Write and Graph Exponential Decay Functions

Transcription

1 8.6 Write and Graph Eponential Deca Functions Before You wrote and graphed eponential growth functions. Now You will write and graph eponential deca functions. Wh? So ou can use a graph to solve a sports problem, as in E. 50. Ke Vocabular eponential deca A table of values represents an eponential function 5 ab provided successive -values are multiplied b b each time the -values increase b. E XAMPLE Write a function rule Tell whether the table represents an eponential function. If so, write a rule for the function. a. 0 } 9 } The value of when 5 0 is } 3, so a 5 } 3. The -values are multiplied b 3 for each increase of in, so the table represents an eponential function of the form 5 ab with b 5 3. The table represents the eponential function 5 } 3 p 3. b. 0 4 } 4 }6 3 } 4 3 } 4 3 } 4 The -values are multiplied b } for 4 each increase of in, so the table represents an eponential function of the form 5 ab with b 5 }. 4 The value of when 5 0 is, so a 5. The table represents the eponential function 5 } 4. GUIDED PRACTICE for Eample. Tell whether the table represents an eponential function. If so, write a rule 0 for the function. 5 } 5 }5 8.6 Write and Graph Eponential Deca Functions 53

2 E XAMPLE Graph an eponential function READ A GRAPH Notice that the graph has a -intercept of and that it gets closer to the positive -ais as the -values increase. Graph the function 5 } and identif its domain and range. STEP Make a table of values. The domain is all real numbers. 0 4 STEP Plot the points. } }4 (, 4) STEP 3 Draw a smooth curve through the points. From either the table or the graph, ou can see the range is all positive real numbers. 3 (, ) (0, ) s, d 5 c s, 4 d E XAMPLE 3 Compare graphs of eponential functions Graph the functions 5 3 p } and 5 } 3 p }. Compare each graph with the graph of 5 }. 5 } 5 3 p } 5 } 3 p } 4 4 } 3 5 c 5 3? c 6 } } 3 } 3 } } ? c } 4 3 }4 } Because the -values for 5 3 p } are 3 times the corresponding -values for 5 }, the graph of 5 3 p } is a vertical stretch of the graph of 5 }. Because the -values for 5} p 3 } are } times the corresponding 3 -values for 5 }, the graph of 5} p 3 } is a vertical shrink with reflection in the -ais of the graph of 5 }. GUIDED PRACTICE for Eamples and 3. Graph 5 (0.4) and identif its domain and range. 3. Graph 5 5 p (0.4). Compare the graph with the graph of 5 (0.4). 53 Chapter 8 Eponents and Eponential Functions

3 COMPARE GRAPHS When a > 0 and 0 < b <, the function 5 ab represents eponential deca. The graph of an eponential deca function falls from left to right. In comparison, the graph of an eponential growth function 5 ab where a > 0 and b > rises from the left. E XAMPLE 4 Classif and write rules for functions Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function. a. b. (, ) (0, 0) (0, 8) (, 4) ANALYZE GRAPHS For the function 5 ab, where 5 0, the value of is 5 a 0 5 a. This means that the graph of 5 ab has a -intercept of a. a. The graph represents eponential growth ( 5 ab where b > ). The -intercept is 0, so a 5 0. Find the value of b b using the point (, ) and a ab Write function. 5 0 p b Substitute.. 5 b Solve. A function rule is 5 0(.). b. The graph represents eponential deca ( 5 ab where 0 < b < ). The -intercept is 8, so a 5 8. Find the value of b b using the point (, 4) and a ab Write function p b Substitute b Solve. A function rule is 5 8(0.5). GUIDED PRACTICE for Eample 4 4. The graph of an eponential function passes through the points (0, 0) and (, 8). Graph the function. Tell whether the graph represents eponential growth or eponential deca. Write a rule for the function. CONCEPT SUMMARY For Your Notebook Eponential Growth and Deca Eponential Growth Eponential Deca 5 ab, a > 0 and b > 5 ab, a > 0 and 0 < b < (0, a) (0, a) 8.6 Write and Graph Eponential Deca Functions 533

4 EXPONENTIAL DECAY When a quantit decas eponentiall, it decreases b the same percent over equal time periods. To find the amount of the quantit left after t time periods, use the following model. KEY CONCEPT For Your Notebook REWRITE EQUATIONS Notice that ou can rewrite 5 ab as 5 a( r) t b replacing b with r and with t (for time). Eponential Deca Model a is the initial amount. r is the deca rate. 5 a( r) t 5 r is the deca factor. t is the time period. The relationship between the deca rate r and the deca factor r is similar to the relationship between the growth rate and growth factor in an eponential growth model. You will eplore this relationship in Eercise 45. E XAMPLE 5 Solve a multi-step problem FORESTRY The number of acres of Ponderosa pine forests decreased in the western United States from 963 to 00 b 0.5% annuall. In 963 there were about 4 million acres of Ponderosa pine forests. a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time. b. To the nearest tenth, about how man million acres of Ponderosa pine forests were there in 00? AVOID ERRORS The deca rate in this eample is 0.5%, or So, the deca factor is 0.005, or 0.995, not a. Let P be the number of acres (in millions), and let t be the time (in ears) since 963. The initial value is 4, and the deca rate is P 5 a( r) t Write eponential deca model. 5 4( 0.005) t Substitute 4 for a and for r. 5 4(0.995) t Simplif. b. To find the number of acres in 00, 39 ears after 963, substitute 39 for t. P 5 4(0.995) 39 < 33.7 Substitute 39 for t. Use a calculator. c There were about 33.7 million acres of Ponderosa pine forests in 00. at classzone.com GUIDED PRACTICE for Eample 5 5. WHAT IF? In Eample 5, suppose the deca rate of the forests remains the same beond 00. About how man acres will be left in 00? 534 Chapter 8 Eponents and Eponential Functions

5 8.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7 and 49 5 STANDARDIZED TEST PRACTICE Es., 9, 36, 45, and 49 5 MULTIPLE REPRESENTATIONS E. 50. VOCABULARY What is the deca factor in the eponential deca model 5 a( r) t?. WRITING Eplain how ou can tell if a graph represents eponential growth or eponential deca. EXAMPLE on p. 53 for Es. 3 6 WRITING FUNCTIONS Tell whether the table represents an eponential function. If so, write a rule for the function } 3 }9 7 3 EXAMPLE on p. 53 for Es. 7 8 GRAPHING FUNCTIONS Graph the function and identif its domain and range } } } } } } (0.3) 4. 5 (0.5) 5. 5 (0.) 6. 5 (0.9) 7. 5 (0.7) 8. 5 (0.5) EXAMPLE 3 on p. 53 for Es MULTIPLE CHOICE The graph of which function is shown? A 5 (0.5) B 5 (0.5) C p (0.5) D 5 4 p (0.5) (0, 4) (, ) COMPARING FUNCTIONS Graph the function. Compare the graph with the graph of 5 } p } p } 4. 5 } p } } p 3 } p } p } p } p } } p } } p 3 } p } p } Write and Graph Eponential Deca Functions 535

6 MATCHING Match the function with its graph (0.) (0.) } (0.) A. B. C POPULATION A population of 90,000 decreases b.5% per ear. Identif the initial amount, the deca factor, and the deca rate. Then write a function that models the population over time. 36. MULTIPLE CHOICE What is the deca rate of the function 5 4(0.97) t? A 4 B 0.97 C 0.3 D ERROR ANALYSIS In 004 a person purchased a car for $5,000. The value of the car decreased b 4% annuall. Describe and correct the error in writing a function that models the value of the car since a( r) t 5 5,000(0.4) t EXAMPLE 4 on p. 533 for Es RECOGNIZING EXPONENTIAL MODELS Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function (0, 6) (, 4.8) (0, 8) (, 4.8) (0, 8) (,.8) at classzone.com 4. REASONING Without graphing, eplain how the graphs of the given functions are related to the graph of f() 5 (0.5). a. m() 5 } 3 p (0.5) b. n() 54 p (0.5) c. p() 5 (0.5) CHALLENGE Write an eponential function of the form 5 ab whose graph passes through the given points. 4. (0, ),, } (, 0), (, 4) 44., 3 },, 3 } WRITING The initial amount of a quantit is a units and the quantit is decaing at a rate of r (a percent per time period). Show that the amount of the quantit after one time period is a( r). Eplain how ou found our answer. 46. CHALLENGE Compare the graph of the function f() 5 4 with the graph of the function g() 5 } p 4. Use properties of eponents to 6 eplain our observation WORKED-OUT SOLUTIONS for on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

7 PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLE 5 on p. 534 for Es CELL PHONES You purchase a cell phone for $5. The value of the cell phone decreases b about 0% annuall. Write a function that models the value of the cell phone over time. Then find the value of the cell phone after 3 ears. Round to the nearest dollar. 48. ANIMAL POPULATION Scientists studied the population of a species of bat in some caves in Missouri from 983 to 003. In 983, there were 4,00 bats living in the caves. That number decreased b about % annuall until 003. a. Identif the initial amount, the deca factor, and the deca rate. b. Write a function that models the number of bats since 983. Then find the number of bats in SHORT RESPONSE In 003 a famil bought a boat for $4000. The boat depreciates (loses value) at a rate of 7% annuall. In 006 a person offers to bu the boat for $3000. Should the famil sell the boat? Eplain. 50. MULTIPLE REPRESENTATIONS There are a total of 8 teams at the start of a citwide 3-on-3 basketball tournament. Half of the teams are eliminated after each round. a. Writing a Model Write a function for the number of teams left after rounds. b. Making a Table Make a table for the function using 5 0,,,..., 7. c. Drawing a Graph Use the table in part (b) to graph the function. After which round are there 4 teams left in the tournament? 5. GUITARS The frets on a guitar are the small metal bars that divide the fingerboard. The distance d (in inches) between the nut and the first fret or an two consecutive frets can be modeled b the function d 5.56(0.9439) f where f is the number of the fret farthest from the nut. Fret Nut a. Identif the deca factor and the deca rate for the model. b. What is the distance between the nut and the first fret? c. The distance between the th and 3th frets is about half the distance between the nut and the first fret. Use this fact to find the distance between the th and 3th frets. Use the model to verif our answer. 8.6 Write and Graph Eponential Deca Functions 537

8 5. CHALLENGE A college student finances a computer that costs $850. The financing plan states that as long as a minimum monthl pament of.5% of the remaining balance is made, the student does not have to pa interest for 4 months. The student makes onl the minimum monthl paments until the last pament. What is the amount of the last pament if the student bus the computer without paing interest? Round our answer to the nearest cent. 53. MULTI-STEP PROBLEM Maimal ogen consumption is the maimum volume of ogen (in liters per minute) that the bod uses during eercise. Maimal ogen consumption varies from person to person and decreases with age b about 0.5% per ear after age 5 for active adults. a. Model A 5-ear-old female athlete has a maimal ogen consumption of 4 liters per minute. Another 5-ear-old female athlete has a maimal ogen consumption of 3.5 liters per minute. Write a function for each athlete that models the maimal consumption each ear after age 5. b. Graph Graph the models in the same coordinate plane. c. Estimate About how old will the first athlete be when her maimal ogen consumption is equal to what the second athlete s maimal ogen consumption is at age 5? MIXED REVIEW PREVIEW Prepare for Lesson 9. in Es Simplif the epression. (p. 96) 54. (3) ( ) ( 4) ( 9) 60. (5 ) 6. (3 4)7 6. ( ) Solve the equation (p. 34) (p. 4) (p. 48) 66. 7( 5) 5 0 (p. 48) QUIZ for Lessons Graph the function } (p. 50). 5 3 p } 4 (p. 53) 3. 5 } 4 p 3 (p. 50) 4. 5 (0.) (p. 53) p 5 (p. 50) (0.4) (p. 53) 7. COINS You purchase a coin from a coin collector for $5. Each ear the value of the coin increases b 8%. Write a function that models the value of the coin over time. Then find the value of the coin after 0 ears. Round to the nearest cent. (p. 50) 538 EXTRA PRACTICE for Lesson 8.6, p. 945 ONLINE QUIZ at classzone.com

9 Etension Use after Lesson 8.6 Ke Vocabular geometric sequence common ratio Relate Geometric Sequences to Eponential Functions GOAL Identif, graph, and write geometric sequences. In a geometric sequence, the ratio of an term to the previous term is constant. This constant ratio is called the common ratio and is denoted b r. A geometric sequence with first term a and common ratio r has the form a, a r, a r, a r 3,.... For instance, if a 5 5 and r 5, the sequence 5, 5 p, 5 p, 5 p 3,..., or 5, 0, 0, 40,..., is geometric. E XAMPLE Identif a geometric sequence Tell whether the sequence is arithmetic or geometric. Then write the net term of the sequence. a. 3, 6, 9,, 5,... b. 8, 64, 3, 6, 8,... a. The first term is a 5 3. Find the ratios of consecutive terms: a } a 5 6 } 3 5 a 3 } a 5 9 } 6 5 } a 4 } a3 5 } 9 5 } 3 a 5 } a4 5 5 } 5 } 4 REVIEW ARITHMETIC SEQUENCES For help with identifing an arithmetic sequence and finding a common difference, see p Because the ratios are not constant, the sequence is not geometric. To see if the sequence is arithmetic, find the differences of consecutive terms. a a a 3 a a 4 a a 5 a The common difference is 3, so the sequence is arithmetic. The net term of the sequence is a 6 5 a b. The first term is a 5 8. Find the ratios of consecutive terms: a } a 5 64 } 8 5 } a 3 } a 5 3 } 64 5 } a 4 } a3 5 6 } 3 5 } a 5 } a4 5 8 } 6 5 } Because the ratios are constant, the sequence is geometric. The common ratio is }. The net term of the sequence is a 6 5 a 5 p } 5 4. E XAMPLE Graph a geometric sequence ANALYZE A GRAPH Notice that the graph in Eample appears to be eponential. To graph the sequence from part (b) of Eample, let each term s position number in the sequence be the -value. The term is the corresponding -value. Then make and plot the points Position, Term, Etension: Relate Geometric Sequences to Eponential Functions 539

10 FUNCTIONS The table shows that a rule for finding the nth term of a geometric sequence is a n 5 a r n. Notice that the rule is an eponential function. Position, n n Term, a n a a r a r a r 3... a r n For the nth term, ou multipl a b r (n ) times. p r p r p r p r KEY CONCEPT For Your Notebook General Rule for a Geometric Sequence The nth term of a geometric sequence with first term a and common ratio r is given b: a n 5 a r n. E XAMPLE 3 Write a rule for a geometric sequence Write a rule for the nth term of the geometric sequence in Eample. Then find a 0. To write a rule for the nth term of the sequence, substitute the values for a and r in the general rule a n 5 a r n. Because a 5 8 and r 5 }, a n 5 8 p } n. The 0th term of the sequence is a p } 0 5 } 4. PRACTICE EXAMPLES,, and 3 on pp for Es. 0 Tell whether the sequence is arithmetic or geometric. Then graph the sequence.. 3,, 48, 9,.... 7, 6, 5, 34, , 8,, 6, , 8, 6,, , 8, 36, 7, , 43, 57, 7,... Write a rule for the nth term of the geometric sequence. Then find a 7. 7., 5, 5, 5, , 6, 5, 04, , 7,,, A chain instructs the recipient to forward the to four more people. The table shows the number of rounds of sending the and the number of new s generated. Write a rule for the nth term of the sequence. Then graph the first si terms of the sequence. Number of rounds sending , n 3 4 Number of new s generated, a n Chapter 8 Eponents and Eponential Functions

11 MIXED REVIEW of Problem Solving Lessons MULTI-STEP PROBLEM The radius of the sun is about 96,600,000 kilometers. The radius of Earth is about 6370 kilometers. a. Write each radius in scientific notation. b. The surface area S of a sphere with radius r is given b S 5 4pr. Assume the sun and Earth are perfect spheres. Find their surface areas. Write our answers in scientific notation. c. What is the ratio of the surface area of the sun to the surface area of Earth? What does the ratio tell ou?. SHORT RESPONSE The graph shows the value of a truck over time. 30,000 (0, 5,000) 0,000 Value (dollars) 0,000 (, 3,750) Time (ears) a. Write an equation for the function whose graphisshown. b. At what rate is the truck losing value? Eplain. 3. GRIDDED ANSWER A new laptop computer costs $000. The value of the computer decreases over time. The value V (in dollars) of the computer after t ears is given b the function V 5 000(0.8) t. What is the deca rate, written as a decimal, of the value of the computer? 4. OPEN - ENDED The value of a house in Iowa increased, on average, at a rate of about 4% per quarter from the first quarter in 00 to the last quarter in 004. Write a function that models the value of the house over time. Choose an initial value of the house and a quarter such that the value of the house is about $75,000. STATE TEST PRACTICE classzone.com 5. EXTENDED RESPONSE A musician is saving mone to bu a new snare drum. The musician puts $00 in a savings account that pas 3% annual interest compounded earl. a. Write a function that models the amount of mone in the account over time. b. Graph the function. c. The musician wants a drum that costs $ Will there be enough in the account after 3 ears? Eplain. 6. MULTI-STEP PROBLEM The graph shows the value of a business over time. Value (dollars) 60,000 80,000 (0, 5,000) (, 9,500) Time (ears) a. Does the graph represent eponential growth or eponential deca? b. Write a function that models the value of the business over time. c. How much is the business worth after 4ears? 7. MULTI-STEP PROBLEM The half-life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient s bloodstream. A certain antibiotic has a half-life of about 8 hours. a. A patient is administered 500 milligrams of the medication. Write a function that models the amount of the medication in the patient s bloodstream over time. b. How much of the 500 milligram dose will be in the patient s bloodstream after 4 hours? Mied Review of Problem Solving 54