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§ 9.1

§ 9.1. Exponential Functions. Exponential Functions. Consider the function. Does this function look different from other functions we have worked with before? If you said, “yes” – you are right. It is different. The difference is that x is in the exponent – not the base.

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§ 9.1

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  1. §9.1 Exponential Functions

  2. Exponential Functions Consider the function Does this function look different from other functions we have worked with before? If you said, “yes” – you are right. It is different. The difference is that x is in the exponent – not the base. Functions like this one are called “exponential functions.” Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.1

  3. Exponential Functions You will need a calculator for evaluating exponential expressions. Any scientific calculator will work for this purpose. Many real-life situations, including population growth, growth of epidemics, radioactive decay, and other changes that involve rapid increase or decrease, can be described using exponential functions. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 9.1

  4. Exponential Functions Page 640 Blitzer, Intermediate Algebra, 5e – Slide #4 Section 9.1

  5. Exponential Functions Page 640 Blitzer, Intermediate Algebra, 5e – Slide #5 Section 9.1

  6. Exponential Functions Page 640 Blitzer, Intermediate Algebra, 5e – Slide #6 Section 9.1

  7. Exponential Functions See graph on page 644. Page 644 Blitzer, Intermediate Algebra, 5e – Slide #7 Section 9.1

  8. Exponential Functions EXAMPLE Graph in the same rectangular coor-dinate system. How is the graph of g related to the graph of f ? SOLUTION We begin by setting up a table showing some of the coordinates for f and g, selecting integers from -2 to 2 for x. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 9.1

  9. Exponential Functions CONTINUED We plot the points for each function and connect them with a smooth curve. Because of the scale on the y-axis, some points on each function are not shown. The graph of g is a reflection of the graph of f across the y-axis. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 9.1

  10. Exponential Functions Pages 645-646 Not on tests. Remember that e is not a variable. It’s just an irrational number. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 9.1

  11. Exponential Functions EXAMPLES from homework Do 2-10 (even) using calculator. Do 12 Blitzer, Intermediate Algebra, 5e – Slide #11 Section 9.1

  12. Exponential Functions EXAMPLES from homework • Do 12-16 • Opposite causes graph to appear in quadrants 3 and 4 since f(x) will always be negative as in 14 and 16 • -1 in number 13 causes graph to cross the y axis at (0, 0) rather than (0, 1) Blitzer, Intermediate Algebra, 5e – Slide #12 Section 9.1

  13. Exponential Functions EXAMPLES from homework Do 18 Blitzer, Intermediate Algebra, 5e – Slide #13 Section 9.1

  14. Exponential Functions EXAMPLES from homework 20 Blitzer, Intermediate Algebra, 5e – Slide #14 Section 9.1

  15. Exponential Functions EXAMPLES from homework 26 Blitzer, Intermediate Algebra, 5e – Slide #15 Section 9.1

  16. Exponential Functions in Application EXAMPLE In college, we study large volumes of information – information that, unfortunately, we do not often retain for very long. The function describes the percentage of information, f(x), that a particular person remembers x weeks after learning the information. Find the percentage of information that is remembered after 4 weeks. Page 646-647 Blitzer, Intermediate Algebra, 5e – Slide #16 Section 9.1

  17. Exponential Functions in Application CONTINUED SOLUTION Because we want to know the percentage of information retained after 4 weeks, we replace x with 4. This is the given function. Replace x with 4. Multiply -0.5 and 4. Evaluate the exponent. Finish simplifying. Therefore, four weeks after learning information, a certain person retains about 30.83% of that information. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 9.1

  18. Exponential Functions Pages 647-648 Blitzer, Intermediate Algebra, 5e – Slide #18 Section 9.1

  19. Exponential Functions in Application EXAMPLE Find the accumulated value of an investment of $5000 for 10 years at an interest rate of 6.5% if the money is (a) compounded semiannually, (b) compounded monthly, (c) compounded continuously. SOLUTION We are trying to determine what the accumulated value of an investment is. Therefore, we are looking for A. We first determine the values for t, P, and r. Since the investment will accumulate for 10 years, t = 10. Since the initial investment is $5000, P = 5000. And since the interest rate is 6.5%, r = 0.065. Now we are ready to use the appropriate formulas to answer the questions. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 9.1

  20. Exponential Functions in Application CONTINUED (a) Since the investment is being compounded semiannually, n = 2. We now solve for A. This is the equation to use. Replace P with 5000, r with 0.065, n with 2 and t with 10. Divide and multiply. Add. Evaluate the exponent. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 9.1

  21. Exponential Functions in Application CONTINUED Multiply. Therefore, the accumulated value of the investment is $9,479.19. (b) Since the investment is being compounded monthly, n = 12. We now solve for A. This is the equation to use. Replace P with 5000, r with 0.065, n with 12 and t with 10. Divide and multiply. Add. Blitzer, Intermediate Algebra, 5e – Slide #21 Section 9.1

  22. Exponential Functions in Application CONTINUED Evaluate the exponent. Multiply. Therefore, the accumulated value of the investment is $9,541.92. (c) Since the investment is being compounded continuously, there is no n value. We now solve for A. This is the equation to use. Replace P with 5000, r with 0.065, and t with 10. Multiply. Evaluate the exponent. Blitzer, Intermediate Algebra, 5e – Slide #22 Section 9.1

  23. Exponential Functions in Application CONTINUED Multiply. Therefore, the accumulated value of the investment is $9,600. You may wish to remember the compound interest formula. Almost everyone needs to either borrow or invest – so that’s a formula that is applicable for many! Blitzer, Intermediate Algebra, 5e – Slide #23 Section 9.1

  24. DONE

  25. Exponential Functions EXAMPLE The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function describes the amount, f(x), in kilograms, of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Find f(80) and determine if Chernobyl will be safe for human habitation by 2066. Blitzer, Intermediate Algebra, 5e – Slide #25 Section 9.1

  26. Exponential Functions CONTINUED SOLUTION In finding f(80), we are finding how many kilograms of cesium-137 are in Chernobyl 80 years after 1986, or in 2066. This is the given function. Replace x with 80. Divide. Evaluate the exponent. Multiply. Chernobyl will not be safe for human habitation by 2066 with approximately 157 kilograms of cesium-137. Blitzer, Intermediate Algebra, 5e – Slide #26 Section 9.1

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