Exponential Functions Why? 35 y Tarantulas (hundreds) 30 Then Tarantulas can appear scary with their large hairy 25 You simplified numerical bodies and legs, but they are harmless to humans. 20 expressions involving The graph shows a tarantula spider population that 15 exponents. (Lesson 1-2) increases over time. Notice that the graph is neither 10 5 linear nor quadratic. Now The graph represents the function y = 3(2) x. This is an -2-10 1 2 3 4 5 6x Graph exponential example of an exponential function. functions. Years Since 2010 Identify data that display exponential Graph Exponential Functions An exponential function is a function of the form behavior. y = ab x, where a 0, b > 0, and b 1. Notice that the base is a constant and the exponent is a variable. Exponential functions are nonlinear and nonquadratic New Vocabulary functions. exponential function For Your Key Concept Exponential Function Math Online glencoe.com Words An exponential function is a function that can be Extra Examples described by an equation of the form y = ab x, Personal Tutor where a 0, b > 0, and b 1. Self-check Quiz x y= _ 1 Homework Help Examples y = 2(3) x y = 4x (2) EXAMPLE 1 Graph with a > 0 and b > 1 a. Graph y = 3 . Find the y-intercept, and state the domain and range. x y x 3x y -2 3 -2 _ 1 9 -1 3 -1 _ 1 3 x 0 3 0 1 y=3 1 31 3 2 32 9 0 x Graph the ordered pairs, and connect the points with a smooth curve. The graph crosses the y-axis at 1, so the y-intercept is 1. The domain is all real numbers, and the range is all positive real numbers. b. Use the graph to approximate the value of 3 0.7. The graph represents all real values of x and their corresponding values of y for y = 3 x. So, when x = 0.7, y is about 2. Use a calculator to confirm this value: 3 0.7 2.157669. Check Your Progress 1A. Graph y = 7 x. Find the y-intercept, and state the domain and range. 1B. Use the graph to approximate the value of y = 7 0.5 to the nearest tenth. Use a calculator to confirm the value. Personal Tutor glencoe.com Lesson 9-6 Exponential Functions 567 The graphs of functions of the form y = ab x, where a > 0 and b > 1, all have the same shape as the graph in Example 1. The greater the base or b-value, the faster the graph rises as you move from left to right on the graph. The graphs of functions of the form y = ab x, where a > 0 and 0 < b < 1, also have the same general shape. StudyTip EXAMPLE 2 Graph with a > 0 and 0 < b < 1 a < 0 If the value of a 1 x is less than 0, the graph a. Graph y = _ (3) . Find the y-intercept, and state the domain and range. will be reflected across the x-axis. x (_ 3) 1 x y y 1 -2 -2 (_ 3) 9 x y= _ 1 (3) 0 0 (_ 1 3) 1 2 2 (_ 1 3) _ 1 9 0 x The y-intercept is 1. The domain is all real numbers, and the range is all positive real numbers. Notice that as x increases, the y-values decrease less rapidly. -1.5 b. Use the graph to approximate the value of _ 1 (3) . When x = -1.5, the value of y is about 5. Use a calculator to confirm this value: KEYSTROKES: ( 1 3 -1.5 ENTER 5.196152. Check Your Progress x 2A. Graph y = _ 1 (2) - 1. Find the y-intercept, and state the domain and range. -2.5 2B. Use the graph to approximate the value of _ 1 () - 1 to the nearest tenth. 2 Use a calculator to confirm the value. Personal Tutor glencoe.com Exponential functions occur in many real world situations. EXAMPLE 3 Use Exponential Functions to Solve Problems SODA The consumption of soda has increased each year since 2000. The function C = 179(1.029) t models the amount of soda consumed in the world, where C is the amount consumed in billions of liters and t is the number of years since 2000. a. Graph the function. What values of C and t are The United States is the meaningful in the context of the problem? largest soda consumer in the world. In a recent year, Since t represents time, t > 0. At t = 0, the the United States consumption is 179 billion liters. Therefore, in the accounted for one third of context of this problem C > 179 is meaningful. the world's total soda consumption. Source: Worldwatch Institute [-50, 50] scl: 10 by [0, 350] scl: 25 568 Chapter 9 Quadratic and Exponential Functions b. How much soda was consumed in 2005? C = 179(1.029) t Original equation = 179(1.029) 5 t=5 = 206.5 Use a calculator. The world soda consumtion in 2005 was approximately 206.5 billion liters. Check Your Progress 3. A certain bacteria doubles every 20 minutes. Beginning with 10 cells in a culture, the population can be represented by the function B = 10(2) t, where B is the number of bacteria cells and t is the time in 20 minute increments. How many will there be after 2 hours? Personal Tutor glencoe.com Identify Exponential Behavior Recall from Lesson 3-3 that linear functions have a constant rate of change. Exponential functions do not have constant rates of change, but they do have constant ratios. EXAMPLE 4 Identify Exponential Behavior Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. Problem-SolvingTip x 0 5 10 15 20 25 Make an Organized y 64 32 16 8 4 2 List Making an organized list of x-values and Method 1 Look for a pattern. corresponding y-values The domain values are at regular intervals of 5. Look for a common factor among is helpful in graphing the range values. the function. It can also help you identify 64 32 16 8 4 2 patterns in the data. _ _ _ _ _ 1 1 1 1 1 2 2 2 2 2 The range values differ by the common factor of _ 1 . 2 Since the domain values are at regular intervals and the range values differ by a positive common factor, the data are probably exponential. Its equation x may involve _ 1 (2) . Method 2 Graph the data. y 64 Plot the points and connect them with a smooth curve. 56 48 The graph shows a rapidly decreasing value of y as x 40 increases. This is a characteristic of exponential 32 behavior in which the base is between 0 and 1. 24 StudyTip 16 8 Checking Answers The graph of an Check Your Progress -50 5 10 15 20 25 30 35x exponential function may resemble part of 4. Determine whether the set of data shown below displays exponential behavior. the graph of a Write yes or no. Explain why or why not. quadratic function. Be sure to check for a x 0 3 6 9 12 15 pattern as well as to y 12 16 20 24 28 32 look at a graph. Personal Tutor glencoe.com Lesson 9-6 Exponential Functions 569 Check Your Understanding Examples 1 and 2 Graph each function. Find the y-intercept, and state the domain and range. Then pp. 567568 use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. 1. y = 2 x; 2 1.5 2. y = -5 x; -5 0.5 x -0.5 1 x 0.5 3. y = - _ 1 ;- _ (5) (5) 1 4. y = 3 _ ;3 _ (4) (4) 1 Graph each function. Find the y-intercept, and state the domain and range. 5. f(x) = 6 x + 3 6. f(x) = 2 - 2 x Example 3 7. BIOLOGY The function f(t) = 100(1.05) t models the growth of a fruit fly pp. 568569 population, where f(t) is the number of flies and t is time in days. a. What values for the domain and range are reasonable in the context of this situation? Explain. b. After two weeks, approximately how many flies are in this population? Example 4 Determine whether the set of data shown below displays exponential behavior. p. 569 Write yes or no. Explain why or why not. 8. x 1 2 3 4 5 6 9. x 2 4 6 8 10 12 y -4 -2 0 2 4 6 y 1 4 16 64 256 1024 = Step-by-Step Solutions begin on page R12. Practice and Problem Solving Extra Practice begins on page 815. Examples 1 and 2 Graph each function. Find the y-intercept, and state the domain and range. Then pp. 567568 use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. 1 x 1.5 1 x _ 0.5 10. y = 2 8 x, 2(8) -0.5 11. y = 2 _ ;2 _ (6) (6) 1 12. y = _ ( 12 ) ( 12 ) ; 1 13. y = -3 9 x, -3(9) -0.5 14. y = -4 10 x, -4(10) -0.5 15. y = 3 11 x, 3(11) -0.2 Graph each function. Find the y-intercept, and state the domain and range. 16. y = 4 x + 3 17. y = _ 1( x 2 - 8) 18. y = 5(3 x) + 1 19. y = -2(3 x) + 5 2 Example 3 20. BIOLOGY A population of bacteria in a culture increases according to the model pp. 568569 p = 300(2.7) 0.02t, where t is the number of hours and t = 0 corresponds to 9:00 a.m. a. Use this model to estimate the number of bacteria at 11 a.m. b. Graph the function and name the p-intercept. Describe what the p-intercept represents, and describe a reasonable domain and range for this situation. Example 4 Determine whether the set of data shown below displays exponential behavior. p. 569 Write yes or no. Explain why or why not. 21 x -4 0 4 8 12 22. x -6 -3 0 3 y 2 -4 8 -16 32 y 5 10 15 20 23. x -8 -6 -4 -2 24. x 20 30 40 50 60 y 0.25 0.5 1 2 y 1 0.4 0.16 0.064 0.0256 570 Chapter 9 Quadratic and Exponential Functions 25 PHOTOGRAPHY Jameka is enlarging a photograph to make a poster for school. She will enlarge the picture repeatedly at 150%. The function P = 1.5 x models the B new size of the picture being enlarged, where x is the number of enlargements. How big is the picture after it has been enlarged 4 times? 26. FINANCIAL LITERACY Daniel invested $500 into a savings account. The equation A = 500(1.005) 12t models the value of Daniel's investment A after t years. How much will Daniel's investment be worth in 8 years? Identify each function as linear, quadratic, or exponential. The world's largest 27. y 28. y 29. y photograph, named The Great Picture, was created by a group of photographers known as The Legacy Project. The 0 x photograph has an area of 0 x 3375 square feet. Source: Photoshop Support 0 x 30. y = 4 x + 3 31. y = 2x(x - 1) 32. 5x + y = 8 33. GRADUATION The number of graduates at a high school has increased by a factor of 1.055 every year since 2001. In 2001, 110 students graduated. The function N = 110(1.055) t models. N, the number of students expected to graduate t year after 2001. How many students will graduate in 2012? C Describe the graph of each equation as a transformation of the graph of y = 2 x. 34. y = 2 x + 6 35. y = 3(2) x 36. y = -_ 1 ( )x 2 4 x 37. y = -3 + 2 x 38. y = _ () 1 39. y = -5(2) x 2 40. DEER The deer population at a national park doubles every year. In 2000, there were 25 deer in the park. The function N = 25(2) t models the number of deer N in the national park t years after 2000. What will the deer population in the park be in 2015? H.O.T. Problems Use Higher-Order Thinking Skills 41. CHALLENGE Write an exponential function that passes through the points at (0, 3) and (1, 6). 42. REASONING Determine whether the graph of y = ab x, where a 0, b > 0, and b 1, sometimes, always, or never has an x-intercept. Explain your reasoning. 43. OPEN ENDED Choose an exponential function that represents a real-world situation, and graph the function. Analyze the graph. 44. REASONING Compare and contrast an exponential function of the form y = ab x + c, where a 0, b > 0, and b 1 and a quadratic function of the form y = ax 2 + c. 45. WRITING IN MATH Explain how to determine whether a set of data displays exponential behavior. Lesson 9-6 Exponential Functions 571Download Link: