1.1 THE CARTESIAN PLANE AND THE DISTANCE FORMULA

Transcription

1 6000_00.qd //05 : PM Pge CHAPTER Funtions, Grphs, nd Limits. THE CARTESIAN PLANE AND THE DISTANCE FORMULA Plot points in oordinte plne nd red dt presented grphill. Find the distne etween two points in oordinte plne. Find the midpoints of line segments onneting two points. Trnslte points in oordinte plne. Qudrnt II Origin Qudrnt III FIGURE. -is Vertil rel line -is Qudrnt I Horizontl rel line -is Qudrnt IV The Crtesin Plne The Crtesin Plne Just s ou n represent rel numers points on rel numer line, ou n represent ordered pirs of rel numers points in plne lled the retngulr oordinte sstem, or the Crtesin plne, fter the Frenh mthemtiin René Desrtes ( ). The Crtesin plne is formed using two rel numer lines interseting t right ngles, s shown in Figure.. The horizontl rel numer line is usull lled the -is, nd the vertil rel numer line is usull lled the -is. The point of intersetion of these two es is the origin, nd the two es divide the plne into four prts lled qudrnts. Eh point in the plne orresponds to n ordered pir, of rel numers nd, lled oordintes of the point. The -oordinte represents the direted distne from the -is to the point, nd the -oordinte represents the direted distne from the -is to the point, s shown in Figure.., (, ) Direted distne from -is Direted distne from -is -is STUDY TIP The nottion, denotes oth point in the plne nd n open intervl on the rel numer line. The ontet will tell ou whih mening is intended. FIGURE. (, ) (, ) FIGURE. (0, 0) (, 0) (, ) EXAMPLE Plotting Points in the Crtesin Plne Plot the points,,,, 0, 0,, 0, nd,. SOLUTION To plot the point, -oordinte -oordinte imgine vertil line through on the -is nd horizontl line through on the -is. The intersetion of these two lines is the point,. The other four points n e plotted in similr w nd re shown in Figure.. TRY IT Plot the points,,,,,, 0,, nd,.

2 6000_00.qd //05 : PM Pge Using retngulr oordinte sstem llows ou to visulize reltionships etween two vriles. It would e diffiult to overestimte the importne of Desrtes s introdution of oordintes to the plne. Tod his ides re in ommon use in virtull ever sientifi nd usiness-relted field. In Emple, notie how muh our intuition is enhned the use of grphil presenttion. SECTION. The Crtesin Plne nd the Distne Formul EXAMPLE Skething Stter Plot The mounts A (in millions of dollrs) spent on snowmoiles in the United Sttes from 99 through 00 re shown in the tle, where t represents the er. Sketh stter plot of the dt. (Soure: Ntionl Sporting Goods Assoition) t A SOLUTION To sketh stter plot of the dt given in the tle, ou simpl represent eh pir of vlues n ordered pir t, A, nd plot the resulting points, s shown in Figure.. For instne, the first pir of vlues is represented the ordered pir (99, 55). Note tht the rek in the t-is indites tht the numers etween 0 nd 99 hve een omitted. Dollrs (in millions) Amounts Spent on Snowmoiles A Yer FIGURE. t TRY IT From 99 to 000, the enrollments E (in millions) of students in U.S. puli olleges re shown, where t represents the er. Sketh stter plot of the dt. (Soure: U.S. Ntionl Center for Edution Sttistis) t E STUDY TIP In Emple, ou ould let t represent the er 99. In tht se, the horizontl is would not hve een roken, nd the tik mrks would hve een leled through 0 (insted of 99 through 00). TECHNOLOGY The stter plot in Emple is onl one w to represent the given dt grphill. Two other tehniques re shown t the right. The first is r grph nd the seond is line grph. All three grphil representtions were reted with omputer. If ou hve ess to omputer grphing softwre, tr using it to represent grphill the dt given in Emple. Dollrs (in millions) Amounts Spent on Snowmoiles A Yer t Dollrs (in millions) Amounts Spent on Snowmoiles A Yer t

3 6000_00.qd //05 : PM Pge CHAPTER Funtions, Grphs, nd Limits + = FIGURE.5 Pthgoren Theorem (, ) d (, ) The Distne Formul Rell from the Pthgoren Theorem tht, for right tringle with hpotenuse of length nd sides of lengths nd, ou hve Pthgoren Theorem s shown in Figure.5. (The onverse is lso true. Tht is, if, then the tringle is right tringle.) Suppose ou wnt to determine the distne d etween two points, nd, in the plne. With these two points, right tringle n e formed, s shown in Figure.6. The length of the vertil side of the tringle is nd the length of the horizontl side is. B the Pthgoren Theorem, ou n write d d d. This result is the Distne Formul. FIGURE.6 Two Points Distne Between The Distne Formul The distne d etween the points, nd, in the plne is d. EXAMPLE Finding Distne (, ) d 5 (, ) Find the distne etween the points, nd,. SOLUTION Let,, nd,,. Then ppl the Distne Formul s shown. d Distne Formul Sustitute for,,, nd. 5 Simplif. 5.8 Use lultor. FIGURE.7 Note in Figure.7 tht distne of 5.8 looks out right. TRY IT Find the distne etween the points, nd,.

4 6000_00.qd //05 : PM Pge 5 SECTION. The Crtesin Plne nd the Distne Formul 5 EXAMPLE Verifing Right Tringle Use the Distne Formul to show tht the points,,, 0, nd 5, 7 re verties of right tringle. 8 (5, 7) SOLUTION The three points re plotted in Figure.8. Using the Distne Formul, ou n find the lengths of the three sides s shown elow. 6 Beuse d d 0 5 d d d d ou n ppl the onverse of the Pthgoren Theorem to onlude tht the tringle must e right tringle. (, ) FIGURE.8 d d d (, 0) 6 TRY IT Use the Distne Formul to show tht the points,, 5, 5, nd 6, re verties of right tringle. The figures provided with Emples nd were not rell essentil to the solution. Nevertheless, we strongl reommend tht ou develop the hit of inluding skethes with our solutions even if the re not required. (50, 5) EXAMPLE 5 Finding the Length of Pss In footll gme, qurterk throws pss from the five-rd line, 0 rds from the sideline. The pss is ught wide reeiver on the 5-rd line, 50 rds from the sme sideline, s shown in Figure.9. How long ws the pss? SOLUTION You n find the length of the pss finding the distne etween the points 0, 5 nd 50, 5. d So, the pss ws 50 rds long. STUDY TIP Distne Formul Simplif. In Emple 5, the sle long the gol line showing distne from the sideline does not normll pper on footll field. However, when ou use oordinte geometr to solve rel-life prolems, ou re free to ple the oordinte sstem in n w tht is onvenient to the solution of the prolem. Line of srimmge (0, 5) FIGURE.9 TRY IT 5 A qurterk throws pss from the 0-rd line, 0 rds from the sideline. The pss is ught wide reeiver on the 0-rd line, 5 rds from the sme sideline. How long ws the pss?

5 6000_00.qd //05 : PM Pge 6 6 CHAPTER Funtions, Grphs, nd Limits The Midpoint Formul To find the midpoint of the line segment tht joins two points in oordinte plne, ou n simpl find the verge vlues of the respetive oordintes of the two endpoints. The Midpoint Formul The midpoint of the segment joining the points, nd, is 6 Midpoint,. (, 0) 6 6 ( 5, ) 6 Midpoint FIGURE.0 (9, ) 9 EXAMPLE 6 Finding Segment s Midpoint Find the midpoint of the line segment joining the points 5, nd 9,, s shown in Figure.0. SOLUTION Let, 5, nd, 9,. Midpoint, 5 9,, 0 TRY IT 6 Find the midpoint of the line segment joining 6, nd, 8. Struks Corportion s Annul Sles Sles (in illions of dollrs) FIGURE. (00,.08) Midpoint (00,.7) (00,.65) Yer EXAMPLE 7 Estimting Annul Sles Struks Corportion hd nnul sles of $.65 illion in 00 nd $.08 illion in 00. Without knowing n dditionl informtion, wht would ou estimte the 00 sles to hve een? (Soure: Struks Corp.) SOLUTION One solution to the prolem is to ssume tht sles followed liner pttern. With this ssumption, ou n estimte the 00 sles finding the midpoint of the segment onneting the points 00,.65 nd 00,.08. Midpoint , 00,.7 So, ou would estimte the 00 sles to hve een out $.7 illion, s shown in Figure.. (The tul 00 sles were $.9 illion.) TRY IT 7 Mtg Corportion hd nnul sles of $. illion in 00 nd $.79 illion in 00. Wht would ou estimte the 00 nnul sles to hve een? (Soure: Mtg Corp.)

6 6000_00.qd //05 : PM Pge 7 SECTION. The Crtesin Plne nd the Distne Formul 7 Trnslting Points in the Plne EXAMPLE 8 Trnslting Points in the Plne Figure.() shows the verties of prllelogrm. Find the verties of the prllelogrm fter it hs een trnslted two units down nd four units to the right. SOLUTION To trnslte eh verte two units down, sutrt from eh -oordinte. To trnslte eh verte four units to the right, dd to eh -oordinte. Originl Point Trnslted Point, 0,, 6,, 0 5,, 7, 0, 6 7,, 5, The trnslted prllelogrm is shown in Figure.(). (, ) 8 (, 6) (, ) 6 (, 0) () FIGURE. 8 (, 6) (, ) (7, ) (, ) (5, ) 6 (, 0) (7, 0) () (5, ) DREAMWORKS/THE KOBAL COLLECTION Mn movies now use etensive omputer grphis, muh of whih onsists of trnsformtions of points in two- nd three-dimensionl spe. The photo ove shows sene from Shrek. The movie s nimtors used omputer grphis to design the sener, hrters, motion, nd even the lighting in eh sene. TRY IT 8 Find the verties of the prllelogrm in Emple 8 fter it hs een trnslted two units to the left nd four units down. T AKE ANOTHER LOOK Trnsforming Points in Coordinte Plne Emple 8 illustrtes points tht hve een trnslted (or slid) in oordinte plne. The trnslted prllelogrm is ongruent to (hs the sme size nd shpe s) the originl prllelogrm. Tr using grphing utilit to grph the trnsformed prllelogrm for eh of the following trnsformtions. Desrie the trnsformtion. Is it trnsltion, refletion, or rottion? Is the trnsformed prllelogrm ongruent to the originl prllelogrm?..,,,,.,,

7 6000_00.qd //05 : PM Pge 8 8 CHAPTER Funtions, Grphs, nd Limits PREREQUISITE REVIEW. The following wrm-up eerises involve skills tht were overed in erlier setions. You will use these skills in the eerise set for this setion. In Eerises 6, simplif eh epression In Eerises 7 0, solve for or EXERCISES. In Eerises 6, () find the length of eh side of the right tringle nd () show tht these lengths stisf the Pthgoren Theorem. *.. (, ) (, 6) (0, 0) (, 0) (, ) (, ).. (, ) (, ) (, ) (7, ) (7, ) (, ) (, ) (, ) (, ) (, ) (, 5) (6, ) *The nswers to the odd-numered nd seleted even eerises re given in the k of the tet. Worked-out solutions to the oddnumered eerises re given in the Student Solutions Guide. In Eerises 7, () plot the points, () find the distne etween the points, nd () find the midpoint of the line segment joining the points. 7.,, 5, 5 8.,,, 9.,,, 5 0.,, 5 6,.,,,., 7,,.,,,., 0, 0, In Eerises 5 8, show tht the points form the verties of the given figure. (A rhomus is qudrilterl whose sides hve the sme length.) Verties Figure 5. 0,,, 7,, Right tringle 6.,,,,, Isoseles tringle 7. 0, 0,,,,,, Rhomus 8. 0,,, 7,,,, Prllelogrm In Eerises 9, use the Distne Formul to determine whether the points re olliner (lie on the sme line). 9. 0,,, 0,, 0. 0,, 7, 6, 5,., 6,,, 5,.,,,, 5, 5 In Eerises nd, find suh tht the distne etween the points is 5.., 0,,.,,, In Eerises 5 nd 6, find suh tht the distne etween the points is , 0,, 6. 5,, 5,

8 6000_00.qd //05 : PM Pge 9 SECTION. The Crtesin Plne nd the Distne Formul 9 7. Use the Midpoint Formul repetedl to find the three points tht divide the segment joining, nd, into four equl prts. 8. Show tht, is one of the points of trisetion of the line segment joining, nd,. Then, find the seond point of trisetion finding the midpoint of the segment joining, nd,. 9. Use Eerise 7 to find the points tht divide the line segment joining the given points into four equl prts. (),,, (),, 0, 0 0. Use Eerise 8 to find the points of trisetion of the line segment joining the given points. (),,, (),, 0, 0. Building Dimensions The se nd height of the trusses for the roof of house re feet nd 5 feet, respetivel (see figure). () Find the distne d from the eves to the pek of the roof. () The length of the house is 0 feet. Use the result of prt () to find the numer of squre feet of roofing. In Eerises nd, use grphing utilit to grph stter plot, r grph, or line grph to represent the dt. Desrie n trends tht pper.. Consumer Trends The numers (in millions) of le television susriers in the United Sttes for re shown in the tle. (Soure: Nielsen Medi Reserh) Yer Susriers Yer Susriers Consumer Trends The numers (in millions) of ellulr telephone susriers in the United Sttes for re shown in the tle. (Soure: Cellulr Teleommunitions & Internet Assoition) Yer Susriers Yer Susriers d 0. Wire Length A gu wire is strethed from rodsting tower t point 00 feet ove the ground to n nhor 5 feet from the se (see figure). How long is the wire? Dow Jones Industril Averge In Eerises 5 nd 6, use the figure elow showing the Dow Jones Industril Averge for ommon stoks. (Soure: Dow Jones, In.) 5. Estimte the Dow Jones Industril Averge for eh dte. () Mrh 00 () Deemer 00 () M 00 (d) Jnur Estimte the perent inrese or derese in the Dow Jones Industril Averge () from April 00 to Novemer 00 nd () from June 00 to Ferur ft Figure for 5 nd 6 5 ft The smol indites n eerise in whih ou re instruted to use grphing tehnolog or smoli omputer lger sstem. The solutions of other eerises m lso e filitted use of pproprite tehnolog. Dow Jones Industril Averge,000 0,600 0,00 9,800 9,00 9,000 8,600 8,00 7,800 Mr. De. M Jn. JFMAMJJASONDJFMAM J J ASONDJ F

9 6000_00.qd //05 : PM Pge 0 0 CHAPTER Funtions, Grphs, nd Limits Constrution In Eerises 7 nd 8, use the figure, whih shows the medin sles pries of eisting one-fmil homes sold (in thousnds of dollrs) in the United Sttes from 987 to 00. (Soure: Ntionl Assoition of Reltors) 7. Estimte the medin sles prie of eisting one-fmil homes for eh er. () 987 () 99 () 997 (d) Estimte the perent inreses in the vlue of eisting one-fmil homes () from 99 to 99 nd () from 00 to 00. Figure for 7 nd 8 Medin Sles Prie (in thousnds of dollrs) Yer Reserh Projet In Eerises 9 nd 0, () use the Midpoint Formul to estimte the revenue nd profit of the ompn in 00. () Then use our shool s lirr, the Internet, or some other referene soure to find the tul revenue nd profit for 00. () Did the revenue nd profit inrese in liner pttern from 999 to 00? Eplin our resoning. (d) Wht were the ompn s epenses during eh of the given ers? (e) How would ou rte the ompn s growth from 999 to 00? (Soure: Wlgreen Compn nd The Ynkee Cndle Compn) 9. Wlgreen Compn Yer Revenue (millions of $) 7,89,505 Profit (millions of $) The Ynkee Cndle Compn Computer Grphis In Eerises nd, the red figure is trnslted to new position in the plne to form the lue figure. () Find the verties of the trnsformed figure. () Then use grphing utilit to drw oth figures... (, ) (0, ) units (, ) (0, 0) units (, ) (, 0) units (, ) units. Eonomis The tle shows the numers of er infetions treted dotors t HMO linis of three different sizes: smll, medium, nd lrge. Cses per Cses per Cses per Numer of smll lini medium lini lrge lini dotors () Show the reltionship etween dotors nd treted er infetions using three urves, where the numer of dotors is on the horizontl is nd the numer of er infetions treted is on the vertil is. () Compre the three reltionships. (Soure: Adpted from Tlor, Eonomis, Fourth Edition) Yer Revenue (millions of $) Profit (millions of $). 7.8 The smol indites n eerise tht ontins mteril from tetooks in other disiplines.