Chapter 3 Exponential and Logarithmic Functions. Section a. In the exponential decay model A. Check Point Exercises

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1 Chaptr Eponntial and Logarithmic Functions Sction. Chck Point Erciss. a. A 87. Sinc is yars aftr, whn t, A. b. A A 87 k() k 87 k 87 k k.4 Thus, th growth function is A t.4t.4t A 87..4t 87.4t 87.4t t 8.4 Africa s population will rach million approimatly 8 yars aftr, or 8.. a. In th ponntial dcay modl A A, A substitut for A sinc th amount prsnt aftr 8 yars is half th original amount. A k8 A 8k 8k 8k / k.48 8 So th ponntial dcay modl is.48t A A b. Substitut 6 for A and for A in th modl from part (a) and solv for t..48t 6.48t 6.48t 6.48t 6 6 t 7.48 Th strontium-9 will dcay to a lvl of grams about 7 yars aftr th accidnt.. a. Th tim prior to larning trials corrsponds to t =..8.() f ().4 Th proportion of corrct rsponss prior to larning trials was.4. b. Substitut for t in th modl:.8 f ().7.() Th proportion of corrct rsponss aftr larning trials was.7. c c. In th logistic growth modl, f() t, bt a th constant c rprsnts th limiting siz that f(t) can attain. Th limiting siz of th proportion of corrct rsponss as continud larning trials tak plac is Copyright 4 Parson Education, Inc.

2 Sction. Eponntial Growth and Dcay; Modling Data 4. a. T = C + (T o C) 8 ( ) k 8 7 k 7 k k 7 k 7 k 7 7 k.67 k T 7.67t b. T = () 48 o Aftr minuts, th tmpratur will b 48 o. c. = t 7.67t.67t 4.67t 4.67t 4 4 t.67 9 t Th tmpratur will rach o aftr 9 min.. A logarithmic function would b a good choic for modling th data. 6. An ponntial function would b a good choic for modling th data although modl choics may vary. 7. a. 97 is yars aftr 949. f( ) f ().74().94.8 g ( ).77(.7) g().77(.7).7 Th ponntial function g srvs as a bttr modl for b. is yars aftr 949. f( ) f ().74() g ( ).77(.7) g().77(.7) 4. Th linar function f srvs as a bttr modl for. y 4(7.8) ( 7.8) 4 Roundd to thr dcimal placs: ( 7.8) y Concpt and Vocabulary Chck.. ;. A ; A. A; c 4. logarithmic. ponntial 6. linar 7. Copyright 4 Parson Education, Inc. 4

3 Chaptr Eponntial and Logarithmic Functions Ercis St.. Sinc is yars aftr, find A whn t :.6t A 7..6 A 7. A 7. A 7. A 7. In, th population of Japan was 7. million.. Sinc is yars aftr, find A whn t :.9t A..9 A. A. A. A. In, th population of Iraq was. million.. Sinc k.9, Iraq has th gratst growth rat at.9% pr yar. 4. Sinc k is ngativ for both Japan and Russia, thy hav dcrasing populations. Th population of Japan is dropping at a rat of.6% pr yar. Th population of Russia is dropping at a rat of.% pr yar.. Substitut A 77 into th modl for India and solv for t: t.8t.8t.8t 77 t 7..8 Th population of India will b 77 million approimatly yars aftr, or. 6. Substitut A 49 into th modl for India and solv for t: t.8t.8t.8t 49 t 7..8 Th population of India will b 49 million approimatly yars aftr, or a. A 6.4. Sinc is yars aftr, whn t, A. b. A A 6.4 k() k 6.4 k 6.4 k k. Thus, th growth function is.t.t A t t t t 4. Now, + 4 = 4, so th population will b 9 million is approimatly th yar Copyright 4 Parson Education, Inc.

4 Sction. Eponntial Growth and Dcay; Modling Data 8. a. A.. Sinc is yars aftr, whn t, A. b. A A. k() k. k. k.. k.6 Thus, th growth function is.6t.6t A t 9..6t t. 9. t 4.6 Now, + 4 = 4, so th population will b 9 million is approimatly th yar 4.. P ( ) k 6.9 4k k k k 4 k.88 Th growth rat is.88.. P ( ) k 4.7 4k k k k 4 k.74 Th growth rat is t P ( ) (4) P(4) (4) P(4) Th population is projctd to b 46. million in..49t P ( ) (4) P(4) (4) P(4) Th population is projctd to b 4.7 million in.. P ( ) k 7. 4k k k k 4 k.9 Th growth rat is.9. Copyright 4 Parson Education, Inc. 47

5 Chaptr Eponntial and Logarithmic Functions 4. P ( ) k.4 4k k k k 4 k.68 Th growth rat is.68. A 6.t. 7 A 6.69 A 6 A 8. Approimatly 8 grams of carbon-4 will b prsnt in 7 yars. 8. Aftr, yars, thr will b 6 8 grams prsnt. Aftr, yars, thr will b 8 4 grams prsnt. Aftr 7, yars, thr will b 4 grams prsnt. Aftr, yars, thr will b gram prsnt. Aftr, yars, thr will b gram prsnt. 9.. A A.t.t.t.t..t. t,679. Th paintings ar approimatly,679 yars old. 6. A 6.t. 4 A 6.8 A 6 A 4. Approimatly 4 grams of carbon-4 will b prsnt in,4 yars. 7. Aftr sconds, thr will b 6 8 grams prsnt. Aftr sconds, thr will b 8 4 grams prsnt. Aftr sconds, thr will b 4 grams prsnt. Aftr 4 sconds, thr will b grams prsnt. Aftr sconds, thr will b gram prsnt... A A t.t.t.t.88.t.88 t 6. In 989, th skltons wr approimatly 6 yars old...t..t...t. t. t.6 Th half-lif is.6 yars. 48 Copyright 4 Parson Education, Inc.

6 Sction. Eponntial Growth and Dcay; Modling Data t..6t..6t. t.6 t. Th half-lif is. yars... 6k. 6k. 6k. k 6 k.48 Th dcay rat is.48% pr yar... 46k. 46k. 46k. k 46 k. Th dcay rat is.% pr yar... 7.k. 7.k. 7.k. k 7. k.968 Th dcay rat is.968% pr day... k. k. k. k k.64 Th dcay rat is.64% pr hour. 7. a. k..k.k k.9. Th ponntial modl is givn by.9t A A. b..9t A A.9t.94A A.9t.94.9t t.94 t.69.9 Th ag of th dinosaur ons is approimatly.69 billion or 6,9, yars old. 8. First find th dcay quation.. 74k. 74k.. 74k. k 74 k.94.94t A Nt us th dcay quation answr qustion..94t A.94t..94t...94t. t.94 t 7.7 It will tak 7.7 yars.. For gratr accuracy, us k. 74. A 74 t givs 74. yars Copyright 4 Parson Education, Inc. 49

7 Chaptr Eponntial and Logarithmic Functions 9. First find th dcay quation... k. k. k. k k.7.7t A Nt us th dcay quation answr qustion. A.8.8.7t.7t.7t.8.7t.8 t.7 t 7. It will tak 7. yars.. First find th dcay quation... 6k. 6k. 6k. k 6 k.94.94t A Nt us th dcay quation answr qustion. A t.94t.94t.9.94t.9 t.94 t. It will tak. hours.. First find th dcay quation... k. k. k. k k.776 A.776t. A A 4 k 4 k k 7 k 7.67 Th ponntial modl is givn by.67t A A. Nt us th dcay quation answr qustion. A t.776t.776t.7.776t.7 t.776 t 6. It will tak 6. hours..67t.67t.67t.67 t t Th population will drop blow birds approimatly 4 yars from now. (This is 9 yars from th tim th population was 4.) 46 Copyright 4 Parson Education, Inc.

8 Sction. Eponntial Growth and Dcay; Modling Data A A t k Th population will doubl in t yars. k A A A A t k Th population will tripl in t yars. k A 4..t a. k., so Nw Zaland s growth rat is %. b..t A 4..t t.t.t t 69. Nw Zaland s population will doubl in approimatly 69 yars. A..t a. k., so Mico s growth rat is.%. b..t A..t...t.t.t t 8. Mico s population will doubl in approimatly 8 yars. 7. a. Whn th pidmic bgan, t =., f () Twnty popl bcam ill whn th pidmic bgan , f 4, 8 About 8 popl wr ill at th nd of th fourth wk. b. 4 c. In th logistic growth modl, c f() t, bt a th constant c rprsnts th limiting siz that f(t) can attain. Th limiting siz of th population that bcoms ill is, popl..7 f( ).6( ) 4..7 f () 6..6() 4. Th function modls th data quit wll..7 f( ).6( ) 4..7 f (6) 6.8.6(6) 4. Th function modls th data quit wll..7 f( ) ( ) ( ) ( ) ( ) ( ) ( ).6( ).6 6 Th world population will rach 7 billion 6 yars aftr 949, or. Copyright 4 Parson Education, Inc. 46

9 Chaptr Eponntial and Logarithmic Functions 4..7 f( ) ( ) ( ) ( ) ( ) ( ) ( ).6( ) t. 7.9.t 7.8.t 7.8.t.8 7.t t 7.t Th world population will rach 8 billion 76 yars aftr 949, or..7 f( ).6( ) 4. As incrass, th ponnt of will dcras. This.6( ) will mak bcom vry clos to and mak th dnominator bcom vry clos to. Thus, th limiting siz of this function is.7 billion. 9.() P().7 7 Th probability that a -yar-old has som coronary hart disas is about.7%. 9.(8) P(8) Th probability that an 8-yar-old has som coronary hart disas is about 88.6% t 48. Th probability of som coronary hart disas is % at about ag ( 7. ) Th probability of som coronary hart disas is 7% at about ag a. = 4 + (7 4) k k k k k k.96 k T 4.96t 46 Copyright 4 Parson Education, Inc.

10 Sction. Eponntial Growth and Dcay; Modling Data b. T = () o Aftr minuts, th tmpratur will b o. c. = t.96t.96t.96t.96t t.96 8 t Th tmpratur will rach o aftr 8 min. 48. a. T = C + (T o C) 7 (4 7) k 8 k k 8 k 8 k 8 8 k.4 k T 7 8.4t b. T = 7 + 8k -.4() o Aftr minuts, th tmpratur will b o. c. 4 = t 7 8.4t 7.4t 8 7.4t 8 7.4t t.4 7 t Th tmpratur will rach 4 o aftr 7 min. 49. T = C + (T o C) 8 7 (8 7) k 7 47 k 7 k 47 7 k 47 7 k k.9 k T t t 47.9t.9t 47.9t 47.9t t.9 6 t Th tmpratur will rach o aftr 6 min.. T = C + (T o C) 6 (4 6) k 4 k k 4 k 4 k 4 4 k.8 k T 6 4.8t 4 = 6 4.8t Copyright 4 Parson Education, Inc. 46

11 Chaptr Eponntial and Logarithmic Functions 4.8t.8t 4.8t 4.8t 4 4 t.8 4 t Th tmpratur will rach 4 o aftr 4 min.. a. Scattr plot: 4. a. Scattr plot: b. A logarithmic function appars to b th bst choic for modling th data.. a. Scattr plot: b. An ponntial function appars to b th bst choic for modling th data.. a. Scattr plot: b. A linar function appars to b th bst choic for modling th data. 6. a. Scattr plot: b. An ponntial function appars to b th bst choic for modling th data.. a. Scattr plot: b. A linar function appars to b th bst choic for modling th data. 7. y = (4.6) is quivat to 4.6 y ; Using 4.6.6,.6 y. b. A logarithmic function appars to b th bst choic for modling th data. 464 Copyright 4 Parson Education, Inc.

12 Sction. Eponntial Growth and Dcay; Modling Data 8. y = (7.) is quivat to 7. y ; Using ,.988 y. 9. y =.(.7) is quivat to.7 y. ; Using.7.7,.7 y.. 6. y = 4.(.6) is quivat to.6 y 4. ; Using.6.,. y Answrs will vary. 7. a. Th ponntial modl is y.(.). Sinc r.999 is vry clos to, th modl fits th data wll. b. y... y..9 y. Sinc k.9, th population of th Unitd Stats is incrasing by about % ach yar. 7. Th logarithmic modl is y Sinc r =.87 is fairly clos to, th modl fits th data okay, but not grat. 7. Th linar modl is y Sinc r.997 is clos to, th modl fits th data wll.. 7. Th powr rgrssion modl is y 9.. Sinc r =.896, th modl fits th data fairly wll. 74. Using r, th modl of bst fit is th ponntial modl y... Th modl of scond bst fit is th linar modl y Using th ponntial modl: Using th linar modl: y According to th ponntial modl, th U.S. population will rach million around th yar 6. According to th linar modl, th U.S. population will rach million around th yar. Both rsults ar rasonably clos to th rsult found in Eampl (). Eplanations will vary. 7. a. Eponntial Rgrssion: y.46(.) ; r.994 Logarithmic Rgrssion: y ; r.67 Linar Rgrssion: y.7.97; r.947 Th ponntial modl has an r valu closr to. Thus, th bttr modl is y.46(.). b. y.46(.). y.46. y.46 Th 6-and-ovr population is incrasing by approimatly % ach yar. Copyright 4 Parson Education, Inc. 46

13 Chaptr Eponntial and Logarithmic Functions 76. Modls and prdictions will vary. Sampl modls ar providd Ercis 47: y.4.78 Ercis 48: y Ercis 49: y 4.4 Ercis : y Ercis : y Ercis : y dos not mak sns; Eplanations will vary. Sampl planation: Sinc th car s valu is dcrasing (dprciating), th growth rat is ngativ. 78. dos not mak sns; Eplanations will vary. Sampl planation: This is not ncssarily so. Growth rat masurs how fast a population is growing rlativ to that population. It dos not indicat how th siz of a population compars to th siz of anothr population. 79. maks sns 8. maks sns 8. tru 8. tru 8. tru 84. tru 8. Us data to find k (8.6 7) k.7.6 k.7 k.6.7 k.6.7 k k.69 k Us k to writ quation (98.6 7).69t t.6.69t t t t Th dath occurrd at 88 minuts bfor 9:, or 8: am. 86. Answrs will vary π π 4 π π 4 π π 8 Th solution st is. 8 7π 7π π π π π 6 π 6 π π 4π π π 4π π 466 Copyright 4 Parson Education, Inc.

Chaptr Rviw Erciss Chaptr Rviw Erciss. This is th graph of f( ) 4 rflctd about th y- ais, so th function is g ( ) 4. 7. Th graph of g() rflcts th graph of f() about th y ais.

14 Chaptr Rviw Erciss Chaptr Rviw Erciss. This is th graph of f( ) 4 rflctd about th y- ais, so th function is g ( ) Th graph of g() rflcts th graph of f() about th y ais.. This is th graph of f( ) 4 rflctd about th - ais and about th y-ais, so th function is h ( ) 4.. This is th graph of f( ) 4 rflctd about th - ais and about th y-ais thn shiftd upward units, so th function is r ( ) This is th graph of f( ) 4.. Th graph of g() shifts th graph of f() on unit to th right. asymptot of f: y asymptot of g: y domain of f = domain of g =, rang of f =, rang of g =, 8. Th graph of g() rflcts th graph of f() about th - ais. asymptot of f: y asymptot of g: y domain of f = domain of g =, rang of f = rang of g =, 6. Th graph of g() shifts th graph of f() on unit down. asymptot of f: y asymptot of g: y domain of f = domain of g =, rang of f = rang of g =, 9. Th graph of g() vrtically strtchs th graph of f() by a factor of. asymptot of f: y asymptot of g: y domain of f = domain of g =, rang of f =, rang of g =, asymptot of f: y asymptot of g: y domain of f = domain of g =, rang of f = rang of g =, Copyright 4 Parson Education, Inc. 467

15 Chaptr Eponntial and Logarithmic Functions..% compoundd smiannually:. A 68.6.% compoundd monthly:. A % compoundd smiannually yilds th gratr rturn.. 7% compoundd monthly:.7 A 4, 8,.6 6.8% compoundd continuously:.68 A4, 7,77.8 7% compoundd monthly yilds th gratr rturn.. a. Whn first takn out of th microwav, th tmpratur of th coff was. b. Aftr minuts, th tmpratur of th coff was about..48 T 7 9. Using a calculator, th tmpratur is about 9. c. Th coff will cool to about 7 ; Th tmpratur of th room is 7. b b 4. Bcaus log, w conclud log 8.. Bcaus, w conclud. 8 log log log 8. log log log 9. Bcaus logb, w conclud log8 8. So, log (log88) log. Bcaus logb w conclud log. Thrfor, log (log88) / y 8 6. log logb 6 4 domain of f = rang of g =, rang of f = domain of g =, 8. log 874 y 9. log 64 bcaus log bcaus.. log 9 is undfind and cannot b valuatd sinc log b is dfind only for. /. log6 4 bcaus domain of f = rang of g =, rang of f = domain of g =,. Bcaus logb b, w conclud log Copyright 4 Parson Education, Inc.

This is th graph of f( ) log shiftd lft units thn rflctd about th y-ais, so th function is h ( ) log( ).. This is th graph of f log. 8. 9. -intrcpt: (, ) vrtical asymptot: = domain:, rang:, 6.

16 Chaptr Rviw Erciss. This is th graph of f log rflctd about th y-ais, so th function is g( ) log( ).. This is th graph of f log shiftd lft units, rflctd about th y-ais, thn shiftd upward on unit, so th function is r ( ) log( ). 4. This is th graph of f( ) log shiftd lft units thn rflctd about th y-ais, so th function is h ( ) log( ).. This is th graph of f log intrcpt: (, ) vrtical asymptot: = domain:, rang:, 6. -intrcpt: (, ) vrtical asymptot: = domain:, rang:, asymptot of f: asymptot of g: domain of f =, domain of g =, rang of f = rang of g =, -intrcpt: (, ) vrtical asymptot: = domain:, rang:, asymptot of f: asymptot of g: domain of f = domain of g =, rang of f = rang of g =, 4. Th domain of f consists of all for which. Solving this inquality for, w obtain. Thus th domain of f is, Copyright 4 Parson Education, Inc. 469

17 Chaptr Eponntial and Logarithmic Functions 4. Th domain of f consists of all for which. Solving this inquality for, w obtain. Thus, th domain of f is,. 4. Th domain of f consists of all for which ( ). Solving this inquality for, w obtain or. Thus, th domain of f is (,) (, ) Bcaus, w conclud Bcaus log, w conclud. log Bcaus, w conclud I R log log I Th Richtr scal magnitud is a. f 768log 76 Whn first givn, th avrag scor was 76. () 768log 67 (4) 768log 4 6 (6) 76 8log 6 6 (8) 76 8log log 6 b. f f f f f c. Aftr, 4, 6, 8, and months, th avrag scors ar about 67, 6, 6, 9, and 6, rspctivly.. log log66 log6 log66 log6 log 6 log 4 log4 log log 4 y y log log log log log y log64 log log y6 4. logb 7 logb log b (7 ) log. b log log log log log 6. 4 y 4 y 4 y 49. Rtntion dcrass as tim passs. t It will tak about 9 wks. 7. y y y 47 Copyright 4 Parson Education, Inc.

18 Chaptr Rviw Erciss log7,48 8. log6 7, log log tru; ( )() ( )() 6. fals; 6. fals; ( 9) log( 9) log( ) log ( ) log 4 4log 6. tru; log log 7 log log 7 log , ,4 8 =,4, = = =.4 4 7, 4 7, 4 7, 4 7, 4 7, 7, ( 4) () (7) Copyright 4 Parson Education, Inc. 47

19 Chaptr Eponntial and Logarithmic Functions 7. 6 or ( ) ( ).99 ( ) dos not ist. Th solution st is, approimatly.. log4 = 4 = 64 = 69 = Th solutions st is {} ( ) 4( ) ( ).4 Th solutions st is. 76. log log 4 log ( )( ) 4 4 log ( 9) = ± = dos not chck bcaus log ( ) dos not ist. Th solution st is {}. 77. log log log 9 = 9( + ) = = dos not chck bcaus log 8 8 dos not ist. Th solution st is ( ) dos not chck and must b rjctd. Th solution st is {} log4 log4 log4 log log log log log dos not chck and must b rjctd. Th solution st is {4}. P t Th pak of Mt. Evrst is about. mils abov sa lvl. 47 Copyright 4 Parson Education, Inc.

20 Chaptr Rviw Erciss 8. St.4.6 t t t t t t This modl projcts that 7.9 million smartphons will b sold 9 yars aftr 4, or. 8. a. f This modl projcts that opposition to homosual rlationships among first-yar collg womn will diminish to 6% yars aftr 979, or. b. g This modl projcts that opposition to homosual rlationships among first-yar collg mn diminish to 4% yars aftr 979, or t.6,, 4 4t,(.6), 4t (.6).6 4t (.6).6 4t t It will tak about 7. yars..7t,,.7t,,.7.7t.7t t It will tak about 4.6 yars. 8. Whn an invstmnt valu tripls, A P. r P P r r r r.97 Th intrst rat would nd to b about % 86. a. k... k.. k.. k... k.6 k A..6t b..6() A. 6.6 In, th population will b about 6.6 million. Copyright 4 Parson Education, Inc. 47

21 Chaptr Eponntial and Logarithmic Functions c..6t t.6t.6t t t Th population will rach 7 million about 9 yars aftr, in Us th half-lif of 4 days to find k. A A k4 4k 4k 4k k 4 k.49 Us A A to find t..49t A A.49t..49t...49t. t.49 t It will tak about days for th substanc to dcay to % of its original amount. 89. a. T = C + (T o C) 6 (8 6) k 9 k 9 k k 4 k 4 4 k.48 k T 6.48t b. = t 4.48t.48t.48t.48t t t Th tmpratur will rach o aftr 8 min. 9. a. Scattr plot: 88. a. b.,.9() f () 499 popl bcam ill whn th pidmic bgan., f (6) 4, (6) 4,4 wr ill aftr 6 wks. b. A linar function appars to b th bttr choic for modling th data. c., popl 474 Copyright 4 Parson Education, Inc.

22 Chaptr Tst 9. a. Scattr plot:. b. An logarithmic function appars to b th bttr choic for modling th data. 9. a. Scattr plot:. log Th domain of f consists of all for which. Solving this inquality for, w obtain. Thus, th domain of f is (,) log 64 log 64 log log 4 b. An ponntial function appars to b th bttr choic for modling th data. 9. y 7.6 y 7 y y 6..4 y 6. y log log log 8 8 log 4 6 6log log y log log y 6 log y Answrs will vary. log7. log 7.74 log Chaptr Tst Copyright 4 Parson Education, Inc. 47

23 Chaptr Eponntial and Logarithmic Functions or.694 Th solution st is, ; log log log log or = = = dos not chck bcaus log( ) dos not ist. Th solution st is {} ( ) dos not chck and must b rjctd. Th solution st is. I D log I log Th loudnss of th sound is dcibls.. Sinc,.. logb b bcaus b b.. log6 bcaus % compoundd smiannually:.6 A, $,687. 6% compoundd continuously:.6 A, $, % compoundd smiannually yilds about $ mor than 6% compoundd continuously. 476 Copyright 4 Parson Education, Inc.

24 Chaptr Tst 4. 4t t. 4 4t. 4t. 4t 4t t.9 4. It will tak approimatly.9 yars for th mony to grow to $8.. r r r r r.69 Th mony will doubl in yars with an intrst rat of approimatly 6.9%. 6. a..4 A 8..4 A8. 8. In, th population of Grmany was 8. million. b. Th population of Grmany is dcrasing. W can tll bcaus th modl has a ngativ k.4. c..t t t t t 8..4 Th population of Grmany will b 79. million approimatly yars aftr in th yar. 7. In, t = and A = 4 In, t = = 4 and A = k 4k k 4k k 4 k Th ponntial growth function is A 4.6t. 8. First find th dcay quation.. 7.k. 7.k.. 7.k. k 7. k t A Nt us th dcay quation answr qustion..967t A.967t..967t...967t. t.967 t. It will tak. days. 4 f 4 9 Fourtn lk wr initially introducd to th habitat. 9. a..6 4 f 9 Aftr yars, about lk ar pctd. b..6 c. In th logistic growth modl, c f() t, bt a th constant c rprsnts th limiting siz that f(t) can attain. Th limiting siz of th lk population is 4 lk. Copyright 4 Parson Education, Inc. 477

25 Chaptr Eponntial and Logarithmic Functions. Plot th ordrd pairs. y Th valus appar to blong to a linar function.. Plot th ordrd pairs. y 7 Th valus appar to blong to a logarithmic function.. Plot th ordrd pairs. y 9 Cumulativ Rviw Erciss (Chaptrs P ). 4 4 = or 4 = = 6 = = Th solution st is,.. b b 4ac a () () 4()() () 6 4i i Th solution st is i. Th valus appar to blong to an ponntial function.. Plot th ordrd pairs. 6 y Th valus appar to blong to a quadratic function. 4. y 96.8 y 96 y p: ±, ± q: ± p :, q ( )( ) ( )[ ( ) ( )] ( )( )( ) ( )( )( )( ) ( )( )( ) or or Th solution st is {,, }. 478 Copyright 4 Parson Education, Inc.

Cumulativ Rviw 4. 96 8 8 8 8.974 8 Th solution st is, approimatly.974. 9. Circl with cntr: (, ) and radius of.

Th solution st is {}.. 6. 4 4 4 4 69 4 69 dos not chck and must b rjctd. Th solution st is. 7. 4 6 4 Th solution st is,4. 8.

26 Cumulativ Rviw Th solution st is, approimatly Circl with cntr: (, ) and radius of. Parabola with vrt: (, ) log ( ) log ( ) 4 log [( )( )] 4 4 ( )( ) 46 4 ( 7)( ) 7 or 7 = 7 dos not chck bcaus log ( 7 ) dos not ist. Th solution st is {} dos not chck and must b rjctd. Th solution st is Th solution st is, and 4 6 and Th solution st is,.. -intrcpts: Th -intrcpts ar, and,. vrtical asymptots: 4 4 Th vrtical asymptots ar = and =. Horizontal asymptot: y =. -intrcpts: or or Th -intrcpts ar, and,. Copyright 4 Parson Education, Inc. 479

27 Chaptr Eponntial and Logarithmic Functions. 4. ( ) 6. m Using (, ) point-slop form: y slop-intrcpt form: y ( ) y y 6 6. f g f g f g y varis invrsly as th squar of is prssd as k y. Th hours, H, vary invrsly as th squar of th numbr of cups of coff, C can b prssd k as H. C Us th givn valus to find k. k H C k 8 k Substitut th valu of k into th quation. k H C H C Us th quation to find H whn C 4. H C H 4 H If 4 cups of coff ar consumd you should pct to slp hours. st () 6t 64t Th ball rachs its maimum hight at b (64) t sconds. a ( 6) Th maimum hight is s (). s() 6() 64() 69 ft. st () 6t 64t Lt st () : 6t 64t Us th quadratic formula to solv. b b 4ac t a (64) (64) 4( 6)() t ( 6) t 4., t. Th ngativ valu is rjctd. Th ball hits th ground aftr about 4. sconds Your normal hourly salary is $ pr hour. 48 Copyright 4 Parson Education, Inc.

CHAPTER 5. Section 5-1

CHAPTER 5. Section 5-1 SECTION 5-9 CHAPTER 5 Sction 5-. An ponntial function is a function whr th variabl appars in an ponnt.. If b >, th function is an incrasing function. If < b