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 shows rapid growth or decay.

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What does the base in an exponential function represent?

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The base in an exponential function represents the growth (if greater than 1) or decay (if between 0 and 1) factor of the function.

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How do you identify exponential growth?

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Exponential growth occurs when the base of the exponential function is greater than 1 (e.g., y = a(1 + r)^x, where r is the growth rate).

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How do you identify exponential decay?

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Exponential decay occurs when the base of the exponential function is between 0 and 1 (e.g., y = a(1 - r)^x, where r is the decay rate).

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What is the y-intercept of an exponential function?

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The y-intercept of an exponential function y = a(b^x) is the value of 'a', which represents the initial amount when x = 0.

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If a function is modeled as A = P(1 + r)^t, what do P, r, and t represent?

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P is the principal amount (initial investment), r is the rate of growth (as a decimal), and t is the time in years.

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What is the formula for calculating the amount after n years in exponential growth?

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The formula is A = P(1 + r)^n, where A is the amount after n years, P is the initial amount, r is the growth rate, and n is the number of years.

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What is the formula for calculating the amount after n years in exponential decay?

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The formula is A = P(1 - r)^n, where A is the amount after n years, P is the initial amount, r is the decay rate, and n is the number of years.

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How do you convert a percentage increase to a decimal for use in exponential functions?

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To convert a percentage increase to a decimal, divide the percentage by 100. For example, a 35% increase becomes 0.35.

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What does the term 'initial amount' refer to in exponential functions?

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The initial amount refers to the starting value of the function, represented by 'a' in the equation y = a(b^x).