11/20 - 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 shows rapid growth or decay.

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What does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when the growth factor (b) is greater than 1 in the function y = a(b^x). This means the value of the function increases rapidly as x increases.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay occurs when the decay factor (b) is between 0 and 1 in the function y = a(b^x). This means the value of the function decreases rapidly as x increases.

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How can you identify exponential growth or decay from an equation?

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To identify exponential growth or decay, look at the base of the exponent: if the base is greater than 1, it is growth; if the base is between 0 and 1, it is decay.

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What is the general form of an exponential decay equation?

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

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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 y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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If a car's value is $20,000 and it depreciates at 10% per year, what is the exponential equation that models this situation?

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y = 20,000(0.90)^t.

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What is the difference between linear and exponential functions?

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Linear functions have a constant rate of change, while exponential functions have a variable rate of change that increases or decreases rapidly.

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How do you convert a percentage rate into a growth or decay factor?

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To convert a percentage rate into a growth factor, use (1 + rate); for a decay factor, use (1 - rate). For example, a 5% growth rate becomes 1.05.

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What is the significance of the initial value 'a' in an exponential function?

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The initial value 'a' represents the starting amount or value of the function at time t=0.