9-1 Skills Practice

NAME DATEPERIODSkills Practice9-1Multiplying and Dividing Rational ExpressionsSimplify each expression.3x8y2(y6)34y(x6)3(x )63. }3 4 x182x 2 69x12x2 2 4(x 2 2)(x 1 1)6. }} }3a2 2 24a a 2 83a 1 12a3m2n}7. }}210(ef)38e f5s2s 2480y449z v3 }11. }2 4 21g25y514z v32z 712. }5 7 4 }12 5 }q2 1 2q6q3xx 2413. } 4 }x(x 2 2)2w2 2 5w 2 24w111s1210s}}10. }?}2517gyn3 mn 268. } ? } }6e9. }2 ? }53x2x122}4. }245. } }24e35fb5ab325a b}2. }2 2Lesson 9-121x3y14x y}1. }2 2q2 2 43qq214. } 4 }}2w2 2 6w 2 7w1315. }} ? }}t2 1 19t 1 844t 2 42t 2 2t 1 9t 1 14t 1 12}16. }} ? }}2(w 2 8)(w 2 7)x2 2 5x 1 42x 2 8117. }} 4 (3x2 2 3x) }19. c2}}2d2}c62}}5d52}Glencoe/McGraw-Hill16a2 1 40a 1 253a 2 10a 2 8(4a 1 5)(a 2 4)4a 1 5a 2 8a 1 16}}18. }}}4 }}2220.a2 2 b2}}4a}a1b}}2a519a2b}Glencoe Algebra 2

NAME DATEPERIODSkills Practice9-2Adding and Subtracting Rational ExpressionsFind the LCM of each set of polynomials.1. 12c, 6c2d 12c 2d2. 18a3bc2, 24b2c2 72a 3b 2c 23. 2x 2 6, x 2 3 2(x 2 3)4. 5a, a 2 1 5a(a 2 1)5. t2 2 25, t 1 5 (t 1 5)(t 2 5)6. x2 2 3x 2 4, x 1 1 (x 2 4)(x 1 1)Simplify each expression.5 5x 1 3yy2c 1 52c 2 739. } 1 4 }125y2 12z 2 2y5yz3w232w 29mm2nmn2m13. } 2 } }}3w 1 715. } 2 }}}22m17. } 2 } }1x 1 2x 1 1x2 1 x 1 1xx1119. }}1 } }}2nn232n 1 2n 2 2n 2 321. } 1 }}2Glencoe/McGraw-Hill5 2 1 5m 2n74gh34h7h 1 3g53b 1 d2 15bd 2 6b 2 2d3bd14. } 2 } }}3t22x5 2 3t5x224zz24z 1 4 5z 2 1 4z 2 16z1116. } 1 } }18. } 1 } }}2x 1 1x254x 2 3x 2 102x 2 1 5x 2 220. } 2 }}}}23y 1 y 2 122y 1 6y 1 822. }}2 }}22n12} 2m n12. } 1 }2 }}a2632a54p q10. }1 } }}211. }2 2 } }}2a121338p q8. }}22 1 }Lesson 9-23x7. } 1 } }y 1 12}}}525Glencoe Algebra 2

NAME DATE9-3PERIODStudy Guide and InterventionGraphing Rational FunctionsVertical Asymptotes and Point Discontinuityp(x)Rational Functionan equation of the form f(x) 5 } , where p(x) and q(x) are polynomial expressions andq(x)q(x) Þ 0Vertical Asymptoteof the Graph of aRational FunctionAn asymptote is a line that the graph of a function approaches, but never crosses.If the simplified form of the related rational expression is undefined for x 5 a,then x 5 a is a vertical asymptote.Point Discontinuityof the Graph of aRational FunctionPoint discontinuity is like a hole in a graph. If the original related expression is undefinedfor x 5 a but the simplified expression is defined for x 5 a, then there is a hole in thegraph at x 5 a.ExampleDetermine the equations of any vertical asymptotes and the values4x2 1 x 2 3x 21of x for any holes in the graph of f(x) 5 }}.2First factor the numerator and the denominator of the rational expression.4x2 1 x 2 3x 21(4x 2 3)(x 1 1)(x 1 1)(x 2 1)f(x) 5 }}5 }}2The function is undefined for x 5 1 and x 5 21.(4x 2 3)(x 1 1)(x 1 1)(x 2 1)4x 2 3x21Since }} 5 } , x 5 1 is a vertical asymptote. The simplified expression isdefined for x 5 21, so this value represents a hole in the graph.Determine the equations of any vertical asymptotes and the values of x for anyholes in the graph of each rational function.4x 1 3x 2 101. f(x) 5 }}2asymptotes: x 5 2,x 5 253x 2 13x 1 5x 2 24. f(x) 5 }}2asymptote: x 5 22;1hole: x 5 }x11x 2 6x 1 57. f(x) 5 }}2asymptotes: x 5 1,x55 Glencoe/McGraw-Hill2x2 2 x 2 102x 2 52. f(x) 5 }}5hole: x 5 }x2 2 6x 2 7x 1 6x 2 75. f(x) 5 }}2asymptotes: x 5 1,x 5 272x2 2 x 2 32x 1 3x 2 98. f(x) 5 }}2asymptote: x 5 23;3hole: x 5 }529x2 2 x 2 12x 2 4x3. f(x) 5 }}2asymptote: x 5 0;hole x 5 43x2 2 5x 2 2x136. f(x) 5 }}asymptote: x 5 23x3 2 2x2 2 5x 1 6x 2 4x 1 39. f(x) 5 }}}2holes: x 5 1, x 5 3Glencoe Algebra 2Lesson 9-3Exercises

Assignment

NAME DATE9-3PERIODSkills PracticeGraphing Rational FunctionsDetermine the equations of any vertical asymptotes and the values of x for anyholes in the graph of each rational function.10x 2 13x 1 363x 2 2x 2 81. f(x) 5 }}22. f(x) 5 }}2asymptotes: x 5 4, x 5 22asymptotes: x 5 4, x 5 9x21x 2 4x 1 3x 1 12x 1 10x 2 243. f(x) 5 }}24. f(x) 5 }}2asymptote: x 5 2; hole: x 5 212asymptote: x 5 3; hole: x 5 1x2 1 8x 1 12x12x2 1 x 2 12x235. f(x) 5 }}6. f(x) 5 }}hole: x 5 22hole: x 5 3Graph each rational function.10x24x8. f(x) 5 }f (x )9. f(x) 5 }f (x )f (x )2Ox2x21OGlencoe/McGraw-HillO12. f(x) 5 }f (x )xO531xx2 2 4x2211. f(x) 5 }f (x ) xxx1210. f(x) 5 }O2f (x )xOxGlencoe Algebra 2Lesson 9-323x7. f(x) 5 }

NAME DATE9-4PracticePERIOD(Average)Direct, Joint, and Inverse VariationState whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.5k1. u 5 8wz joint; 8 2. p 5 4s direct; 4 3. L 5 } inverse; 5 4. xy 5 4.5 inverse; 4.5Cd5. } 5 pdirect; p6. 2d 5 mn1joint; }1.25g7. } 5 hinverse; 1.2534x8. y 5 }3inverse; }Find each value.9. If y varies directly as x and y 5 8 when x 5 2, find y when x 5 6. 2410. If y varies directly as x and y 5 216 when x 5 6, find x when y 5 24. 1.511. If y varies directly as x and y 5 132 when x 5 11, find y when x 5 33. 3965612. If y varies directly as x and y 5 7 when x 5 1.5, find y when x 5 4. }13. If y varies jointly as x and z and y 5 24 when x 5 2 and z 5 1, find y when x 5 12and z 5 2. 28814. If y varies jointly as x and z and y 5 60 when x 5 3 and z 5 4, find y when x 5 6and z 5 8. 24015. If y varies jointly as x and z and y 5 12 when x 5 22 and z 5 3, find y when x 5 4and z 5 21. 86416. If y varies inversely as x and y 5 16 when x 5 4, find y when x 5 3. }17. If y varies inversely as x and y 5 3 when x 5 5, find x when y 5 2.5. 618. If y varies inversely as x and y 5 218 when x 5 6, find y when x 5 5. 221.619. If y varies directly as x and y 5 5 when x 5 0.4, find x when y 5 37.5. 320. GASES The volume V of a gas varies inversely as its pressure P. If V 5 80 cubiccentimeters when P 5 2000 millimeters of mercury, find V when P 5 320 millimeters ofmercury. 500 cm321. SPRINGS The length S that a spring will stretch varies directly with the weight F thatis attached to the spring. If a spring stretches 20 inches with 25 pounds attached, howfar will it stretch with 15 pounds attached? 12 in.22. GEOMETRY The area A of a trapezoid varies jointly as its height and the sum of itsbases. If the area is 480 square meters when the height is 20 meters and the bases are28 meters and 20 meters, what is the area of a trapezoid when its height is 8 meters andits bases are 10 meters and 15 meters? 100 m2 Glencoe/McGraw-Hill538Glencoe Algebra 2

NAME DATEPERIODStudy Guide and Intervention9-5Classes of FunctionsIdentify GraphsYou should be familiar with the graphs of the following functions.FunctionDescription of GraphConstanta horizontal line that crosses the y-axis at aDirect Variationa line that passes through the origin and is neither horizontal nor verticalIdentitya line that passes through the point (a, a), where a is any real numberGreatest Integera step functionAbsolute ValueV-shaped graphQuadratica parabolaSquare Roota curve that starts at a point and curves in only one directionRationala graph with one or more asymptotes and/or holesInverse Variationa graph with 2 curved branches and 2 asymptotes,x 5 0 and y 5 0 (special case of rational function)ExercisesIdentify the function represented by each graph.2.y3.yOOOxxxquadratic4.rationaldirect variation5.yOyxconstant7.6.yOOxabsolute value8.yOidentityGlencoe/McGraw-Hillxgreatest integer9.yyOxO yxxsquare root541inverse variationGlencoe Algebra 2Lesson 9-51.

NAME DATEPERIODStudy Guide and Intervention9-5(continued)Classes of FunctionsIdentify EquationsYou should be able to graph the equations of the following functions.FunctionGeneral EquationConstanty5aDirect Variationy 5 axIdentityy5xGreatest Integerequation includes a variable within the greatest integer symbol, v bAbsolute Valueequation includes a variable within the absolute value symbol,ax 2Quadraticy5Square Rootequation includes a variable beneath the radical sign, Ï·Rationaly5}Inverse Variationy5} 1 bx 1 c, where a Þ 0p(x)q(x)axExercisesIdentify the function represented by each equation. Then graph the equation.6x431. y 5 } inverse variation 2. y 5 } x direct variationyyyOOOxxx4. y 5 3x 2 1 absolutevalue9 2x 02x5. y 5 2 } inverse variation 6. y 5 }Ox7. y 5 Ïwx 2 2 square rootOGlencoe/McGraw-HillyxO9. y 5 }} rationalyxO542xx2 1 5x 1 6x128. y 5 3.2 constantyOgreatestintegeryy x223. y 5 2 } quadraticyxOxGlencoe Algebra 2

NAME DATEPERIODSkills Practice9-6Solving Rational Equations and Inequalities12xx2193x2. 2 5 } 1 } }2z2624. 3 2 z 5 } 1, 23. } 5 } 212d111d22s2355. } 5 } 52x 1 3x113212y8. 2 } 5 y 2 7 3, 4x11x 1 1010. } 2 } . 0 k . 03k5v12. n 1 } , } n , 23 or 0 , n , 39. } 5 } 83v11. 2 2 } , } 0 , v , 412m3m5213. } 2 } , 2 } 0 , m , 115x9x 2 7x122qq115n 292n2319. } 1 }5 } 242x282x 1 2x2x 1 22ee 241e222x 2 3x1121. } 1 } 5 } [2e1223. }1 } 5 } 262 Glencoe/McGraw-Hill3n12x12n32x14. } , } 2 1 0 , x , }b12b2116. } 5 4 2 } 417. 2 5 } 1 } 251n1343k3b 2 2b1115. } 1 } 5 9 352q8s6. } 5 } 25, 87. } 5 } 23x22x141 1234n1. } 5 } 21Lesson 9-6Solve each equation or inequality. Check your solutions.4z8z 2 8 2z1218. 8 2 } 5 } }1w121w224w 2420. } 1 } 5 }[212s 1 19s 1 7s 1 123s135s1422. }}2}5} 228t 294t132t2324. }1}5} 52549Glencoe Algebra 2

NAME6DATEPERIODChapter 6 Cumulative Review(Chapters 1–6)1. Find the value of 12 1 36 4 4 2 (5 2 7)2. (Lesson 1-1)1.2. Find the slope of the line that is parallel to the line withequation 3x 1 4y 5 10. (Lesson 2-3)2.3. Describe the system 2x 2 3y 5 21 and y 2 5 5 }2}x as3.3consistent and independent, consistent and dependent, orinconsistent. (Lesson 3-1)4. Find the coordinates of the vertices of the figure formedby the system of inequalities. (Lesson 3-3)x 22x1y#6y 22x 1 y 225. Find the value of6. Solve322) 26)5 12. (Lesson 4-5)44 34 34.5.44 21a11?5by using inverse matrices.b21336.(Lesson 4-8)7. Use synthetic division to find(2x4 1 5x3 2 x2 1 10x 1 4) 4 (x 1 3). (Lesson 5-3)7.48. Use a calculator to approximate Ï983w to threedecimal places. (Lesson 5-5)8.9. Solve Ïxw2 2 1 1 5 8. (Lesson 5-8)9.10. PHYSICS An object is thrown straight up from the top ofa 100-foot platform at a velocity of 48 feet per second. Theheight h(t) of the object t seconds after being thrown isgiven by h(t) 5 216t2 1 48t 1 100. Find the maximumheight reached by the object and the time it takes toachieve this height. (Lesson 6-1)10.11.y11. Solve x2 5 2x 1 3 by graphing. (Lesson 6-2)O12. Solve 4x2 2 4x 5 24 by factoring. (Lesson 6-3)12.13. Find the value of the discriminant for 7x2 1 5x 1 1 5 0.Then describe the number and type of roots for theequation. (Lesson 6-5)13.14. Write y 5 x2 2 7x 1 5 in vertex form. (Lesson 6-6)14. Glencoe/McGraw-Hill372xGlencoe Algebra 2

NAME9DATEPERIODChapter 9 Cumulative Review(Chapters 1–9)341 51. Determine whether C 5and D 523 1inverses. (Lesson 4-7)2. Simplify the expression341}}16are2}5}1621 }}}} 5w 3 . (Lesson 5-7)1 22.3. Solve x2 1 2x 1 2 5 0 by completing the square.4. Graph y x2 2 4x.1.(Lesson 6-4)3.4.(Lesson 6-7)yO5. Use synthetic substitution to find f(3) forf(x) 5 3x3 2 7x2 1 5x 2 10. (Lesson 7-4)5.6. List all of the possible rational zeros of2x4 2 5x3 1 3x2 2 12x 2 6. (Lesson 7-6)6.7. Write an equation for a circle with center at (0, 23) thatpasses through (5, 7). (Lessons 8-1 and 8-3)7.8. Write an equation for the ellipse whose major axis is10 units long and parallel to the x-axis, whose minor axis is6 units long, and whose center is at (1, 22). (Lesson 8-4)8.9. State whether the graph of 5x2 1 5y2 2 10x 1 15y 5 10 is anellipse, circle, parabola, or hyperbola. (Lesson 8-6)9.10. Simplify9y2 2 36}}5y2 1 10y}} . (Lesson 9-1)6y 2 12}}10y2 1 20yx10.11. Suppose y varies jointly as x and z. Find y when x 5 16 andz 5 5, if y 5 9 when x 5 3 and z 5 12. (Lesson 9-4)11.12. Evita adds a 75% acid solution to 8 milliliters of solutionthat is 15% acid. The function that represents the percent12.8(0.15) 1 x(0.75)of acid in the resulting solution is f(x) 5 }},81xwhere x is the amount of 75% acid solution added. Howmuch 75% acid solution should be added to create a solutionthat is 50% acid? (Lesson 9-6) Glencoe/McGraw-Hill570Glencoe Algebra 2

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. 1. u 5 8wz joint; 8 2. p 5 4s direct; 4 3. L 5 inverse; 5 4. xy 5 4.5 inverse; 4.5 5. 5 p 6. 2d 5 mn 7. 5 h 8. y 5 direct; p joint; inverse; 1.25 inverse; Find each value. 9. If y varies directly as x and y 5 8 when x 5 2, find y .