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Avon High School Section: 4.1 ACE COLLEGE ALGEBRA II - NOTES Exponential Functions Mr. Record: Room ALC-129 Day 46 Just browsing? Take your time. Researchers know, to the nearest dollar, exactly what the average amount consumer spends per minute at the shopping mall. And the longer you stay, the more you spend. Data compiled from a 2006 research study is reflected in the bar graph to the right which can be modeled with the function f ( x) 42.2(1.56) x , where f ( x) is the average amount spent, in dollars, at a shopping mall after x hours. The above function is called an exponential function. Do you see what makes it different from the other functions we’ve discussed? Definition of the Exponential Function The exponential function f with base b is defined by or where b is a positive constant other than 1 (b 0 and b 1) and x is any real number. Here are some examples of exponential functions: By contrast, the following functions are not exponential functions. Example 1 Evaluating an Exponential Function Using the function from the top of the page, what is the average amount a consumer would spend, to the nearest dollar, after spending four hours at the mall? Graphing Exponential Functions Example 2 Graphing an Exponential Function Graph f ( x) 2 x y x Example 3 Graphing an Exponential Function 1 Graph f ( x ) 2 x y x TI-Nspire Activity: “Follow that Exponential Function” Characteristics of Exponential Functions of the Form f (x )=b 1. 2. 3. 4. 5. 6. x The domain of f ( x) b x consists of all real numbers: (, ) . The range of f ( x) b x consists of all positive real numbers: (0, ) . The graphs of all exponential functions of the form f ( x) b x pass through the point (0,1) because f (0) b0 1 (b 0) . The y-intercept is 1. There is no x-intercept. If b 1, f ( x) b x has a graph that goes up to the right and is an increasing function. The greater the value of b, the steeper the increase. If b 1 , f ( x) b x has a graph that goes down to the right and is a decreasing function. The smaller the value of b, the steeper the decrease. f ( x) b x is one-to-one and has an inverse that is a function. The graph of f ( x) b x approaches, but does not touch, the x-axis. The x-axis, or y 0 , is a horizontal asymptote. Transformations of Exponential Functions The graphs of exponential functions can be translated vertically or horizontally, reflected, stretched, or shrunk just as polynomial and rational functions. Table 4.1 on page 416 of your text summarizes these transformations. Example 4 Transformations Involving Exponential Functions y Use the graph of f ( x) 2 to sketch a graph of f ( x) 2 x x2 1 x The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential function. But just what is this value….e ? Complete the chart: n 1 2 5 10 100 1000 10000 1000000 ∞ 1 1 n n Example 5 The Gray Wolf Population An insatiable killer. That’s the reputation the gray wolf acquired in the United States in the nineteenth and early twentieth centuries. Although the label was undeserved, an estimated two million wolves were shot, trapped or poisoned. By 1960, the population was reduced to 800 wolves. The figure to the right shows the rebounding population in two recovery areas after the gray wolf was declared an endangered species and received federal protection. The exponential function W (t ) 1.26e0.247t models the gray wolf population of the Northern Rocky Mountains, W (t ) , t years after 1978. If the wolf is not removed from the endangered species list and trends shown in the chart continue, predict its population in the recovery area for this current year. Compound Interest The old equation P rt (Principal equals rate time time) is a bogues formula in the real world and you should all be glad. It doesn’t take into consideration compound interest which is interest computed on your original investment PLUS any accumulated interest. Suppose a sum of money, called the principal, P, is invested at an annual percentage rate r, in decimal form, compounded once per year. Because the interest is added to the principal at year’s end, the accumulated value, A, is A P P r P(1 r ) t Do this over t years, you get A P(1 r ) Most finanacial institutions compound interest more frequently than once a year. In general, when compounding interest n times a year, we say there are n compounding periods per year. The formula above can be adjusted to look like nt r A P 1 n Now, what if we were to use continuous compounding where the number of compounding periods increases infinitely (like compounding every trillionth of a second or quadrillionth of a second, etc). Let’s see what happens to the balance, A, as n Formulas for Compound Interest After t years, the balance, A, in an amount with principal, P and annual interest r (in decimal from) is given by the following formulas: 1. For n compounding per year: 2. For continuous compounding: Example 6 Choosing Between Investments You decide to invest $8000 for 6 years and you have a choice between two accounts. The first account pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?