Lesson 5.1 Exponential Functions

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1 Lesson.1 Eponential Functions 1. Evaluate each function at the given value. Round to four decimal places if necessar. a. r (t) 2(1 0.0) t, t 8 b. j() 9.(1 0.09), Record the net three terms for each sequence. Then write an eplicit function for the sequence. a. a 0 12 b. u 0 0. a n 0.8a n 1 where n 1 u n 2.1u n 1 where n 1. Evaluate each function at 0, 1, and 2 and write a recursive formula for the pattern. Then, indicate whether each equation is a model for eponential growth or deca. a. f () 2000(0.9) b. f () 000( ) c. f () 0.1(1 0.) 4. Calculate the ratio of the second term to the first term, and epress the answer as a decimal value. State the percent increase or decrease. a. 80, 60 b. 6, 2 c. 6, Rohit bought a new car for $17,00. The value of the car is depreciating at a rate of 16% a ear. a. Write a recursive formula that models this situation. Let u 0 represent the purchase price, u 1 represent the value of the car after 1 ear, and so on. b. Define variables and write an eponential equation that models this situation. Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

2 Lesson.2 Properties of Eponents and Power Functions 1. Rewrite each epression as a fraction without eponents or as an integer. Using our calculator, verif that our answer is equivalent to the original epression. a. 2 b. 7 c. 4 4 d. ( ) e. 2 f Rewrite each epression in the form n or a n. a b c. 9 9 d e f Solve. a b c Solve each equation for positive values of. If answers are not eact, approimate to two decimal places. a b c d e f CHAPTER Discovering Advanced Algebra More Practice Your Skills 2010 Ke Curriculum Press

3 Lesson. Rational Eponents and Roots 1. Identif each function as a power function, an eponential function, or neither of these. (The function ma be translated, stretched, or reflected.) a. f () 0. 4 b. f () 1 c. f () Rewrite each epression in the form b in which is a rational eponent. a. c b. d 4 c. 1 r. Solve each equation for positive values of. If answers are not eact, approimate to the nearest hundredth. a. 27 b c d. 2 1 e f Each of the following graphs is a transformation of the power function 2. Write the equation for each curve. c. a. e. d. b. f. Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

4 Lesson.4 Applications of Eponential and Power Equations 1. Solve each equation for positive values of. If answers are not eact, approimate to the nearest hundredth. a. 2.6 b c d e f Rewrite each epression in the form a n. a b c d e. 100, f Give the average annual rate of inflation for each situation described. Give our answers to the nearest tenth of a percent. a. The cost of a movie ticket increased from $6.00 to $8.0 over 10 ears. b. The monthl rent for Hector s apartment increased from $60 to $77 over 4 ears. 4. The population of a small town has been declining because jobs have been leaving the area. The population was 2,000 in 2002 and 18,70 in Assume that the population is decreasing eponentiall. a. Define variables and write an equation that models the population in this town in a particular ear. b. Use our model to predict the population in c. According to our model, in what ear will the population first fall below 12,000? 0 CHAPTER Discovering Advanced Algebra More Practice Your Skills 2010 Ke Curriculum Press

5 Lesson. Building Inverses of Functions 1. Each of the functions below has an inverse that is also a function. Find four points on the graph of each function f, using the given values of. Use these points to find four points on the graph of f 1. a. f () 4; 2, 0, 4, 4 b. f () 2;, 1, 2, 2. For each function below, determine whether or not the inverse of the function is a function. Find the equation of the inverse and graph both equations on the same aes. a. 2 b. c Balloons and Laughs Inc. is a small compan that entertains at children s birthda parties. B & L uses a complicated formula to calculate its prices, taking into account all of its costs. The price equation is p() 4 (8 ) 2 2, where is the number of person-hours supplied for the part at a price of p(). For eample, if 4, four clowns will come for one hour, two clowns will come for two hours, or one clown will come for four hours. a. What is the price if two clowns come to a part for 90 minutes? b. Man customers want to know what the can get for a particular amount of mone. Rewrite the price equation for B & L so that the compan can input the amount of mone a customer wants to spend and the output will be the number of person-hours he or she will get for the mone. Call the new function p 1 (). c. B & L s Ultimate Part costs $12. How man person-hours do ou get at an Ultimate Part? Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

6 Lesson.6 Logarithmic Functions 1. Rewrite each logarithmic equation in eponential form using the definition of logarithm. Then solve for. a. log 1 81 b. log c. log d. log 1 e. log 12 f. log Find the eact value of each logarithm without using a calculator. Write answers as integers or fractions in lowest terms. a. log 81 b. log c. log 1 d. log e. log 8 4 f. log 1,000,000,000. Each graph is a transformation of either 10 or log. Write the equation for each graph. a. b. c. 4. Use the change-of-base propert to solve each equation. (Round to four decimal places.) a. log 120 b. log 0.9 c d e f CHAPTER Discovering Advanced Algebra More Practice Your Skills 2010 Ke Curriculum Press

7 Lesson.7 Properties of Logarithms 1. Use the properties of logarithms to rewrite each epression as a single logarithm. a. log 21 log 7 b. 4 log 2 c. 2 log 4 log 2. Write each epression as a sum or difference of logarithms (or constants times logarithms). Simplif the result if possible. a. log a b c 4 b. log 4 _ r _ s 4 t c. log abc 4. Determine whether each equation is true or false. log 2 a. log 8 b. log log 4 9 log 2 log 4. c. log 1 1 log d. log 4 2 log 8 e. log 1 log f. log log 2 9 g. log 6 2 log 6 h. log 1 log 1 4. Change the form of each epression below using definitions or properties of logarithms or eponents. Name each definition or propert ou use. a. log r log s b. 1 a b c. q a b d. log b m e. (cd ) m f. log b g. r _ s m h. c m n i. log a log a Discovering Advanced Algebra More Practice Your Skills CHAPTER 2010 Ke Curriculum Press

8 Lesson.8 Applications of Logarithms 1. Solve each equation. Round answers that are not eact to four decimal places. a. 19,68 b. 9.(8 ) 220 c (0.72) 2. Suppose that ou invest $,000 in a savings account. How long would it take ou to double our mone under each of the following conditions? a. % interest compounded annuall b..6% interest compounded monthl. The Richter scale rating of the magnitude of an earthquake is given b the formula log I I 0, where I 0 is a certain small magnitude used as a reference point. (Richter scale ratings are given to the nearest tenth.) a. A devastating earthquake, which measured 7.4 on the Richter scale, occurred in western Turke in Epress the magnitude of this earthquake as a multiple of I 0. b. Another earthquake occurred in 1998, centered in Adana, Turke. This earthquake measured 6. on the Richter scale. Epress the magnitude of this earthquake as a multiple of I 0. c. Compare the magnitudes of the two earthquakes. 4. The population of an animal species introduced into an area sometimes increases rapidl at first and then more slowl over time. A logarithmic function models this kind of growth. Suppose that a population of N deer in an area t months after the deer are introduced is given b the equation N 2 log (4t 2). How long will it take for the deer population to reach 800? Round to the nearest whole month. 4 CHAPTER Discovering Advanced Algebra More Practice Your Skills 2010 Ke Curriculum Press

9 LESSON 4.8 Compositions of Functions 1. a. {, 2, 1, 1,, } b. { 2, 1, 0} c. 1 d. 1 e. 6 f a. (2 2) b. 260 c. 8 d. a 4 4a 2 4. a. f () 1.06 b. g () 0.1 c. g (f ()) 0.1(1.06) 0.19 d. Marla s wa: $7.9; Shamim s wa: $7.0 LESSON.1 Eponential Functions 1. a. r (8) b. j (10) a. a 1 9.6, a , a 6.144; 12(0.8) b. u , u , u ; 0.(2.1). a. f (0) 2000, f (1) 1800, f (2) 1620; u , u n 0.9u n 1 where n 1; eponential deca b. f (0) 000, f (1) 00, f (2) ; u 0 000, u n 1.001u n 1 where n 1; eponential growth c. f (0) 0.1, f (1) 0.0, f (2) 0.02; u 0 0.1, u n 0.u n 1 where n 1; eponential deca 4. a. 0.7; decrease of 2% b. 0. _ 8 ; decrease of 11. _ 1 % c. 1.6; increase of 60%. a. u , u n 0.84u n 1 where n 1 b. Let represent the number of ears after the car was purchased and represent the value of the car. 17,00(0.84) LESSON.2 Properties of Eponents and Power Functions 1. a. 1 9 d b. 1 4 e. 2 9 c f a. b. 2 c a..62 b. 0.9 c d e. 1. f LESSON. Rational Eponents and Roots 1. a. Power b. Eponential c. Power 2. a. c 2 b. d 4 c. r 2. a. 24 b c d. 2 e f a. 2 b. 2 c. ( 2) 2 d. 4 ( ) 2 e. 2 2 f. 0.( ) 2 LESSON.4 Applications of Eponential and Power Equations 1. a b. 62 c d e. 0.0 f a. 4 6 b c. 7 d e f.. a..% b..9% 4. a. Let t represent the ear and P represent the population. P 2,000(0.96 ) t 2002 b. 16,92 c LESSON. Building Inverses of Functions 1. a. f : ( 2, 10), (0, 4), 4, 0, (4, 8); f 1 : ( 10, 2), ( 4, 0), 0, 4, (8, 4) b. f : (, 29), ( 1, ), (2, 6), (, 12); f 1 : ( 29, ), (, 1), (6, 2), (12, ) 2. a. Function; b. Not a function; a. 6 8 b c. 18 d e f. 12 1, or ANSWERS Discovering Advanced Algebra More Practice Your Skills 2010 Ke Curriculum Press

10 c. Not a function; 4. a. $61 b. p 1 () c. 1 1_ 4 person-hours (or 1 person-hours if fractional person-hours are not allowed) LESSON.6 Logarithmic Functions 1. a b ; 12 c. 4 2; 2. d ; 10 e. 12; f ; a. 4 b. 1 2 c. 1 d. e. 2 f. 9. a. 10 b. log( 1) c a b c..147 d e f LESSON.7 Properties of Logarithms 1. a. log b. log a. log a 2 1 log b 4 log c b. 1 2 log 4 r 1 log 4 s 4 log t c. log 2 c. 1 log a 1 log b 1 log c 1 4 log. a. False b. True c. False d. True e. True f. True g. False h. True 4. a. log r_ s ; quotient propert of logarithms b. a b ; definition of negative eponents c. q a q b ; product propert of eponents d. m log b ; power propert of logarithms e. c m d m ; power of a product propert f. log b log b ; product propert of logarithms g. r m s m ; power of a quotient propert n h. c m ; definition of rational eponents i. log ; change-of-base propert LESSON.8 Applications of Logarithms 1. a. 9 b c a r b. 19. r. a. 2,118,864 I 0 b. 1,99,262I 0 c. The intensit of the 1999 earthquake was about 12.6 times as great as that of the 1998 earthquake ears (72 months) LESSON 6.1 Matri Representations 1. a b Voter c a. (, ), (2, 0), (4, 6) (, ) b (2, 0) (4, 6) 1 6 c. 1 1 LESSON 6.2 Matri Operations 1. a. 12, 0, z Nonvoter b. n 11 19, n 12 26, n 21 9., n a b a c. Impossible because the inside dimensions aren t the same. d ( 1, ) b. 2 ( 2, ) (2, 0) 1 Discovering Advanced Algebra More Practice Your Skills ANSWERS Ke Curriculum Press

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of