Graphing Exponential Functions and Their Inverses

Day 6-1: Graphing Exponential Functions and Their Inverses Exponential Function - а function where the base is а constant and the exponent is an independent variaЬ!e (ех. у= 5 х ) Asymptote - а line that the graph of an exponential function approaches Exponential Curves - Equation with а variaЬ!e in the exponent. - Set up а tаЬ!е of values or use the graphing calculator. � (\(Wtf 1""'-'С� <, What the graph will look like: - When х is the exponent, the curve will Ье in Quadrants I and II. - The curve will go through the point (О, 1). - The curve will never touch the x-axis. - If an interval is given, the graph will not have aпows. If по interval is given, the graph will have arrows. 1) Which of the following is not in the domain of у= zx ? (2) -2 (1) 1 � They are all in the domain. (3) 3 2) Which of the following is not in the range of у= zx ? (3) 3 (4) Тhеу are all in the range. (1) 1 � -2 3) Graph у = 3" in the interval -2 S х S 2. . у ," �,, 'у �� 3Л>< х 1 �- d- .\ _\ 1) __\ ,..,,,, J ----- � • \ q / - / х - - - -r ,- Day 6-1: Grapblng Exponential Functions and Their Inverses Graph у= 2' + 1. State the domain and range. х 4) �1 )( -s d "' ---1 - ! ----- 'i) -; ) \/ -- � ,_ � j А - ' 1 1 1 1 - х - - ,;1. 'З .s- � А/' ,1-I 1, 63\ 3 !.()!,�' -ц .,..-· -5 1. Н ;i.. - ,1 ..У - - - а. Graph у= ( ¼ J in the interval -2 � х � 2. Ь. Оп the same set of axes, sketch the graph of у= х+ (�J =х+ (-,, 1 f ,;s iJ с. Use your calculator to find the value of х that would satisfy the equation 5. 4 ь 5. - -- 1 ! ,,:,ь,{' 4) у ,., "' 1, -?) 1.,1 1/ ', - 1 1/ , - - - - - ·" х Day 6-1: Graphing Exponential Functions and Their Inverses Exponential Functions and Their Inverses - If а function is reflected in the line у = х, the result is called the inverse function. - The notation for the inverse of f(x) is 1-' (х). - The rule for reflecting over the line у = х is to switch the х and у values. Therefore, the inverse ofthe inverse function у= ь х can Ье written as х = Ь У . а. Sketch the graph of у = 2 х . Ь. On the same set of axes, sketch its inverse. с. Write an equation for the graph drawn in part Ь 6) ' ,У х \/ -3 -� • \Э-S__ .� s .--=··�--•·� �-· " �! ' • - t__,.___ -· 1 � , . J_ 1 \ 4 1 � . . - ff Exponential Curves vs. Logarithmic Curves Exponential Curves - f(x) = ь х or у= ь х - Lies in Quadrants I and II - Never touches the x-axis - Passes through the point (О, 1) - Domain: { all real numbers} - Range: { у > О } � - --- 3 / Log Curves - 1-'(х) = logь х or у= logь х - Lies in Quadrants I and IV - Never touches the y-axis - Passes through the point (1, О) - Domain: { х > О } - Range: { all real numbers} - -� х Day 6-1: Graphing Exponential Functions and Their Inverses Traditionally, we prefer to write equations in the form II у = 11• In order for us to write the inverse of an xponential function in this form, we have to introduce logarithms. А logarithm (log) is another name for exponent. ALL exponential functions сап Ье rewritten using logs. *****Grapblng Log Functions - Туре: у= LOG ("last term") + LOG ("subscript") 7) у Sketch the graph of у = log 4 х. \]1 -=- )� 'х ;! Х � \/ ь ':, 3 ,,4J.€ \ ?-1 !!• н , . ""i ч ь - � - - - - - - 1. ,- "!.,-5 . . -. 1 "' 8) Determine the equation ofthe function shown in the graph below. Lч,,) (1) у= Iog 2 х (2) у= 3" 1, -:::3'1 (3) у= Iog 4 x ' )< ч -: у �\ ч (4) у= log 6 х / х Day 6-2: Logarithms and Logarithmic Functions Use the following formula to convert ftom log form to exponential form, and vice versa: Base Exponent = Answer f--7 log Base Answer = Exponent Log Form of an Eguation - Both logarithmic and exponential functions give information about а power. - All log equations can Ье rewritten in exponential form using the following rule: logь с = а 1. Write log 3 81 = 4 in exponential form 2. Write 8 2 3. Solve for х: log x 8 = 3 = � 64 in logarithmic form. .;'' i/-'$l1;• 'А. ' з): '- ? J. ''3 '8\· ��-- 1 )(:: 4. J ) Find the value of log 5 25 := :Х 5 '\ lS'" 5 \( с: 5 5. :,.. If /(х) = log, х, evaluate /(4). § 1()'115 f'-L/.{\ -:: х 2 :: У х :/ ; и ) С,· )_ 3х -= ;;,., �зl 6(<. "о! oit lo 'i 0 .>'\ ' j ! !о о f; Day 6-2: Logarithms and Logarithmic Functions Solve the following equations for х. 1og 6 х = 2 � (о х 7. 1og 2 х = -4 с: [3(Q с:: х J 8. 2 ' юg 27 х = 3 �J::, J·1 '= 9. х � 10. х = 1og 4 8 11. 12. Write an equation for the inverse 13. of y = (½J· f-:(�)\J 1 ()С {) '/i х -:' ' Write an equation for the inverse of y=!Ox-i. Day 6-3: Properties ofLogarithms Logarithmic Rules - Since logaritluns are exponents, all rules of exponents apply to logarithms as well. - Note: The domain oflogarithms is х > О. Product Rule: log ь АВ Ouotient Rule: А log, - Radical Rule: log ь в = = log, А + log, В log, А - log, В 1 ✓А = log, А.!. = -log А 2 2 ь 3 ,г::;: o also, logь vл= l gь А�4 = -log А 4 ь The expression log 3 а 5 Ь is equivalent to 1. (1) 5log3 аЬ 5 (2) log 3 5аЬ (4) log 3 5а + log3 Ь If х 2. = Jь, express log х2 3. 10 х in terms of log 10 а and log 10 Ь. If L = k, then JogL is equal to 1) 4) ,1og:!. k 2(1ogx - logk) 21ogx-Jogk 21ogx logk Day 6-3: Common Logarithms 4) The expression N = � is equivalent to � logN = ¼(2logx+ logy-logz) 1 2) N=-(2logx+logy)-logz 4 3) N=-log2x+-logy-logz 1 4 1 4 4) logN = -logx+-log(y-z) 4 4 2 1 5. If !оgь х = log, _Р + log, t - ½ log, q, which expression represents х? (1) pt..jq у -= !\ (2) р + t - q 2 (4) j1_ tq 6. The expre.ssion 1) 2) 1 1 Jogm- 2!ogn is equivaJent !о Jog( т-2п) 1og[ ; :1 1 3) !og(¼z-n ) 2 -г;, 1оо о ¾ 2т п Day 6-3: Common Logarithms А Ыасk hole is а region in space where objects seem to disappear. А formu!a used in the snidy of Ыасk holes is the Schwarzschild foпnula, 7. 2а� R =-,-. Based on the la\\'S ofloganthms, ]og.R: c- can Ье represented Ъу 1) 2юgG+1ogM-log2c 2) ]og2G+ !оа,,М-]оg2с 3) ]og2+]ogG+]ogМ-2]ogC 4) 2юgGM-21ogC 8. Find the value of log, 12 + log6 3. Jс:,д19 [ \?,з) )б�l,(31>J (д V.-; 3 lt! 9. Evaluate log 2 32 - log 2 1 . lь()' � � с2 )( -:, з ?-­ ;) х-: ;) S' 1 О .If log ь 3 а. = g and !оg ь 2 log ь 6 ' • \6j Ь.) 1,-- \O_jl, oi-(\-ir� '-" = �s] h , express each of the followiпg in tem1s of g апd h: ь. logь 12 () ?, •.3 J IL>j 61 1- \ ,,j 3 �\ � l'-._:-1J с. Day 6-3: Common Logarithms Common Logarithms - Since there are many different exponents and bases availaЬ!e in the world ofreal numbers, there are also many different logarithms. One of the most often used is the common logarithm, in which the base is 1О. The absence of а base in а logarithmic expression or equation signals that the log is in base 10. - Unless the number we are dealing with is an integral power of 10, its logarithm will Ье show in two parts: • characteristic - an integer that gives us а rough idea ofthe number • mantissa - а decimal that narrows it down to а specific nнmber - А characteristic of 1 tells us that the number is between 101 and 102 , or between 10 and 100, while а characteristic of 3 tells us the number is between 103 and 104 , or between 1,000 and 10,000. - Оп the graphing calculator, the LOG key is used to display the common logarithm of а number. For example, to find log 2.53, enter: LOG 2.53 ENTER. Your display should read 0.4031205212. - The calculator сап also Ье used to find а nшnber if we kпow the common logarithm. If х log N = а, theп N = 10 ° . Therefore, the key that is used to fiпd N is labeled 1 О (2 nd LOG). For example, to find N, if log N = 2.5716, eпter !О х 2.5716 ENТER. Your display should read 372.90654. 8. Fiпd log0.01277 to the nearest ten-thousandth. - \. 8q3 �D"! 10:;, %9 -С �-:;:?') ---� (_-- 9. \ф If logx = 3.78860, fiпd х to the nearest integer. !,;,Р·Н... iDS-"3Ч, � 10. Fiпd log-2.3423. о 11. "3b'1Go .. + \,..,, L If logN = 3.36486, find Nto the nearest hundred. 2 з 11.о (,,L/ 773 JЗоо Day 6-3: Exponential and Logarithmic Equations Exponential Equations - In the previous chapter, we solved exponential equations Ьу rewriting the expression so they shared the same base. However, in some equations, the bases cannot Ье made equal. In cases like this, we could solve Ьу graphing or we could use logarithms to solve for х. 1. Take the logarithm of each side. 2. Rewrite using the power rule (Recall: logA' = с - logA). 3. Solve for the variaЫe 12. Solve Т = 83 for х to the nearest ten-thousandth. 13. Solve х'·' = 112 for х to the nearest tenth. 1 '°Jx ь, 'J.. ::: S�� l д .s, J--- х r� J::s-J f--:: ;/, ч 7 7$'81J & 7 7 Day 6-4: Solving Logarithmic Equations Logarithmic Eguations - Log equations are solved Ьу "undoing" the logarithmic operations and replacing the given equation with its algebraic format. Examples: 1) Solve for х: log ь 3 + 1оgь х = 1оg ь 12 Зх= 1;.;х .3 3 ��� l 2) If 21og3 9 + log3 х = log, 27, find х. с;•i/в ' " ,,,., _, ·~ ···-·••-"'"''\ 1 Day 6-4: Solving Logarithmic Equations 4) Solve log2 (x2 -4) = log2 Зх. {)?-4) о:3)( ?- Х _., 3 ,,:. -Ч � о �=-�f JS-ill.,�. о �­ Х- Ч "'D /. X-1,i -=- D [f:i;J 5) [���,--1]�, �lj Solve for х: 21ogx-log(x-l) = log4 ;).. . I&.� 1S-.. х-1 х '"� ч )< "'- \ь eJ Ч + -.1 6) ч�Q х-� ):о Solveforx: log(x-l)+log(2x-3) = 1 160 LX-1 ') L �)\- 3) =- ) О -:: ( 1 х- 1 ) {Jx-3) d.x. ?.-3 x. -)-х +-3 J ,,, ?- -;s ;,s - *1 ()3 / О -= Qx -..Гх-1 } о D ----(2'!:: ::.,J} " + I -:: ;;)._ ()л -sx-1 -:с Xt/ =-И - ! -1 Гл=��l Day 6-4: Solving Logarithmic Equations ) ) Solve for х: log2 х + log, (х - 4) = 5. х - -3:l-= О ( х- i) � 1--f) � о :l 1..\)( ----·�----х+Ч::о $''С' с, [i�� §) rej Day 6-5: Base е and Natural Logarithms е - irrational number, approximately 2.71828183 ln - natural log f(x) = е х is called the natural exponential function f(x)= lnx is called the natural logarithmic function These two functions are inverses of each other. Therefore, lnx = log,x *** lnex = х *** Finding keys on the Graphing Calculator: ln ех 1) ln 1 = -= \ ь� е..... above the division key (hit the 2nd key, then the + key) left side, next to the 4 above the ln key (hit the 2nd key, then the ln key) е Find: /n о =\ 2) lп е 4) е2 1аз Writing Equivalent Expressions - you can write an equivaleнt base е exponentia! equation for а natural logarithmic equation Ьу using the fact that lп х = log, х - for example: ln 4 = х----; log, 4 = х ➔ ех = 4 Write each exponential equation in logarithmic form: 5) ех = 8 6) е5 = х 8 .,. '1' lь�е. 11'1 � "' х 101-е.. х --:;.s !У\ х -=s х 7) е = 9 !,:,�,:: q -:, )';. 1 1'\ q-::-:x 8) е7 =х 1 ь� е. Х. ::: l l"X = l - you can also write an equivalent natural logarithm equation for а natural base е exponential equation Write each logarithmic equation in exponential form: 9) lпх z 0.7741 \ь� с.Х-::, 1•'-l \ e_f714! -::: )\ 10) 11110 = х "' t \о v lь'\с р '- х ,, 10 11) lпх z 2.1438 J. �ч 3 8' ID<\e Х-:: v (7 ;1,�с1З&' :х 12) ln18 = х 1�0 el&� Х е ,; : 18 Day 6-5: Base е and Natural Logarithms Logarithmic Properties Remain the SАМЕ When Working With the Natural Log а ln - = ln а - ln Ь ь ln аЬ = ln а+ ln Ь 1 &, =(lna)2 =-lna 1 Inaь =Ьlna 2 Write each expression as а single logarithm: 13) З!n!О -ln 8 '!, 1 У\ )D ln i ооо 14) 1 In40+2ln-+lnx 2 1 t\ i ч 1 V) { 8" 1 5) о , i ±) х, ) :). ч () { '/t.1 j ' )( lh !Dx 11"1 1?.,!Г Using the properties of logarithms, write each ofthe following in expanded form: 17) In.Jsinxecosx Evaluate: - substitute 18) Given the function f(x) = 1 .6е 025х, find f( 14) to the nearest thousandth. -tLl'-1)= l,LPe D,a;;{•4) Day 6-5: Base е and Natural Logarithms Solve each ofthe following: 19) ех =30 20) \с,о.,_ 3о х -=- 21) '3 '-1 е ь.о�-.. " � 8 + 2ех = 12 -15 -8 22) " i? е..�-::.;),,, 4е_,х - 5 = 3 to the nearest ten-thousandth. 24) :---&- 25 ) -:-)- Х ""' °if )(':С 8' .: 4 1< .,, 4 ч ·:: х [х-::-,3Ч� f !i e"'L/>, - 1 с,с,,6 е )(. --:::: Ч .3 � 3Ч b!i73.Г �х =24, to the nearest ten-thousandth ') 3 1"' 4 \nx= 4 (.? � '1 li;')t/nе ;t;:1-::-J "'"-J.x )( l --:г li'\ &s -:::х 4. = 40 ·: � х �-�3 11"'1 l;o .• � •l""�e "'.,,." Х. 23) е2х+3 \ь�е �0" d,1,.,\,3 Q еу.. 0 Ч ;:;.. 150е005х = 340 /.$ о 1 ,S(,) ц )(--:: 74S, 23q 4�r,,'ff ��� 26) ln(2x+4)=x2 (usingthe graphing calculator) Day 6-6: Real World Applications Real-World Applications - Exponential Functions are used to represent growth and decay. Exponential Growth and Decay Growth: A(t) = a(l + r)' Decay: A(t) = а(!- r)' а = initial amount r = the percent ofincrease per time period t = time periods ***HINTS: - "О" in front of the decimal indicates decay, decrease, depreciate, etc. - "1" in front of the decimal indicates growth, increase, etc. - change the percent to а decimal Ьу moving the decimal point two places to the Ieft - ***for decay/decrease, FIRST subtract the given percent from 100% THEN move the decimal point two places to the left Example - Exponential Growth: ,) The first U.S. Census was conducted in 1790. At that time, the popu!ation was 3,929,214. Since then, the U.S. population has grown Ьу approximately 2.03% annually. Write an equation to represent this situation. А{ -:с 3{J,P tCJ!4 { 1 + , uJo3) x Example - Exponential Decay: 2) А сир oftea contains 35 milligrams of caffeine. The average teen сап eliminate approximately 12.5% ofthe caffeine from their system per hour. Write an equation to represent this situation. 3) Given the function G(t) =24(.75)', where G(t) is the number of geese in 2008, in hundreds, at Allegheny State Park, and t represents the number of year since 2008. What is the initial nшnber of geese at the park, and at what rate is the population decreasing? Day 6-6: Real World Applications In general, exponential functions are in the form у = аЬ х а is the initial value Ь is the growthldecay factor х is the time у is the ending amount - There are numerous applications of exponential functions in the real world. One of the most familiar examples of exponential growth is the concept of compound interest. о Example: If y ou l1ad $100 in the bank gaining 4% annual interest, compounded annually, the equation f(x) = 100(1.04) х would model the amount ofmoney you would have in the bank after х years. (Note: 1.04 represents 104%) Examples: PQPULATION: 4) The population of the town of Smithville in 2003 was estimated to Ье 35,000 people with an annual rate of increase (growth) of about 2.4% а) What is the growth factor for Smithville? ' { 1+- ,oJ4) §) Ь) Write an equation to model future growth. с) Use your equation to estimate the population in 2007 to the nearest hundredpeople. А{+;:)"' ' 1.s1 j.$:; DQO { 1. (X,t 11) ;;. �gч&;;. - ) 1А �� р�,�, Day 6-6: Real World Applications MONEY: ,) Marisa invests $300 at а bank that offer 5% interest compounded annually. а) What is the growth factor for the investment? Ь) Write an equation to model the growth ofthe investment. ,с fJ.(ь)-:; 'Зо6[1-оs-)" с) How many years will it take for the initial investment to douЫe? &�ьа "' з о"::,L 1. D: :/ '3DD '$�7' .;() --�J А-:: jl.j. ФsJ) ь fi_: ;L/ ,1,t,f?) DEPRECIATION: 6) ь ,q � D t Matt bought а new саг at а cost of $25,000. �iates approximately 15% ofits value each year. а) What is the decay factor for the value ofthe саг? (1-.,s) :,- о&& Ь) Write an equation to model the decay value ofthis car. ,-,[¾:с) с) -а. What ,,vill the саг Ье worth in 1 О years? А -I f) -= J� J 'S • t!,,:;o {@ l?sJ / \ ;,:;, ::IS, ()00 l &ftS°) 1 Day 6-6: Real World Applications 7) As cars get older they are worth less than when they are new. This is called depreciation. Assume that you bought а new 2004 car for $13,700 and that it depreciates at approximately 20% per year. An equation that will model the value ofyour car, V(t), where t is the number of years since 2004, is V(t) = 13,700(0.8)'. а. Find V(l). Explain what this means in terms ofthe value ofyour car. \jc..\"',; 1 е;;,г е,., fW' 1. Ь. Find t, to the nearest thousandth, such that V(t) terms ofthe car. j010o0 "'""""'"� '° i 3, ~n•i,, }е:,с� t [5 \/ tь.Г = 10,000 and explain what this means in -- f об N R ,б Ь. ·"t/'' ,J'A;: •[,, Ч'ti с. After how many years will the car Ье worth only halfof what you paid for it? J... � {11,-н::ioj "F: (;;.8'...ro 1.,ио� 1 7,,1c,r,, i, в J lь� ,S"::: iotJ,� х:: :,, /{){,,J.f, . . , ��:) (J(г' YtiQ•,.;; Day 6-7: Real World Applications (continued) Compound Interest - interest paid on the principal of an investment and any previously earned interest Formula: А = amount in account after t years Р = principal amount invested r = annual interest rate (written as а decimal) п = number of compounding periods PER YEAR Examples: 1) An investment account pays 4.2% annual interest compounded monthly. If$2500 is invested in this account, what will Ье the balance after 15 years? ,&ч;�. \'"l1s) 11 П -; SЬ6 1 1- ---;;: _) 'J ( \ 1ft,::, :: -;Js-ьь (_ ооз�,J i, :: Чь&&, fs'ь ьН,{,. 2) Uk/��(5 ,;�) Diego decided to invest his $500 tax refund rather than spending it. Не found а bank that would рау him 4% interest, compounded quarterly. Ifhe deposits the entire $500 and does not deposit or withdraw any other amount, how long will it take him to douЫe his money in the account? Use А= Р(1 + -;;J', where А is t amount in the account after t years, Р is the principal/initial amount invested, r is the annual interest rate and п is the number of compounding periods per year. /Dj ;) -= 11'� \ D 1 L/ х ! oj /. О 11>� 1 о 1 1 Day 6-7: Real World Applications (continued) Continuous Growth or Decay - exponential functions with the base е are often used to describe continuous or decay General Forrnula: Уо = initial amount k = rate (written as а decimal) t = time у = ending amount Related Forrnulas: Continuous Compounding (Money) А=Ре'' Population Growth у= ае'' Р = initial amount а = amount at time t r = interest rate (written as а decimal) а = initial population у = population at time t k = rate of continuous growth (as а dec.) Examples: 3) When Angelina was bom, her grandparents deposited $3000 into а college savings account paying 4% interest compounded continuously. What will the balance Ье after 10 years? �-:: 3000 е. ,r:,1{,10' ) � Ч4iS, 111irt>C1'3 4) �� Tamika invests $500 at а bank offering 10% interest compounded continuously. Find, to the nearest year, how many years it will take for Tamika's investment to douЬ!e. - Day 6-7: Real World Applications (continued) EXPONENTIAL: ) After an oven is tumed on, its temperature, Т, is represented Ьу the equation Т = 400-350(3.2)-D.im, where т represents the number ofminutes after the oven is tumed on and Trepresents the temperature ofthe oven, in degrees Fahrenheit. How many minutes does it take for the oven's temperature to reach 300 ° F? Round your answer to the nearest minute. lьt �. -� -, t 1 .,,� з ;i_ 1� � ',; -, lf\'\ _. н"''Ъ з J LOGARITHMIC: 6) The temperature, Т, ofа given cup of hot chocolate after it has been cooling for t minutes сап best Ье modeled Ьу the function below, where Т0 is the temperature ofthe room and k is а constant. ln(T- Т0 ) = -kt+ 4.718 А сир ofhot chocolate is placed in а room that has а temperature of 68 ° . After 3 minutes, the temperature ofthe hot chocolate is 150 ° . Compute the value ofk to the nearest thousandth. [Only an algebraic solution сап receive full credit.] Using this value ofk, find the temperature, Т, ofthis сир ofhot chocolate ifit has been sitting in this room for а total of 10 minutes. Express your answer to the nearest degree. [Only an algebraic solution can receive full credit.] т� ь&) "ф, 1641 tм�) • !n L i- t.a) � -з. "' 1 ь, 11'\ { l:,{&.1.) � -з. к�ч. 1� - L/. '11 8 ' l(Je_ е. -3 ...\ с/ ·bl!J"" 5, 1,'! [§ , = Т-ь lf