Writing Exponential Functions from Word Problems

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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. It models situations where a quantity grows or decays at a constant percentage rate.

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

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Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals. The base 'b' in the function f(x) = a(b^x) is greater than 1.

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

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Exponential decay occurs when a quantity decreases by a fixed percentage over equal time intervals. The base 'b' in the function f(x) = a(b^x) is between 0 and 1.

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How do you write an exponential function for depreciation?

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

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How do you write an exponential function for growth?

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To write an exponential function for growth, use the formula y = a(1 + r)^t, where 'a' is the initial value, 'r' is the rate of growth, and 't' is time.

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What is the scale factor in an exponential function?

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The scale factor is the base 'b' in the exponential function f(x) = a(b^x). It determines how much the quantity increases (if b > 1) or decreases (if 0 < b < 1) with each time interval.

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

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y = 20000(0.90)^t.

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If a population of bacteria triples every hour, how would you express this as an exponential function?

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f(h) = a(3^h), where 'a' is the initial number of bacteria.

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What is the difference between linear and exponential functions?

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Linear functions increase or decrease by a constant amount, while exponential functions increase or decrease by a constant percentage.

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How do you determine if an exponential function represents growth or decay?

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Check the base 'b': if b > 1, it represents growth; if 0 < b < 1, it represents decay.