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NOMBRE11FECHAPERÍODOEjercicios preparatoriosExpresiones y ecuaciones racionalesPaso 1Antes de comenzar el Capítulo 11 Lee cada enunciado. Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado. Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,escribe NS (No estoy seguro(a)).PASO 1A, D, o NSPASO 2AoDEnunciado1. Dado que una variación directa se puede escribir como y kx,xuna variación inversa se puede escribir como y .k2. Una expresión racional es una fracción algebraica que contieneun radical.2xy223c3. Para multiplicar dos expresiones racionales, como y ,3c5ymultiplica los numeradores y los denominadores.4. Al resolver problemas con unidades de medida, el análisisdimensional es el proceso de determinar las unidades de larespuesta final, de manera que las unidades pueden ignorarsemientras se desarrollan los cálculos.2a56. Para sumar y , primero suma los(3a - 4)(3a - 4)numeradores y luego los denominadores.7. El mínimo común denominador de dos expresiones racionalesserá el mínimo común múltiplo de los denominadores.8. Una fracción compleja contiene una fracción en su numeradoro denominador.9.( ab )acse puede volver a plantear como .La fracción cbd( d )10. Las soluciones extrínsecas son soluciones que se puedeneliminar porque son extremadamente altas o bajas.Paso 2Después de completar el Capítulo 11 Vuelve a leer cada enunciado y completa la última columna con una A o una D. ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna? En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con losenunciados que marcaste con una D.Capítulo 114Álgebra 1 de GlencoeCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.5. Para dividir (4x2 12x) entre 2x, divide 4x2 entre 2x y 12x entre 2x.

NAMEDATE11-1PERIODStudy Guide and InterventionInverse VariationIdentify and Use Inverse Variations An inverse variation is an equation in thekform of y x or xy k. If two points (x1, y1) and (x2, y2) are solutions of an inverse variation,then x1 y1 k and x2 y2 k.x1 y1 x2 y2Product Rule for Inverse VariationxyExampleIf y varies inversely as x and y 12 when x 4, find x when y 18.Method 1 Use the product rule.x 1 y1 x2 y2Product rule for inverse variation4 12 x2 18x1 4, y1 12, y2 1848 x2Method 2 Use a proportion.xDivide each side by 18.188 x23y12 x2 y1Proportion for inverse variation184 x x1 4, y1 12, y2 1821248 18x28 x2Simplify.3Cross multiply.Simplify.8Both methods show that x2 when y 18.3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ExercisesDetermine whether each table or equation represents an inverse or a directvariation. Explain.1.xy3651081612242. y 6x3. xy 15Assume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then solve.4. If y 10 when x 5,find y when x 2.5. If y 8 when x -2,find y when x 4.6. If y 100 when x 120,find x when y 20.7. If y -16 when x 4,find x when y 32.8. If y -7.5 when x 25, find y when x 5.9. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois,in 4 hours if they drive an average of 45 miles per hour. How long would it take them ifthey increased their average speed to 50 miles per hour?10. GEOMETRY For a rectangle with given area, the width of the rectangle varies inverselyas the length. If the width of the rectangle is 40 meters when the length is 5 meters, findthe width of the rectangle when the length is 20 meters.Chapter 115Glencoe Algebra 1Lesson 11-112From the product rule, you can form the proportion x2 y1 .

NAMEDATE11-1PERIODStudy Guide and Intervention(continued)Inverse VariationGraph Inverse Variations Situations in which the values of y decrease as the valuesof x increase are examples of inverse variation. We say that y varies inversely as x, or y isinversely proportional to x.Inverse Variation Equationan equation of the form xy k, where k 0Example 1Suppose you drive200 miles without stopping. The timeit takes to travel a distance variesinversely as the rate at which youtravel. Let x speed in miles per hourand y time in hours. Graph thevariation.Example 2Graph an inversevariation in which y varies inversely asx and y 3 when x 12.Solve for k.xy kInverse variation equation12(3) kx 12 and y 336 kSimplify.Choose values for x and y, which have aproduct of 36.The equation xy 200 can be used torepresent the situation. Use various speedsto make a table.yyxy102030 6 62010 3 12 2 18218312663040550460206.71020O3.34060xy241224 x12OExercisesGraph each variation if y varies inversely as x.1. y 9 when x -324y32y-24 -12 O1224 x3. y -25 when x 51001612-32 -16 O1632 x-100 -50 O-16-50-24-32-10020y10-20 -10 O5. y -18 when x -936y20 x50x1006. y 4.8 when x 5.47.21810y50-124. y 4 when x 5Chapter 112. y 12 when x 4y3.6-36 -18 O1836 xx-7.2 -3.6 O-10-18-3.6-20-36-7.263.67.2Glencoe Algebra 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.x

NAMEDATE11-1PERIODSkills PracticeInverse VariationDetermine whether each table or equation represents an inverse or a directvariation. Explain.xy0.5814224122. xy 3. -2x y 03Assume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then graph the equation.4. y 2 when x 585. y -6 when x -6y1684-8-44Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.O-16 -8 O8x-4-8-8-166. y -4 when x -1216O816 x7. y 15 when x 3y208-16 -8yy108-20 -1016 xO-8-10-16-201020 xSolve. Assume that y varies inversely as x.8. If y 4 when x 8,find y when x 2.9. If y -7 when x 3,find y when x -3.10. If y -6 when x -2,find y when x 4.11. If y -24 when x -3,find x when y -6.12. If y 15 when x 1,find x when y -3.13. If y 48 when x -4,find y when x 6.114. If y -4 when x , find x when y 2.2Chapter 117Glencoe Algebra 1Lesson 11-11.

NAMEDATE11-1PERIODPracticeInverse VariationDetermine whether each table or equation represents an inverse or a directvariation. Explain.1.2.yx3. x -3yxy0.2540-20.52000252-881.254-1674. y x8Asssume that y varies inversely as x. Write an inverse variation equation thatrelates x and y. Then graph the equation.5. y -2 when x -1216y24O7. y 2.5 when x 2yy128-16 -86. y -6 when x -5816 x12-24 -12 O-8-12-16-2424 xOx8. If y 124 when x 12, find y when x -24.9. If y -8.5 when x 6, find y when x -2.5.10. If y 3.2 when x -5.5, find y when x 6.4.11. If y 0.6 when x 7.5, find y when x -1.25.12. EMPLOYMENT The manager of a lumber store schedules 6 employees to take inventoryin an 8-hour work period. The manager assumes all employees work at the same rate.a. Suppose 2 employees call in sick. How many hours will 4 employees need to takeinventory?b. If the district supervisor calls in and says she needs the inventory finished in 6 hours,how many employees should the manager assign to take inventory?13. TRAVEL Jesse and Joaquin can drive to their grandparents’ home in 3 hours if theyaverage 50 miles per hour. Since the road between the homes is winding andmountainous, their parents prefer they average between 40 and 45 miles per hour.How long will it take to drive to the grandparents’ home at the reduced speed?Chapter 118Glencoe Algebra 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Write an inverse variation equation that relates x and y. Assume that y variesinversely as x. Then solve.

NAME11-1DATEPERIODWord Problem PracticeInverse Variation4. BUSINESS In the manufacturing of acertain digital camera, the cost ofproducing the camera varies inversely asthe number produced. If 15,000 camerasare produced, the cost is 80 per unit.Graph the relationship and label thepoint that represents the cost per unit toproduce 25,000 cameras.300Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Price per Unit ( )Find the illumination produced 8 feetfrom the same source.Yearsto DoubleMoney414.4512610.2971020Units Produced(thousands)x30a. Write an equation that representsthe relationship between frequency fand length . Use k for the constantof variation.b. If you have two different lengthstrings, which one vibrates morequickly (that is, which string has agreater frequency)?3. ELECTRICITY The resistance, in ohms,of a certain length of electric wire variesinversely as the square of the diameterof the wire. If a wire 0.04 centimeter indiameter has a resistance of 0.60 ohm,what is the resistance of a wire of thesame length and material that is0.08 centimeter in diameter?Chapter 111005. SOUND The sound produced by a stringinside a piano depends on its length. Thefrequency of a vibrating string variesinversely as its length.Annual InterestRate(percent)1820002. MONEY A formula called the Rule of72 approximates how fast money willdouble in a savings account. It is basedon the relation that the number of yearsit takes for money to double variesinversely as the annual interest rate.Use the information in the table to writethe Rule of 72 formula.yLesson 11-11. PHYSICAL SCIENCE The illumination Iproduced by a light source variesinversely as the square of the distance dfrom the source. The illuminationproduced 5 feet from the light source is80 foot-candles.Id2 k80(5)2 k2000 kc. Suppose a piano string 2 feet longvibrates 300 cycles per second. Whatwould be the frequency of a string4 feet long?9Glencoe Algebra 1

NAMEDATE11-1PERIODEnrichmentDirect or Indirect VariationFill in each table below. Then write inversely, or directly to complete eachconclusion.1. 24816322. Hours2456W44444Speed55555555DistanceAFor a set of rectangles with a widthof 4, the area variesas the length.3.Oat Bran1 cup2 cup1 cupWater1 cup2 cup3 cup123Servings3For a car traveling at 55 mi/h, thedistance covered variesas the hours driven.4.Hours of WorkRateHours128248Hours per Person100100100100202550100A job requires 128 hours of work. Thenumber of hours each person worksvariesas the numberof people working.6.5For a 100-mile car trip, the time thetrip takes variesas theaverage rate of speed the car travels.b3456h10101010A15For a set of right triangles with a heightof 10, the area variesas the base.Use the table at the right.7. x variesas y.8. z variesas y.9. x variesas z.Chapter 1110x11.522.53y23456z6040302420Glencoe Algebra 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Miles128People WorkingThe number of servings of oat branvariesas the numberof cups of oat bran.5.128

NAMEDATE11-2PERIODStudy Guide and InterventionRational Functions10The function y x is an example of a rational function.Because division by zero is undefined, any value of a variable that results in a denominatorof zero must be excluded from the domain of that variable. These are called excludedvalues of the rational function.Identify Excluded ValuesExampleState the excluded value for each function.3a. y xThe denominator cannot equal zero.The excluded value is x 0.4b. y x-5x-5 0Set the denominator equal to 0.x 5Add 5 to each side.The excluded value is x 5.Lesson 11-2ExercisesState the excluded value for each function.21. y x12. y x-33. y 44. y x5. y 56. y - 3x - 27. y x-18. y 9. y xx-22x - 4x 33xx 15x 10x-710. y x-511. y x-212. y 713. y 3x - 414. y x15. y 2x 83x 216xx 11x 47x - 3516. DINING Mya and her friends are eating at a restaurant. The total bill of 36 is split36among x friends. The amount each person pays y is given by y x , where x is thenumber of people. Graph the function.3632Bill per Person ( )Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.x 1x-4282420161284012345678Number of PeopleChapter 1111Glencoe Algebra 1

NAMEDATE11-2Study Guide and InterventionPERIOD(continued)Rational FunctionsIdentify and Use AsymptotesBecause excluded vales are undefined, they affectthe graph of the function. An asymptote is a line that the graph of a function approaches.aA rational function in the form y c has a vertical asymptote at the x-value thatx-bmakes the denominator equal zero, x b. It has a horizontal asymptote at y c.1 2 . Then graph the function.Identify the asymptotes of y Examplex-1Step 1 Identify and graph the asymptotes using dashed lines.vertical asymptote: x 1horizontal asymptote: y 2Step 2 Make a table of values and plot the points.Then connect them.x–1023y1.5132.5yy 2x01y x-1 2x 1ExercisesIdentify the asymptotes of each function. Then graph the function.43. y x 1-22. y x0x24. y x -3x25. y -26. y x 1x-3Chapter 11yxx00y0yyy012yx0xGlencoe Algebra 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.31. y x

NAMEDATE11-2PERIODSkills PracticeRational FunctionsState the excluded value for each function.61. y x22. y x3. y x-34. y 3x - 55. y -56. y x7. y x-18. y 99. y x-2x 4x 6x 83x 212x - 149x - 365x 40Identify the asymptotes of each function. Then graph the function.311. y x212. y x 1Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.y313. y x-2yxyx013x0115. y 3x 1yChapter 11x0214. y -1x-20yyx0Lesson 11-2110. y x0xGlencoe Algebra 1

NAMEDATE11-2PERIODPracticeRational FunctionsState the excluded value for each function.2x3. y -11. y x32. y x-14. y 5. y x-5x 5x 12x 312x 3616. y 5x - 2Identify the asymptotes of each function. Then graph the function.38. y x17. y xyx-1yxx0111. y 2x 2212. y -1x-3y0x 1yxx0Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.210. y yyxx013. AIR TRAVEL Denver, Colorado, is located approximately1000 miles from Indianapolis, Indiana. The average speed of a1000plane traveling between the two cities is given by y x ,where x is the total flight time. Graph the function.0Average Speed (mph)029. y 1000800600400200012345Total Flight TimeChapter 1114Glencoe Algebra 1

NAME11-2DATEPERIODWord Problem PracticeRational Functions17,900x 1.2y 100, where x is the age ofthe car. What are the asymptotes of thefunction? Explain why x 0 cannot bean asymptote.3002502001501005001234Time (hours)2. DRIVING Peter is driving to hisCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.4. USED CARS While researching cars topurchase online, Ms. Jacobs found thatthe value of a used car is inverselyproportional to the age of the car. Theaverage price of a used car is given bygrandparents’ house 40 miles away.During the trip, Peter makes a 30-minutestop for lunch. The average speed of40, wherePeter’s trip is given by y x 0.55. FAMILY REUNION The Gaudet familyis holding their annual reunion atWatkins Park. It costs 50 to get apermit to hold the reunion at the park,and the family is spending 8 per personon food. The Gaudets have agreed to splitthe cost of the event evenly amongall those attending.a. Write an equation showing the costper person y if x people attend thereunion.x is the total time spent in the car. Whatare the asymptotes of the function?b. What are the asymptotes of theequation?c. Now assume that the family wants tolet a long-lost cousin attend for free.Rewrite the equation to find the newcost per paying person y.3. ERROR ANALYSIS Nicolas is graphing20- 6 and draws athe equation y x 3graph with asymptotes at y 3 andx – 6. Explain the error that Nicolasmade in his graph.d. What are the asymptotes for the newequation?Chapter 1115Glencoe Algebra 1Lesson 11-2Average Speed (km/h)1. BULLET TRAINS The Shinkansen, orJapanese bullet train network, provideshigh-speed transportation throughoutJapan. Trains regularly operate at speedsin excess of 200 kilometers per hour. Theaverage speed of a bullet train travelingbetween Tokyo and Kyoto is given by515y x , where x is the total travel timein hours. Graph the function.

NAMEDATE11-2PERIODEnrichmentInequalities involving Rational FunctionsInequalities involving rational functions can be graphed much like those involving linearfunctions.1Graph y x.ExampleyStep 1 Plot points and draw a smooth solid curve. Because theinequality involves a greater than or equal to sign,1solu

Chapter 11 Resource Masters Bothell, WA Chicago, IL † Columbus, OH † New York, NY 000i_ALG1_A_CRM_C11_TP_660285.indd 10i_ALG1_A_CRM_C11_TP_660285.indd 1 PDF Pass 112/20/10 11:29 PM2/20/10 11:29 PM