Exponential Growth and Decay Functions (Exponent of t) Read 6.1 Examples 1-3

Transcription

1 CC Algebra II HW #42 Name Period Row Date Section Vocabulary In the eponential growth model Eponential Growth and Decay Functions (Eponent of t) Read 6.1 Eamples 1-3 y = 2.4(1.5), identify the initial amount, the growth factor, and the percent increase. 2. Which One Doesn t Belong? Which characteristic of an eponential decay function does not belong with the other three? Eplain your reasoning. Evaluate the epression for (a) = 2 and (b) = Tell whether the function represents eponential growth or eponential decay. Then graph the function. (See Eample 1.) y = y = (0.6)

2 Analyzing Relationships Use the graph of f () = b to identify the value of the base b Modeling with Mathematics The value of a mountain bike y (in dollars) can be approimated by the model t y = 200(0.75), where t is the number of years since the bike was new. (See Eample 2.) a. Tell whether the model represents eponential growth or eponential decay. b. Identify the annual percent increase or decrease in the value of the bike. c. Estimate when the value of the bike will be $ Modeling with Mathematics In 2006, there were approimately 233 million cell phone subscribers in the United States. During the net 4 years, the number of cell phone subscribers increased by about 6% each year. (See Eample 3.) a. Write an eponential growth model giving the number of cell phone subscribers y (in millions) t years after Estimate the number of cell phone subscribers in b. Estimate the year when the number of cell phone subscribers was about 278 million.

3 CC Algebra II HW #43 Name Period Row Date Section 6.1 Evaluate the epression for (a) = 2 and (b) = Eponential Growth and Decay Functions Read 6.1 Eamples Problem Solving When a plant or animal dies, it stops acquiring carbon-14 from the atmosphere. The amount y (in grams) of carbon-14 in the body of an organism after t years is y = a(0.5) 5730, where a is the initial amount (in grams). What percent of the carbon-14 is released each year? (See Eample 4.) t Rewrite the function in the form y = a( 1+ r) or t t 29. y = a(2) y = a(0.5) 12 t t y = a( 1 r). Then state the growth or decay rate. 37. Problem Solving You deposit $5000 in an account that pays 2.25% annual interest. Find the balance after 5 years when the interest is compounded quarterly. (See Eample 5.) 39. Error Analysis You invest $500 in the stock of a company. The value of the stock decreases 2% each year. Describe and correct the error in writing a model for the value of the stock after t years.

4 Use the given information to find the amount A in the account earning compound interest after 6 years when the principal is $ r = 1.83%, compounded daily 49. Problem Solving The population p of a small town after years can be modeled by the function What is the average rate of change in the population over the first 6 years? Justify your answer. p = 6850(1.03).

5 CC Algebra II HW #44 Name Period Row Date The Natural Base e Read 6.2 Eamples 2-3 Section Writing Tell whether the function 4 1 f ( ) = e represents eponential growth or eponential decay. Eplain. 3 Tell whether the function represents eponential growth or eponential decay. Then graph the function. (See Eample 2.) y = 0. 5e 21. y = 0.4e Using Structure Use the properties of eponents to rewrite the function in the form Then find the percent rate of change. 0.25t 0.4t 27. y = e 29. y = 2e y = a( 1+ r) or t y = a( 1 r). t Using Tools Use a graphing calculator to graph the function. Then identify the domain and range. 33. y = 2 e + 1 (Make a sketch of the graph here.)

6 43. Problem Solving The growth of Mycobacterium tuberculosis bacteria can be modeled by the function 0.166t N( t) = ae, where N is the number of cells after t hours and a is the number of cells when t = 0. a. At 1:00 p.m., there are 30 M. tuberculosis bacteria in a sample. Write a function that gives the number of bacteria after 1:00 p.m. b. Use a graphing calculator to graph the function in part (a). (Make a sketch of the graph here.) c. Describe how to find the number of cells in the sample at 3:45 p.m.

7 CC Algebra II HW #45 Name Period Row Date Logarithms Read 6.3 Eamples 1-4 Section Complete the Sentence The epression log 3 9 is read as. Rewrite the equation in eponential form. (See E. 1.) 9. log 16 = Rewrite the equation in logarithmic form. (See E. 2.) = 25 Evaluate the logarithm. (See Eample 3.) 19. log log Writing Eplain why the epressions log 2 ( 1) and log 1 1 are not defined. Evaluate the logarithm using a calculator. Round your answer to three decimal places. (See Eample 4.) ln Modeling With Mathematics Skydivers use an instrument called an altimeter to track their altitude as they fall. The altimeter determines altitude by measuring air pressure. The altitude h (in meters) above sea level is related to the air pressure P (in pascals) by the function shown in the diagram. What is the altitude above sea level when the air pressure is 57,000 pascals?

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9 CC Algebra II HW #46 Name Period Row Date Section 6.3 Simplify the epression. (See Eample 5.) ln e 39. Logarithmic Functions Read 6.3 Eamples log Error Analysis Describe and correct the error in rewriting 1 4 = 64 3 in logarithmic form. Find the inverse of the function. (See Eample 6.) 43. y = = log 47. y = ln( 1) y Problem Solving The wind speed s (in miles per hour) near the center of a tornado can be modeled by s = 93 log d + 65, where d is the distance (in miles) that the tornado travels. a. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado. b. Find the inverse of the given function. Describe what the inverse represents.

10 Graph the function. (See Eample 7.) 57. y = log 1 3 Using Tools Use a graphing calculator to graph the function. Determine the domain, range, and asymptote of the function. 63. y = ln( ) (Make a sketch of the graph here.) 71. Critical Thinking Evaluate the logarithm. (Hint: For each logarithm b. log 8 32 log b, rewrite b and as powers of the same base.) Maintaining Mathematical Proficiency Identify the function family to which f belongs. Compare the graph of f to the graph of its parent function. (Section 1.1) 78.

11 CC Algebra II HW #47 Name Period Row Date Transformations of Eponential and Logarithmic Functions Read 6.4 Eamples 1-6 Section 6.4 h 1. Writing Given the function f ( ) = ab + k, describe the effects of a, h, and k on the graph of the function. In Eercises 7 16, describe the transformation of f represented by g. Then graph each function. (See Eamples 1 and 2.) 7. f ( ) = 3, g ( ) = f ( ) = e, g( ) = e + 6 Describe the transformation of f represented by g. Then graph each function. (See Eample 3.) f ( ) = 2, g( ) = 2

12 Describe the transformation of f represented by g. Then graph each function. (See Eample 4.) 27. ( ) = log, g ( ) = 3log 4 5 f 4 Write a rule for g that represents the indicated transformations of the graph of f. (See Eample 5.) 35. f ( ) = 5 ; translation 2 units down, 37. f ( ) = e ; horizontal shrink by a factor of 1 2, followed by a reflection in the y-ais followed by a translation 5 units up Write a rule for g that represents the indicated transformation of the graph of f. (See Eample 6.) 39. ( ) = log ; vertical stretch by a factor of 6, followed by a translation 5 units down f 6 Using Structure Describe the transformation of the graph of f represented by the graph of g. Then give an equation of the asymptote. 47. f ( ) = ln, g ( ) = ln( + 6)

13 CC Algebra II HW #48 Name Period Row Date Section Complete the Sentence To condense the epression Properties of Logarithms Read 6.5 Eamples 1-6 log log3 y, you need to use the Property of Logarithms. 2. Writing Describe two ways to evaluate log 7 12 using a calculator. Use log and log to evaluate the logarithm. (See Eample 1.) 4. log log 7 4 Epand the logarithmic epression. (See Eample 2.) log ln 3y Error Analysis Describe and correct the error in epanding the logarithmic epression. 21. Condense the logarithmic epression. (See Eample 3.) log log ln 2 + 7ln + 4ln y 3

14 y 31. Reasoning Which of the following is not equivalent to log5? Justify your answer. 3 (A) 4log5 y log5 3 (B) 4 log5 y log5 3 + log5 (C) 4log5 y log5 3 log5 4 (D) log5 y log5 3 log5 4 Use the change-of-base formula to evaluate the logarithm. (See Eamples 4 and 5.) 33. log log Thought Provoking Determine whether the formula log b ( M + N) = logb M + logb N is true for all positive, real values of M, N, and b (with b 1 ). Justify your answer.

15 CC Algebra II HW #49 Name Period Row Date Solving Eponential Equations Read 6.6 Eamples 1-2 Section 6.6 In Eercises 5 16, solve the equation. (See Eample 1.) = = = ( ) 49 + = = e = 15

16 17. Modeling With Mathematics The length! (in centimeters) of a scalloped hammerhead shark can be modeled by the function! = e 0. 05t where t is the age (in years) of the shark. How old is a shark that is 175 centimeters long? Use Newton s Law of Cooling to solve the problem. (See Eample 2.) 19. You are driving on a hot day when your car overheats and stops running. The car overheats at 280 F and can be driven again at 230 F. When it is 80 F outside, the cooling rate of the car is r = How long do you have to wait until you can continue driving?

17 CC Algebra II HW #50 Name Period Row Date Solving Logarithmic Equations Read 6.6 Eamples 3-4 Section Writing When do logarithmic equations have etraneous solutions? Solve the equation. (See Eample 1.) = 33 In Eercises 21 32, solve the equation. (See Eample 3.) 21. ln( 4 7) = ln( + 11) 25. log 2 (4 + 8) = = 31. log ( 6) 2 Solve the equation. Check for etraneous solutions. (See Eample 4.) log 2 + log 2 ( 2) = log3 3 + log3 3 = 2

18 43. Problem Solving You deposit $100 in an account that pays 6% annual interest. How long will it take for the balance to reach $1000 for each frequency of compounding: b. quarterly d. continuously Using Tools Use a graphing calculator to solve the equation. Round your answer to the nearest hundredth ln 2 = 3

19 CC Algebra II HW #51 Name Period Row Date Modeling with Eponential and Logarithmic Functions Read 6.7 Eamples 1, 5, 6 Section 6.7 Determine the type of function represented by the table. Eplain your reasoning. (See Eample 1.) Error Analysis Describe and correct the error in determining the type of function represented by the data Using Tools A store sells motorized scooters. The table shows the numbers y of scooters sold during the th year that the store has been open. Use a graphing calculator to find an eponential model for the data. Then use the model to predict the number of motorized scooters sold in the tenth year. (See Eample 5.)

20 33. Using Tools An object at a temperature of 160 C is removed from a furnace and placed in a room at 20 C. The table shows the temperatures d (in degrees Celsius) at selected times t (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form t = a + bln d that represents the data. Estimate how long it takes for the object to cool to 50 C. (See Eample 6.) 35. Drawing Conclusions The table shows the average weight (in kilograms) of an Atlantic cod that is years old from the Gulf of Maine. a. Show that an eponential model fits the data. Then find an eponential model for the data. b. By what percent does the weight of an Atlantic cod increase each year in this period of time? Eplain. 37. Making an Argument Your friend says it is possible to find a logarithmic model of the form d = a + b ln t for the data in Eercise 33. Is your friend correct? Eplain.