College Algebra LinC
College Algebra LinC MAC 1105
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This 5 page Class Notes was uploaded by Deshawn Dickens on Thursday October 29, 2015. The Class Notes belongs to MAC 1105 at Valencia College taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/231223/mac-1105-valencia-college in Calculus and Pre Calculus at Valencia College.
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Date Created: 10/29/15
MA 1105 Handout 4 Graphs oiExponential and Logarithmic Functions Tn HM Hand u functions T rlthml The Expunenlial Functiun The ocporiential function with baseb is defined by x ow sirice o abquot o o is w ii 1 b gtl 1 ltl ori x 013 is all the real numbers x The range ofthe exponential function is all the values ofy 2 o ocporiential decay The domain othe exponential functl ornpare the graphs of x Zxand fx10 o o and type in the equations 2 Z Xand lg low and selectmzl ZDeClmal Ifwe change the styles ofthe curves we obtain the graph picture where e X raph in buld is the functionYz Example G Jay Notice thatboth functions are othe foirri x ab sirice y Z x with o l and b z and Y2 IOAX Wtha1 andb10 forboLh ofthem since a l forboth functions their viritercept x 10 grows fasterthari x 2 asth x 10 decays fasterthan x 2 as the riegativevalues ofx decrease We can say that the largerthe base b the faster the function grows 1 e positive values ofx increase and Page 1 of5 Example 2 Compare the graphs of x 23 and fx 2 andtype inthe equations Y 2m and Y 2 xand select L L c L ZDecimal 39 the graph in bold is the functionlr2 22 x J since a 2 for v and a l for vi the yintercept ofv is 2 while the yintercept ofy2 the bold curve is 1 Example 3 Compare the graphs of x 2 and fx2 Go to and type in the equations in 2 x and Y2 e xl and select 1 4 ZDecimal am can be obtained by pressing and the s lfwe change the styles ofthe curves we obtain the picture below where the graph in bold is the function Y X w o aaf Ifwe compare the two grepls we see that both graphs cross the yaxis at y Sincee s 2718 is larger than 2 we see that x e grows faster than fx 2 All the exponential functions with base b gt1 will have the same shape The difference will be that the higher the wlue ofthe base the faster the function will increase Example 4 Compare the graphs of x 2 and x 2 2 t 4 and the L keys To enter eAxl E Ifwe change the styles ofthe curves we obtain the picture below where the graph in bold is the function v2 2quot L Page2 of5 since o 1for both functions their yintercept 1 fx 2 is an increasing function and fa is a decreasing function We can see that fx w is a decreasing 1 1 function ifwe rewrite the function asfx 1 Smcez s 2718 is a number 2 2 between zero and one The Logarithmic Function L 394 4L fxlogbxwi The can be either greater than 1 17 gt1 or between 0 and 1 0 lt 17 lt 1 Example 5 Compare the graph of y 1og2 x andy 1ogm x and type in the equations 31 log xlog 2 and Y2 1og 1 and select 39 4 4 ZDecimal 1fwe changethe sty1es ofthe curves we obtain the picture below where the picture in buld is the function Y2 1og 1 Asx increases y logmx gets closer to the xax isthany 1og2xforxgt1 Asx approaches 0 logmx gets c1osertotheyaxis thany 1og2rforo ltxlt1 We can say that the larger the base the slower the 1oganthrnic function grows Example 6 Compare the graph of y 2 and y lngtc to see ifthey are inverse functions Go to E and type in the equations Y1 equotgtc and Y2 In at using the window x 8 a anoiy5 5 sown be1ow since the graphs are symmetric about the line y x the bold line we can see that y 2 and y lngtc are inverse functions Page 3 of5 Graphical Snllltinn nfExpnnEntial and Lngarithmic Equatinns Example Fmdthe solution of the equamh 2M 17x graphmany by usmg the immmhm feature nf ynllr calculztnr Round to three decimal pl aces ole quot Let Y1 2M and Y2 17x Selectthe wmeWX 78 8 andy 75 5 shown below 7 A In rsection 555555 Example 8 Fmdthe solution of the equamh 1hx171x 0 graphmally Let Y1 mama and Y2 0 521mm wmeWX 78 8 andy 75 5 shown urves and a E choosmg ophon 5 Intersect andthe The solution to three decimal lanes 5 Intersection x5571 ss Zero vo x5571 ss vo Page4of5 Example 9 Ftndthe solutton to the equatton log2xx 0 graphlcally Roundto tluee deeunal plaees ogxlog2 x Seleet m 4 By usmg the ehange of base formula let 11 1 chooslng opuon 2 Zero an ZDeeunal Wlndow By pressmg d The soluuon to the equateon to three deeunal plaees ls x 641 Zero xsmasrt v F amnlnl yxg x r nht H plaees Letyl new Seleetmzt ZDeclmalwlndow To ndtherelatlvemaxlmumwe useCALC Press andselectoptlon4 the two anow heads Ihatwtll glve you the relative maxlmum potnt 1 000 0 368 Note Should you need further revlew ftndlng maxunuhn and hntnunuhn valuesquot VJth the grapherquot please revlew handout 3 E3675 Nate Same calcnlztmswill give x 1 instead nl39 10000004 Other caleulatm will give 9999999 It all depends nn haw the calculatnr rnnnds the numbers End at Handnut 4 Page 5 of5