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Exponential Functions At the beginning of the quarter we saw how certain relationships between variables could be modeled with a variety of functions. When you studied linear functions, you saw how growth always proceeded at a constant rate, regardless of the size of the variables. This rate was called slope. Sometimes however, the amount of growth changes as the size of the population changes. This occurs when there is a feedback loop. A feedback loop occurs when you have two (or more) variables and each variable affects the other. An example of a positive feedback loop is what occurs between the size of a population and the number of births in that population. More people leads to more births which lead to more people, etc. Another example is money in an account. As the account gets larger, the amount of interest earned increases, which leads to more money in the account, which leads to more interest, etc. An exponential function is used to model situations in which there is a positive feedback loop. An exponential function is shown as y = abx or f(x) = abx, where a is the initial size of the population, b is the growth factor which indicates the proportion by which the population will increase every period. The variable x, which is an exponent, represents the number of periods. The variable y or f(x) is the total amount that the population becomes after x periods. As an example, consider a small business that sees a hundred percent growth every year. This corresponds to a doubling of their first year’s earnings. If the small business earned $1,000 in profit the first year, then the exponential equation that would model this is f(x) = 1,000(2)x-1, where x is the number of years the business has existed. Number of years in business Substitution into function 1 2 3 4 f(x) = 1,000(2)1-1 f(x) = 1,000(2)0 f(x) = 1,000(2)2-1 f(x) = 1,000(2)1 f(x) = 1,000(2)3-1 f(x) = 1,000(2)2 f(x) = 1,000(2)4-1 f(x) = 1,000(2)3 Total Amount of profit after that many years in business f(x) = 1000 f(x) = 2000 f(x) = 4000 f(x) = 8000 Notice how the total amount of profit made doubles each year. The variable b in the function determines if there is exponential growth or exponential decay. When b is greater than 1, the function will grow. When b is between 0 and 1, the function will decay. These are shown in the graphs below. Note that b cannot be equal to 1, nor can it be negative. f(x) = 2x f(x) = (1/2)x Exponential equations are used for the growth of investments in which money r remains in the account leading to compounded interest. The formula is Pt P0 1 k Where Pt is the value after t years Compounding P0 is the starting value Annual r is the annual interest rate (APR) Quarterly t is the number of years for the entire investment Monthly k is the number of compounding periods in a year Daily tk Example: $5,000 is invested at 3.1% interest, compounded monthly. How much money is in the account after 8 years? 0.031 Pt 5000 1 12 Pt 6405.25 812 k 1 4 12 365