Ready To Go On? Skills Intervention 7-1 Exponential Functions, Growth, and Decay

Transcription

1 7A Find these vocabular words in Lesson 7-1 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 7-1 Eponential Functions, Growth, and Deca eponential growth eponential deca asmptote base Graphing Eponential Functions A. Tell whether the function f () 2. shows growth or deca. Then graph. What is the value of the base? Is the base greater than one or between one and zero? Does the function show growth or deca? Complete the table of values: Graph the function using the table of values. B. Tell whether the function g () 2(0.7 ) shows growth or deca. Then graph. Find the value of the base. Is the base greater than one or between one and zero? Does the function show growth or deca? Complete the table of values: Graph the function using the table of values. 108 Holt Algebra 2

2 7A Read To Go On? Problem Solving Intervention 7-1 Eponential Functions, Growth, and Deca A function of the form f () a b, where a is greater than 0 and b is greater than 1, is an eponential growth function which increases as increases. When b is between 0 and 1 the function is called an eponential deca function, which decreases as decreases. The value of a new car is $24,00, and its value decreases 9% each ear. a. Write an eponential function representing the value of the car. b. Graph the function on a calculator. c. Use the graph to predict when the car s value will fall to $10,000. Understand the Problem 1. What is the initial value of the car? 2. Determine whether the function will show growth or deca. 3. Describe the growth factor or deca factor of the car s value. Make a Plan 4. What do ou need to determine?. Let A(t ) represent the final value of the car. Write a function to model the value of the car. Time t Final amount A(t ) Initial Amount a 1 plus rate of increase or 1 minus rate of decrease (1 r ) ( ) 6. Simplif the function in Eercise. Solve 7. Graph the function in Eercise 6 on our calculator. 8. Use the graph to predict when the value of the car will fall below $10,000. Use the trace feature. It will take ears for the car s value to drop to $10,000. Look Back 9. To check our solution, substitute the solution ou predicted for t in Eercise 8 into the equation ou wrote in Eercise 6. Let A(t ) equal 10,000. ( ) 10. Does our solution make the equation true? 109 Holt Algebra 2

3 7A Read To Go On? Skills Intervention 7-2 Inverses of Relations and Functions Find these vocabular words in Lesson 7-2 and the Multilingual Glossar. Vocabular inverse relation inverse function Writing Inverse Functions b Using Inverse Operations A. Use inverse operations to write the inverse of f () 2. 2 Set f (). 2 Switch and. Solve for. Write in format. Write the inverse b substituting f 1 () for. f 1 () Simplif. Check: Since (1, 7) satisfies f (), does (7, 1) satisf f 1 ()? 2 4 B. Use inverse operations to write the inverse of f () Set f () Switch and. 3 4 Solve for Write in format. Write the inverse b substituting f 1 () for. f 1 () Simplif. Check our answer: Since (2, 0) satisfies f (), does (0, 2) satisf f 1 ()? 110 Holt Algebra 2

4 7A Read To Go On? Problem Solving Intervention 7-2 Inverses of Relations and Functions To write the inverse of a function, switch and in the original function and solve for. Ruth rents an apartment in the cit for a $0 initial realtor fee and a rate of $700 per month. The total amount spent on the apartment can be epressed as a function of months,, b f () Find the inverse function. Then, use the inverse function to find the number of months Ruth rented the apartment if she spent a total of $13,10. Understand the Problem 1. Describe the fees Ruth spent on the apartment. Make a Plan 2. What do ou need to determine? 3. Use inverse operations to write the inverse of f () that models months as a function of the total amount spent on the apartment Set f () Switch and. 700 Solve for. Write in format and substitute f 1 () for. f 1 () Simplif. Solve 4. Evaluate the inverse function for $13,10. f 1 () (13,10) Ruth rented the apartment for months. Look Back. To check our solution, substitute the number of months into the original function. f () 0 700( ) 6. Does our solution make the function equal $13,10? 111 Holt Algebra 2

5 7A Read To Go On? Skills Intervention 7-3 Logarithmic Functions Find these vocabular words in Lesson 7-3 and the Multilingual Glossar. Vocabular logarithm common logarithm logarithmic function Converting from Eponential to Logarithmic Form Remember a logarithm is an eponent: b a lo g b a ; lo g A. Convert 3 12 to logarithmic form. Find the value of the base b. Find the value of the eponent. Find the value of a. Write in the form lo g b a. B. Convert to logarithmic form. Find the value of the base b. Find the value of the eponent. Find the value of a. Write in the form lo g b a. Converting from Logarithmic to Eponential Form A. Convert lo g to eponential form. Find the value of the base b. Find the value of the eponent. Find the value of a. Write in the form b a. B. Convert lo g to eponential form. Find the value of the base b. Find the value of the eponent. Find the value of a. Write in the form b a. 112 Holt Algebra 2

6 7A Read To Go On? Skills Intervention 7-4 Properties of Logarithms Adding and Subtracting Logarithms A. Epress lo g 6 3 lo g 6 72 as a single logarithm. Simplif, if possible. Use the Propert of Logarithms to simplif this epression. lo g 6 Appl the appropriate propert of logarithms. lo g 6 Simplif. Think 6? 216. B. Epress lo g lo g 2 7 as a single logarithm. Simplif, if possible. Use the lo g 2 Propert of Logarithms to simplif this epression. Appl the appropriate propert of logarithms. lo g 2 Simplif. Think 2? 32. Simplifing Logarithms with Eponents A. Epress log1 0 4 as a single logarithm. Simplif, if possible. Use the log Propert of Logarithms to simplif this epression. Appl the appropriate propert of logarithms. ( ) Simplif. Think 10? 10. B. Epress lo g as a single logarithm. Simplif, if possible. Use the lo g 3 Propert of Logarithms to simplif this epression. Appl the appropriate propert of logarithms. ( ) Simplif. Think 3? 1 3. Recognizing Inverses A. Simplif lo g Use the Propert of Logarithms to simplif this epression. lo g 7 Appl the appropriate propert of logarithms. ( ) Simplif. Think 7? 7. B. Simplif lo g lo g 8 Appl the Inverse Propert of Logarithms. ( ) Simplif. 113 Holt Algebra 2

7 7A Read To Go On? Quiz 7-1 Eponential Functions, Growth, and Deca Tell whether the function shows growth or deca. Then graph. 1. f () f () 1 3 (4 ) 3. The population of a town is 20,000 and, it increases at a rate of 2% per ear. Predict the town s population after ears. 7-2 Inverses of Relations and Functions 4. Graph the relation and connect the points. Then graph its inverse Graph each function. Then write and graph the inverse.. f () 6. f () 4 3 f 1 () f 1 () 114 Holt Algebra 2

8 7A Read To Go On? Quiz continued 7. Sarah bought a set of bowls for a wedding present. She spent a total of $37.80, which included a shipping charge of $6.0 and % sales ta. What was the price of the bowls, including ta? 7-3 Logarithmic Functions Write the eponential function in logarithmic form Write the logarithmic function in eponential form. 12. lo g lo g lo g lo g e Use the given -values to graph f () 0. ; 2, 1, 0, 1, 2. Then graph the inverse function. 7-4 Properties of Logarithms Epress as a single logarithm. Simplif, if possible. 17. lo g 4 64 lo g lo g lo g Simplif each epression. 19. lo g log 9 0. Evaluate. 21. log lo g Holt Algebra 2

9 7A Read To Go On? Enrichment The Richter Scale The Richter scale is used to measure the magnitude, or size, of earthquakes. It is a logarithmic function given b the formula: M 2 3 log E 10, where M is the magnitude and E is the number of ergs of 11.8 energ released. Some of the largest earthquakes in the world are shown below. Use the table to answer the following questions. Location Year Magnitude Chile Prince William Sound, Alaska Coast of Northern Sumatra Kamchatka Coast of Ecuador Northern Sumatra, Indonesia Rat Island, Alaska How much energ was released b the earthquake in Chile? 2. How much energ was released b the 200 earthquake in Northern Sumatra? 3. How man times as much energ is released b an earthquake with a magnitude of 9.0 than b an earthquake with a magnitude of 7.0? 4. On Jul 26, 200, an earthquake in western Montana registered a magnitude of.6 on the Richter scale. Find the energ released b the earthquake. 116 Holt Algebra 2

10 7B Read To Go On? Skills Intervention 7- Eponential and Logarithmic Equations and Inequalities Find these vocabular words in Lesson 7- and the Multilingual Glossar. Vocabular eponential equation logarithmic equation Solving Eponential Equations A. Solve is a power of. Rewrite each side with the same base. 6 To raise a power to a power, eponents. 6 Set the eponents equal. B. Solve Solve for Take the of both sides. log 42 log Appl the Propert of Logarithms. log 42 log 9 Divide both sides b log 9. 1 Divide. Solve for. Solving Logarithmic Equations A. Solve lo g lo g 8 4 Appl the Propert of Logarithms. lo g 8 Divide both sides b to isolate lo g 8. 8 Appl the definition of a logarithm. 2 3 B. Solve log 00 log. 2 Simplif. log Appl the Propert of Logarithms. 10 Appl the definition of a logarithm. 10 Solve for. 117 Holt Algebra 2

11 7B Read To Go On? Problem Solving Intervention 7- Eponential and Logarithmic Equations and Inequalities You can use eponential functions to predict population growth. The population of a small French village, currentl 120, grows at a rate of 2% per ear. This growth can be epressed b the eponential equation P 120( ) t, where P is the population after t ears. Find the number of ears it will take for the population to eceed Understand the Problem 1. Describe the growth of the village s population. Make a Plan 2. What do ou need to determine? 3. Write an inequalit that models the situation. P Define P. 120( ) t 2000 Substitute known values in the equation. Solve 4. Solve the inequalit for t. 120( ) t Write the inequalit from Eercise ( ) t ( ) t Simplif. Divide both sides b 120. log( ) log1.6 Take the log of both sides. Beginning in ear log( ) log1.6 Appl the Power Propert of Logarithms. log 1.6 t Isolate t. log t Solve for t. Round to a whole number., the village s population will eceed 2000 people. Look Back. To check our solution, substitute our answer for t into the original eponential equation. P 120( ) t 120(1 0.02) 6. Does our solution make the epression eceed 2000? 118 Holt Algebra 2

12 7B Read To Go On? Skills Intervention 7-6 The Natural Base, e Find these vocabular words in Lesson 7-6 and the Multilingual Glossar. Vocabular natural logarithm natural logarithmic function half-life Simplifing Epressions with e or ln. A. Simplif ln e 3. Appl the Propert of Logarithms. ln e 3 ln 3( ) Simplif. Think e? e. B. Simplif e ln( 2). Appl the Propert of Eponents. e ln( 2) Simplif. Think b lo g b. C. Simplif e 10ln. Appl the reverse of the Propert of Logarithms. e 10ln e ln Appl the Propert of Eponents. e ln Simplif. Think b lo g b. D. Simplif 7ln e 0. Appl the Propert of Logarithms. 7ln e 0 7( )ln 7( )( ) Simplif. Think e? e. E. Simplif ln e 6t. Appl the Propert of Logarithms. ln e 6t 6 ln 6 ( ) Simplif. Think e? e. 119 Holt Algebra 2

13 7B Read To Go On? Problem Solving Intervention 7-6 The Natural Base, e The half-life of a substance is the time it takes for half of the substance to break down or convert to another substance during the process of deca. Neptunium-239, a radioactive isotope, has a half-life of 2.4 das. Use the deca function N(t ) N 0 e kt to determine the amount of a 100-gram sample that remains after 20 das. Understand the Problem 1. Describe the deca of neptunium What do ou need to determine? Make a Plan 3. Find the deca constant, k, for neptunium-239. Remember that half of the initial quantit will remain after 2.4 das. N(t ) N 0 e kt e k Substitute 1 for N 0, 2.4 for t, and 1 for N(t ). 2 ln e k Simplif and take the natural log of both sides. ln k ln e Appl the Power Propert of Logarithms. ln k Simplif and isolate k. k Solve for k. Round to 4 decimal places. Solve 4. Write the deca function using our value for k and solve for N(t ). N(t ) N 0 e kt N(t ) e ( ) Substitute 100 for N 0, for t, and our value for k. N(t ) Solve for N(t ). Look Back. To check our solution, substitute our answers for N(t ) and k into the deca function. N(t ) N 0 e kt e t t 6. Do our answers for N(t ) and k result in t equaling 20 das? 120 Holt Algebra 2

14 7B Read To Go On? Skills Intervention 7-7 Transforming Eponential and Logarithmic Functions Find these vocabular words in Lesson 7-7 and the Multilingual Glossar. Vocabular transformation parent function Writing Transformed Eponential Functions A. f () 7 is translated 4 units left and stretched verticall b a factor of. To translate a function 4 units horizontall to the left should ou add or subtract 4 from? f () Start b identifing the parent function. f () 7 To translate 4 units left, replace with 4. f () 7 Stretch verticall b multipling b. B. f () 11 is horizontall compressed b a factor of 1 and reflected across 4 the -ais. To reflect a function across the -ais, should ou change the sign on the coefficient or the eponent? f () Start b identifing the parent function. f () 11 Horizontall compress b multipling b 4. f () 11 Reflect across the -ais b replacing with. Writing Transformed Logarithmic Functions A. f () lo g 2 is verticall compressed b a factor of 1 and translated units down. 3 To translate a function units down, should ou add or subtract from? f () Start b identifing the parent function. f () lo g 2 Verticall compress b multipling the right side b 1 3. f () lo g 2 To translate units down, subtract from the right side. B. f () ln is translated 1 unit right and reflected across the -ais. f () Start b identifing the parent function. f () ln( ) To translate 1 unit right, replace with 1. f () ln ( ) Reflect across the -ais b multipling the right side b Holt Algebra 2

15 7B Read To Go On? Skills Intervention 7-8 Curve Fitting with Eponential and Logarithmic Models Find this vocabular word in Lesson 7-8 and the Multilingual Glossar. Identifing Eponential Data A. Determine whether f is an eponential function of. If so, find the constant ratio f () Vocabular eponential regression For linear functions, For eponential functions, the are constant. of each -value and the previous value is constant. Using the table of values: a. Find the first differences. b. Find the ratios of the f () terms. Is the function linear or eponential? If the function is eponential, what is the constant ratio? Use linear or eponential regression to find a function that models the data. f () B. Determine whether f is an eponential function of. If so, find the constant ratio f () Using the table of values: a. Find the first differences. b. Find the ratios of the f () terms. Is the function linear or eponential? If the function is eponential, what is the constant ratio? Use linear or eponential regression to find a function that models the data. f () 122 Holt Algebra 2

16 7B Read To Go On? Quiz 7- Eponential and Logarithmic Equations and Inequalities Solve lo g 4 ( 6) log 0 log 2 3. A lotter winner can choose a prize of either $00,000 or one penn on the first da, quadruple that (4 cents) on the second da, and so on for 30 das. On what da would the lotter winner receive more than the original $00,000 prize? 7-6 The Natural Base, e 6. Graph f () 2 e. 7. Graph f () e 1 2. Simplif. 8. ln e e ln(2 1) 10. e 7ln ln e 12. ln e ln e What is the total amount after ears for an investment of $2000 invested at 4% compounded continuousl? 1. Use the deca function N t N 0 e kt to determine how much of 20 grams of carbon-14 will remain after 00 ears. Carbon-14 s half-life is 730 ears. 123 Holt Algebra 2

17 7B Read To Go On? Quiz continued 7-7 Transforming Eponential and Logarithmic Functions Graph the function. Find the -intercept and asmptote. Describe how the graph is transformed from the graph of the parent function. 16. g () log( 1) 17. h() e 3 2 -intercept: asmptote: transformation: -intercept: asmptote: transformation: Write the transformed function. 18. f () is verticall stretched b 4 and reflected across the -ais. 19. f () ln(3) is horizontall compressed b 1 and verticall translated 1 unit up Curve Fitting with Eponential and Logarithmic Models Determine whether is an eponential function of. If so, find the constant ratio and use eponential regression to find a function that models the data Holt Algebra 2

18 7B Read To Go On? Enrichment Compounding Continuousl When investing mone at a compounded interest rate, the interest is paid on the original principal and on the accumulated interest. Recall that compound interest is computed using the formula: A P 1 r n nt As n increases, the interest approaches that of continuousl compounded interest. The formula for continuousl compounded interest is: A P e rt Compare compound interest intervals b completing the table. 1. Emmett invests $2000 for 10 ears at a % compounded interest rate. Complete the table. Principal (P ) Rate (r ) Time in ears (t ) Compound Interval (n) 2000 % 10 Semi-annuall n % 10 Quarterl n % 10 Monthl n % 10 Dail n % 10 Continuousl Amount (A) 2. Which interval earns more mone for an investment? Answer each of the following questions. 3. How long will it take an investment of $1000 to triple in value if it is invested at a rate of 8% compounded monthl? 4. How long will it take an investment of $200 to double in value if it is invested at a rate of 6% compounded continuousl?. Caroln has $ in her savings account. She invested her mone at a 4% interest rate for ears compounded continuousl. How much did she originall invest? 12 Holt Algebra 2