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.

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What does it mean for a quantity to decay exponentially?

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Exponential decay occurs when a quantity decreases at a rate proportional to its current value, often modeled by the function f(t) = a(1 - r)^t, where 'r' is the decay rate.

Front

What is the decay rate in the function y = a(1 - r)^t?

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The decay rate 'r' represents the percentage decrease of the quantity per time unit.

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How do you determine the starting value in an exponential function?

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

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What is the formula for exponential growth?

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The formula for exponential growth is f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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How do you write an exponential function for a depreciating asset?

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To write an exponential function for a depreciating asset, use the formula V(t) = a(1 - r)^t, where 'a' is the initial value and 'r' is the depreciation rate.

Front

What does the base 'b' represent in an exponential function?

Back

In an exponential function f(x) = a(b)^x, the base 'b' represents the growth (if b > 1) or decay (if 0 < b < 1) factor.

Front

If a car's value is $20,000 and it depreciates at 10% per year, what is the exponential function modeling its value?

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V(t) = 20000(0.90)^t, where 0.90 represents the remaining value after 10% depreciation.

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What is the difference between exponential growth and exponential decay?

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Exponential growth increases over time at a rate proportional to its current value, while exponential decay decreases over time at a rate proportional to its current value.

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How do you find the value of an investment after 't' years with a growth rate of 'r'?

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Use the formula A = P(1 + r)^t, where A is the amount after 't' years, P is the principal amount, and r is the growth rate.