Exponential Functions

Transcription

1 6. Eponential Functions Essential Question What are some of the characteristics of the graph of an eponential function? Eploring an Eponential Function Work with a partner. Cop and complete each table for the eponential function = 6(). In each table, what do ou notice about the values of? What do ou notice about the values of? = 6() = 6() 3 6 JUSTIFYING CONCLUSIONS To be proficient in math, ou need to justif our conclusions and communicate them to others. 5 Eploring an Eponential Function Work with a partner. Repeat Eploration for the eponential function = 6 ( ). Do ou think the statement below is true for an eponential function? Justif our answer. As the independent variable changes b a constant amount, the dependent variable is multiplied b a constant factor. Graphing Eponential Functions Work with a partner. Sketch the graphs of the functions given in Eplorations and. How are the graphs similar? How are the different? Communicate Your Answer. What are some of the characteristics of the graph of an eponential function? 5. Sketch the graph of each eponential function. Does each graph have the characteristics ou described in Question? Eplain our reasoning. a. = b. = (3) c. = 3(.5) d. = ( ) e. = 3 ( ) f. = ( 3 ) Section 6. Eponential Functions 73

2 6. Lesson What You Will Learn Core Vocabular eponential function, p. 7 Previous independent variable dependent variable parent function Identif and evaluate eponential functions. Graph eponential functions. Solve real-life problems involving eponential functions. Identifing and Evaluating Eponential Functions An eponential function is a nonlinear function of the form = ab, where a, b, and b >. As the independent variable changes b a constant amount, the dependent variable is multiplied b a constant factor, which means consecutive -values form a constant ratio. Identifing Functions Does each table represent an eponential function? Eplain. a. 3 b STUDY TIP In Eample b, consecutive -values form a constant ratio. =, 6 =, 3 6 = a As increases b, is not multiplied b a constant factor. So, the function is not eponential. b As increases b, is multiplied b. So, the function is eponential. Evaluating Eponential Functions Evaluate each function for the given value of. a. = (5) ; = 3 b. = 3(.5) ; = a. = (5) Write the function. b. = 3(.5) = (5) 3 Substitute for. = 3(.5) = (5) Evaluate the power. = 3() = 5 Multipl. = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Does the table represent an eponential function? Eplain.. 3. Evaluate the function when =,, and = (9). =.5() 7 Chapter 6 Eponential Functions and Sequences

3 Graphing Eponential Functions The graph of a function = ab is a vertical stretch or shrink b a factor of a of the graph of the parent function = b. When a <, the graph is also reflected in the -ais. The -intercept of the graph of = ab is a. Core Concept Graphing = ab When b > Graphing = ab When < b < STUDY TIP a > a > The graph of = ab approaches the -ais but never intersects it. (, a) (, a) (, a) (, a) a < a < Graphing = ab When b > Graph f () = (). Compare the graph to the graph of the parent function. Describe the domain and range of f. 6 f() = ( ) Step Make a table of values. Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. f ( ) 6 g() = The parent function is g() =. The graph of f is a vertical stretch b a factor of of the graph of g. The -intercept of the graph of f,, is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is >. Graphing = ab When < b < Graph f () = ( ). Compare the graph to the graph of the parent function. Describe the domain and range of f. g() = ( ) Step Make a table of values. Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. f ( ) f() = ( ) The parent function is g() = ( ). The graph of f is a reflection in the -ais of the graph of g. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is <. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. 5. f () = () 6. f () = ( ) Section 6. Eponential Functions 75

4 To graph a function of the form = ab h + k, begin b graphing = ab. Then translate the graph horizontall h units and verticall k units. Graphing = ab h + k Graph = () 3 +. Describe the domain and range. 6 = () = () 3 + Step Graph = (). This is the same function that is in Eample 3, which passes through (, ) and (, ). Step Translate the graph 3 units right and units up. The graph passes through (3, 6) and (, ). Notice that the graph approaches the line = but does not intersect it. From the graph, ou can see that the domain is all real numbers and the range is >. Comparing Eponential Functions An eponential function g models a relationship in which the dependent variable is multiplied b.5 for ever unit the independent variable increases. Graph g when g() =. Compare g and the function f from Eample 3 over the interval = to =. You know (, ) is on the graph of g. To find points to the right of (, ), multipl g() b.5 for ever unit increase in. To find points to the left of (, ), divide g() b.5 for ever unit decrease in. Step Make a table of values. 3 g() STUDY TIP Note that f is increasing faster than g to the right of =. Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. Both functions have the same value when =, but the value of f is greater than the value of g over the rest of the interval. 6 f g Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Describe the domain and range. 7. = (3) +. f () = (.5) WHAT IF? In Eample 6, the dependent variable of g is multiplied b 3 for ever unit the independent variable increases. Graph g when g() =. Compare g and the function f from Eample 3 over the interval = to =. 76 Chapter 6 Eponential Functions and Sequences

5 Solving Real-Life Problems For an eponential function of the form = ab, the -values change b a factor of b as increases b. You can use this fact to write an eponential function when ou know the -intercept, a. The table represents the eponential function = (5) Modeling with Mathematics Bacterial Population The graph represents a bacterial population after das. 7 (, 76) a. Write an eponential function that represents the population. b. Find the population after 5 das. Population 6 5 (, 3) 3 (, ) (3, 9) (, ) Da. Understand the Problem You have a graph of the population that shows some data points. You are asked to write an eponential function that represents the population and find the population after a given amount of time.. Make a Plan Use the graph to make a table of values. Use the table and the -intercept to write an eponential function. Then evaluate the function to find the population. 3. Solve the Problem a. Use the graph to make a table of values The -intercept is 3. The -values increase b a factor of as increases b. So, the population can be modeled b = 3(). b. To find the population after 5 das, evaluate the function when = 5. = 3() Write the function. = 3() 5 Substitute 5 for. = 3() Evaluate the power. = 37 Multipl. There are 37 bacteria after 5 das.. Look Back The graph resembles an eponential function of the form = ab, where b > and a >. So, the eponential function = 3() is reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. A bacterial population after das can be represented b an eponential function whose graph passes through (, ) and (, ). (a) Write a function that represents the population. (b) Find the population after 6 das. (c) Does this bacterial population grow faster than the bacterial population in Eample 7? Eplain. Section 6. Eponential Functions 77

6 6. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. OPEN-ENDED Sketch an increasing eponential function whose graph has a -intercept of.. REASONING Wh is a the -intercept of the graph of the function = ab? 3. WRITING Compare the graph of = (5) with the graph of = 5.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = 3 f () = () f () = ( 3) = 5(3) Monitoring Progress and Modeling with Mathematics In Eercises 5, determine whether the equation represents an eponential function. Eplain. 5. = (7) 6. = 6 7. = 3. = 3 USING STRUCTURE In Eercises, match the function with its graph.. f () = (.5). = (.5) 3. = (). f () = () 9. = 9( 5). = () In Eercises, determine whether the table represents an eponential function. Eplain. (See Eample.).. A. B C. D In Eercises 5 3, graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. (See Eamples 3 and.) 5 5. f () = 3(.5) 6. f () = In Eercises 5, evaluate the function for the given value of. (See Eample.) 5. = 3 ; = 6. f () = 3() ; = 7. = (5) ; =. f () =.5 ; = 3 9. f () = 3 (6) ; = 3. = () ; = 5 7. f () = (7). f () = 6 ( 3 ) 9. f () = () 3. f () = 3 (.5) In Eercises 3 36, graph the function. Describe the domain and range. (See Eample 5.) 3. f () = 3 3. f () = Chapter 6 Eponential Functions and Sequences

7 33. = = ( ) = (.75) f () = 3(6) 5 In Eercises 37, compare the graphs. Find the value of h, k, or a f() = g() = a() 3. g() = 3 h f() = 3. g() =.5 + k f() =.5 f() = (6) 3 g() = (6) 3 h. ERROR ANALYSIS Describe and correct the error in evaluating the function. g() = 6(.5) ; = g( ) = 6(.5) = 3 = 9. ERROR ANALYSIS Describe and correct the error in finding the domain and range of the function. The domain is all real numbers, and the range is <. g() = (.5). An eponential function h models a relationship in which the dependent variable is multiplied b for ever unit the independent variable increases. The value of the function at is MODELING WITH MATHEMATICS You graph an eponential function on a calculator. You zoom in repeatedl to 5% of the screen size. The function =.5 represents the percent (in decimal form) of the original screen displa that ou see, where is the number of times ou zoom in. a. Graph the function. Describe the domain and range. b. Find and interpret the -intercept. c. You zoom in twice. What percent of the original screen do ou see? 6. MODELING WITH MATHEMATICS A population of cootes in a national park triples ever ears. The function = 5(3) represents the population, where is the number of -ear periods. a. Graph the function. Describe the domain and range. b. Find and interpret the -intercept. c. How man cootes are in the national park in ears? In Eercises 7 5, write an eponential function represented b the table or graph. (See Eample 7.) In Eercises 3 and, graph the function with the given description. Compare the function to f () =.5() over the interval = to =. (See Eample 6.) 3. An eponential function g models a relationship in which the dependent variable is multiplied b.5 for ever unit the independent variable increases. The value of the function at is. 9. (, ) 6 (,.5) (, ) (3, ) 5. (, ) (, ) (, ) (3, ) Section 6. Eponential Functions 79

8 5. MODELING WITH MATHEMATICS The graph represents the number of visitors to a new art galler after months. Visitors Art Galler (, 9) (, 6) (, ) (3, 35) Month 5. HOW DO YOU SEE IT? The eponential function = V() represents the projected value of a stock weeks after a corporation loses an important legal battle. The graph of the function is shown. Stock price (dollars) Stock Week a. Write an eponential function that represents this situation. b. Approimate the number of visitors after 5 months. 5. PROBLEM SOLVING A sales report shows that 33 gas grills were purchased from a chain of hardware stores last ear. The store epects grill sales to increase 6% each ear. About how man grills does the store epect to sell in Year 6? Use an equation to justif our answer. a. After how man weeks will the stock be worth $? b. Describe the change in the stock price from Week to Week USING GRAPHS The graph represents the eponential function f. Find f (7). (,.5) (, 3) 53. WRITING Graph the function f () =. Then graph g() = 3. How are the -intercept, domain, and range affected b the translation? 6 (, 6) 5. MAKING AN ARGUMENT Your friend sas that the table represents an eponential function because is multiplied b a constant factor. Is our friend correct? Eplain. 6. THOUGHT PROVOKING Write a function of the form = ab that represents a real-life population. Eplain the meaning of each of the constants a and b in the real-life contet WRITING Describe the effect of a on the graph of = a when a is positive and when a is negative. 56. OPEN-ENDED Write a function whose graph is a horizontal translation of the graph of h() =. 57. USING STRUCTURE The graph of g is a translation units up and 3 units right of the graph of f () = 5. Write an equation for g. Maintaining Mathematical Proficienc Write the percent as a decimal. (Skills Review Handbook) Chapter 6 Eponential Functions and Sequences 6. REASONING Let f () = ab. Show that when is f ( + k) increased b a constant k, the quotient is f () alwas the same regardless of the value of. 6. PROBLEM SOLVING A function g models a relationship in which the dependent variable is multiplied b for ever units the independent variable increases. The value of the function at is 5. Write an equation that represents the function. 63. PROBLEM SOLVING Write an eponential function f so that the slope from the point (, f ()) to the point (, f ()) is equal to. Reviewing what ou learned in previous grades and lessons 6. % % 66. % 67. 5%