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 initial value in an exponential function?

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

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What is the y-intercept of an exponential function?

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The y-intercept of an exponential function is the point where the graph intersects the y-axis, which occurs at (0, a).

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If a function is modeled as A = 1200(0.85)^6, what type of model is it?

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This model represents exponential decay because the base (0.85) is less than 1.

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What is the formula for calculating the amount after 't' years in an exponential growth model?

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The formula is A = P(1 + r)^t, where A is the amount after t years, P is the initial amount, r is the growth rate, and t is the time in years.

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What is the formula for calculating the amount after 't' years in an exponential decay model?

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The formula is A = P(1 - r)^t, where A is the amount after t years, P is the initial amount, r is the decay rate, and t is the time in years.

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How do you convert a percentage increase to a growth factor?

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To convert a percentage increase to a growth factor, add 1 to the percentage expressed as a decimal. For example, a 35% increase becomes 1 + 0.35 = 1.35.