Exponential Functions Flashcard

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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 find the y-intercept of an exponential function?

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The y-intercept of an exponential function f(x) = a * b^x is found by evaluating f(0), which equals a.

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What is the general form of an exponential growth equation?

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The general form of an exponential growth equation is P(t) = P0 * e^(rt), where P0 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 P(t) = P0 * e^(-rt), where P0 is the initial amount, r is the decay rate, and t is time.

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How do you calculate the population after a certain number of years with a constant growth rate?

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To calculate the population after a certain number of years with a constant growth rate, use the formula P = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate (as a decimal), and t is the number of years.

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What is the significance of the constant 'e' in exponential functions?

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The constant 'e' (approximately 2.718) is the base of natural logarithms and is used in continuous growth or decay models.

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How do you round values in exponential equations?

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To round values in exponential equations, follow standard rounding rules: if the digit after the rounding place is 5 or greater, round up; if it is less than 5, round down.

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What is the relationship between exponential functions and logarithms?

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Exponential functions and logarithms are inverse functions. If y = b^x, then x = log_b(y), where log_b is the logarithm base b.

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How can you determine the best-fitting exponential equation for a set of data?

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To determine the best-fitting exponential equation for a set of data, use methods such as regression analysis to find the parameters that minimize the difference between the observed data and the model.