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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How do you identify the initial value in an exponential function from a word problem?

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The initial value is the starting amount before any growth or decay occurs. It is usually given in the problem as the quantity at time t=0.

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What does the base in an exponential function represent?

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The base represents the growth (if greater than 1) or decay (if between 0 and 1) factor of the function. For example, a base of 1.05 indicates a 5% growth rate.

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How do you convert a percentage growth rate into a base for an exponential function?

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To convert a percentage growth rate into a base, use the formula: base = 1 + (percentage rate as a decimal). For example, a 3.5% growth rate becomes 1.035.

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How do you convert a percentage decay rate into a base for an exponential function?

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To convert a percentage decay rate into a base, use the formula: base = 1 - (percentage rate as a decimal). For example, a 5% decay rate becomes 0.95.

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Write an exponential function for an investment of $1500 increasing at a rate of 3.5%.

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B(t) = 1500(1.035)^t.

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Write an exponential function for a college tuition of $12,000 increasing at a rate of 6% per year.

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C(t) = 12000(1.06)^t.

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Write an exponential function for a soda machine starting with 75 gallons of cola decreasing at a rate of 5% each hour.

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f(h) = 75(0.95)^h.

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Write an exponential function for a culture of bacteria starting with 5000 cells dying by 30% each year.

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y = 5000(0.7)^x.

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What is the general form of an exponential decay function?

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