as a decimal), and t is time.

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What does the base of an exponential function represent?

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

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If a quantity doubles every time period, how can this be expressed in an exponential function?

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If a quantity doubles, it can be expressed as f(t) = C_0(2)^t, where C_0 is the initial quantity and t is the number of time periods.

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What is the formula to calculate the future value of an investment with continuous compounding?

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The formula for continuous compounding is A = Pe^(rt), where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and e is Euler's number (approximately 2.71828).

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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 the value of the function when time (t) is zero, represented as C_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 periods that have passed, affecting the growth or decay of the quantity.

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How can you find the amount after a certain number of years with a given growth rate?

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Use the formula C(t) = C_0(1 + r)^t, substituting C_0 for the initial amount, r for the growth rate, and t for the number of years.

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What is the difference between exponential growth and linear growth?

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Exponential growth increases at a rate proportional to its current value, leading to faster increases over time, while linear growth increases by a constant amount.