Advanced Functions and Relations

Transcription

1 Advanced Functions and Relations Focus Use myriad tools to interpret and solve challenging real-world problems. CHAPTER 8 Rational Epressions and Equations Evaluate and simplify rational epressions with polynomial denominators. CHAPTER 9 Eponential and Logarithmic Relations Utilize the properties of logarithms to evaluate eponential and logarithmic relations. CHAPTER 0 Conic Sections Recognize and distinguish between equations of different conic sections. Manipulate and analyze conic sections. 438 Unit 3

2 Algebra and Earth Science Earthquake! Have you ever felt an earthquake? Earthquakes are very frightening and can cause great destruction. An earthquake occurs when the tectonic plates of the Earth split or travel by each other. Some areas of the Earth seem to have more earthquakes than others. San Francisco s Great Quake of 906 almost destroyed the city with a 7.9 magnitude earthquake, 6 aftershocks, and one of the worst urban fires in American history. In this project, you will eplore how functions and relations are related to locating, measuring, and classifying earthquakes. Log on to ca.algebra.com to begin. CRBIS Unit 3 Advanced Functions and Relations 439

3 8 Rational Epressions and Equations Standard 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. (Key) Key Vocabulary continuity (p. 457) direct variation (p. 465) inverse variation (p. 467) rational epression (p. 44) Real-World Link Intensity of Light The intensity, or brightness, of light decreases as the distance between a light source, such as a star, and a viewer increases. You can use an inverse variation equation to epress this relationship. Rational Epressions and Equations Make this Foldable to help you organize your notes. Begin with a sheet of plain 8 by paper. Fold in half lengthwise leaving a margin at the top. Fold again in thirds. pen. Cut along the folds on the short tab to make three tabs. Label as shown. 440 Chapter 8 Rational Epressions and Equations NASA

4 GET READY for Chapter 8 Diagnose Readiness You have two options for checking Prerequisite Skills. ption ption Take the Quick Check below. Refer to the Quick Review for help. Take the nline Readiness Quiz at ca.algebra.com. Solve each equation. Write in simplest form. (Lesson -3). 8 5 = t = = 5 a = 9m b = s = r = n = 7 9. INCME Jamal s allowance is 7 8 of Syretta s allowance. If Syretta s allowance is $0, how much is Jamal s allowance? (Lesson -3) 0. BAKING Marc gave away of the cookies 3 he baked. If he gave away 4 cookies, how many cookies did he bake? (Lesson -3) EXAMPLE Solve 3 7 = 4 k. Write in simplest form (43) = 4k Multiply each side by = 4k Simplify = k Divide each side by 4. (4) = k Simplify. Since the GCF of 559 and 697 is, the solution is in simplest form. Solve each proportion. (Prerequisite Skill). 3 4 = r = 5 y = m a = v 9 = 8 4. t 3 = = b p = = 3 z = r. REAL ESTATE A house which is assessed for $00,000 pays $3000 in taes. What should the taes be on a house in the same area that is assessed at $350,000? (Prerequisite Skill) EXAMPLE Solve the proportion 4 7 = u = u 5 Write the equation. 4(5) = 7u Find the cross products. 60 = 7u Simplify. 60 = u Divide each side by 7. 7 Since the GCF of 60 and 7 is, the answer is in simplified form. u = 60 7 or Chapter 8 Get Ready for Chapter 8 44

5 8- Multiplying and Dividing Rational Epressions Main Ideas Simplify rational epressions. Simplify comple fractions. Standard 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. (Key) The Goodie Shoppe sells candy and nuts by the pound. ne item is a miture made with 8 pounds of peanuts and 5 pounds of cashews. Add lb of peanuts. 8 8 Therefore, or of the lb of peanuts 5 lb of cashews 3 lb of nuts 3 miture is peanuts. If the store manager adds an additional pounds of peanuts to the 8+ miture, then of the 3 + miture will be peanuts. New Vocabulary rational epression comple fraction Simplify Rational Epressions A ratio of two polynomial epressions 8+ such as is called a rational epression. Because variables in 3 + algebra often represent real numbers, operations with rational numbers and rational epressions are similar. To write a fraction in simplest form, you divide both the numerator and denominator by their greatest common factor (GCF). To simplify a rational epression, you use similar techniques. EXAMPLE a. Simplify Simplify a Rational Epression ( - 5). ( - 5)( - ) Look for common factors. ( - 5) -5 = ( - 5)( - ) - = - -5 How is this similar to simplifying? 0 5 Simplify. b. Under what conditions is this epression undefined? Ecluded Values Numbers that would cause the epression to be undefined are called ecluded values. Just as with a fraction, a rational epression is undefined if the denominator is equal to 0. To find when this epression is undefined, completely factor the original denominator. ( - 5) ( - 5) = ( - 5)( - ) ( - 5)( - )( + ) - = ( - )( + ) The values that would make the denominator equal to 0 are 5,, or -. So the epression is undefined when = 5, =, or = Chapter 8 Rational Epressions and Equations

6 Simplify each epression. Under what conditions is the epression undefined? 3y (y + 6) A. B. 4 3 ( - 7-8) (y + 6) ( y - 8y + ) ( - 64) Use the Process of Elimination For what value(s) of is undefined? A -4, -3 B -4 C 0 D -4, 3 Eliminating Choices Sometimes you can save time by looking at the possible answers and eliminating choices, rather than actually evaluating an epression or solving an equation. Read the Item You want to determine which values of make the denominator equal to 0. Solve the Item Notice that if equals 0 or a positive number, must be greater than 0. Therefore, you can eliminate choices C and D. Since both A and B contain -4, determine whether the denominator equals 0 when = = (-3) + 7 (-3) + = -3 = or 0 Multiply and simplify. Since the denominator equals 0 when = -3, the answer is A.. For what values of is undefined? F -7, - G -7, H -, 7 J, 7 Personal Tutor at ca.algebra.com Sometimes you can factor out - in the numerator or denominator to help simplify rational epressions. EXAMPLE Simplify by Factoring ut Simplify z w - z z 3 - z 3 w. z w - z z 3 - z 3 w = z (w - ) z 3 ( - w) = z (-)( - w) 3 ( - w) z = - z or - z Factor the numerator and the denominator. w - = - (-w + ) or - ( - w) Simplify. Simplify each epression. y - 3 3A. 3 - y 3B. - y - 4y Etra Eamples at ca.algebra.com Lesson 8- Multiplying and Dividing Rational Epressions 443

7 Remember that to multiply two fractions, you multiply the numerators and multiply the denominators. To divide two fractions, you multiply by the multiplicative inverse, or reciprocal, of the divisor. Multiplication Division = = = 3 3 or = or 3 The same procedures are used for multiplying and dividing rational epressions. Multiplying Rational Epressions Words Rational Epressions To multiply two rational epressions, multiply the numerators and the denominators. d = ac, if b 0 and d 0. Symbols For all rational epressions a b and c d, a b c bd Dividing Rational Epressions Words To divide two rational epressions, multiply by the reciprocal of the divisor. Symbols For all rational epressions a b and c d, a b c d = a b d c = ad, if b 0, bc c 0 and d 0. The following eamples show how these rules are used with rational epressions. EXAMPLE Multiply and Divide Rational Epressions Alternative Method When multiplying rational epressions, you can multiply first and then divide by the common factors. For instance, in Eample 4, 4a 5b 5 b 6 a = 60a b 3 80 a 3 b. Now divide the numerator and denominator by the common factors. 3b 60a b 80 a 3 b = 3b 4 a 4 a Simplify 4a each epression. a. 5b 5 b 6 a 3 4a 5b 5 b 6 a = a 3 5 b b 3 Factor. 5 b a a a 3 b = Simplify. a a = 3b 4 a Simplify. 4 y b. 5 a 3 b y 3 3 5a b 4 y 5 a 3 b y 3 5ab = 4 y 3 5 a 3 b 5ab 3 Multiply by the reciprocal of the divisor. 3 y y 5 a b b b = Factor. 3 5 a a a b b b y y = Simplify. 3 a a y = 3 a Simplify. y 444 Chapter 8 Rational Epressions and Equations

8 4A. 8 t s 5 r 5sr t 3 s 8 ab 4C. 9b 5 y 3 0y 4B. 9 m n 3 6ab 8 a b 4 7 m 5 n 4 pq 4D. 5 w 7 z p 3 q 3 35 w 3 z 8 Sometimes you must factor the numerator and/or the denominator first before you can simplify a product or a quotient of rational epressions. EXAMPLE Polynomials in the Numerator and Denominator Simplify each epression. a = + 4) ( - ) 3 ( + ) ( + 3) ( + ) ( - ) 3 ( + 4) = ( + 3) b. a + a + 3 a + a - a - 9 a + a + 3 a + a - a - 9 = = a + a + 3 a - 9 a + a - = + ) + 3) (a - 3) (a + 3) (a + 4) (a - 3) = a + a + 4 Factor. Simplify. Distributive Property Multiply by the reciprocal of the divisor. Factor. Simplify. 5A. y - 5y + 5 y + 5y + 6 y + 4y - 5 5B. b + b - 35 b - 4 b - 5 b + Simplify Comple Fractions A comple fraction is a rational epression whose numerator and/or denominator contains a rational epression. The epressions below are comple fractions. Animation ca.algebra.com a 5 3b 3 t t + 5 m m p + 3 p - 4 To simplify a comple fraction, rewrite it as a division epression, and use the rules for division. Lesson 8- Multiplying and Dividing Rational Epressions 445

9 EXAMPLE Simplify r r - 5 s = r 5s - r r r - 5 s r 5s - r Simplify a Comple Fraction r r - 5 s. r 5s - r Epress as a division epression. r = r - 5 s 5s - r r Multiply by the reciprocal of the divisor. = r r (-)(r - 5s) Factor. (r + 5s) (r - 5s) r -r = r + 5s or - r Simplify. r + 5s Simplify each epression. ( + 3) 6A B. y - 7 y - 3 y - 49 y + 4y - Eample (pp ) Eample (p. 443) Simplify each epression.. 45m n 3 0 n a + b. a - b c 3 d 54 cd 5 5. STANDARDS PRACTICE Identify all values of y for which is undefined. y - 4 y - 4y - A -, 4, 6 B -6, -4, C -, 0, 6 D -, 6 Eample 3 (p. 443) Eample 4 (pp ) Eample 5 (p. 445) Eample 6 (p. 446) Simplify each epression. 6. 9y - 6y 3 y + 5y - 8. a 5 b c 3bc 3 8 a p + 6p (p + ) 4. c 3 d 3 a c d a 6p - 3 p b3 - a 3 a - b 9. 3t + 6 7t y 3 4t - 4 5t y y y y - 4y Chapter 8 Rational Epressions and Equations

10 HELP HMEWRK For See Eercises Eamples 6 9 0, , 35 Simplify each epression bc b 3 9. c + 5 c yz 4z 6 3 y 5. p 3 q -p 4q 8. 4t (t + ) 3t + 3 t - p 3 q mn m n 0. 3t t 3. -4ab c 4 c 8 a t t + t + t 3p - p - 49 p - 7p 3p m + n p - m + n 4q Under what conditions is ( + 5) ( - ) undefined? 35. For what values is d (d + ) (d + ) ( d - 4) undefined? 8. 5t - 5 t t t + t - 3 5d ( - 9 5df) 7. 4w w m 3 3n m n + y - y + y + y 36. GEMETRY A parallelogram with an area of square units has a base of 3-5 units. Determine the height of the parallelogram. PRACTICE EXTRA See pages 907, 933. Self-Check Quiz at ca.algebra.com 37. GEMETRY Parallelogram F has an area of square meters and a height of + 3 meters. Parallelogram G has an area of square meters and a height of 3 - meters. Find the area of right triangle H. Simplify each epression. 38. (-3 y) 3 9 y 4. y + 4y y + 5y a a a b 3 6 y 8y 0 a 3 b y z 5 5. ( 4y 39. ( -rs ) r s 3 4. a + a + a + 3a b - 4b b - b cd 8w ( -4w) 5c 40. ( -5m n ) 3 5 m n ) 50. w - w + 4 z 3 w - 8w + 80 w - 5w + 50 w - 9w + 0 r + r - 8 r + 4r + 3 r - 3r Lesson 8- Multiplying and Dividing Rational Epressions 447

11 a + ab + b a - b 53. Under what conditions is undefined? BASKETBALL For Eercises 54 and 55, use the following information. At the end of the season, the Los Angeles Clippers Elton Brand had made 4060 field goals out of 8098 attempts during his NBA career. 54. Write a ratio to represent the ratio of the number of career field goals made to career field goals attempted by Elton Brand at the end of the season. 55. Suppose Elton Brand attempted a field goals and made m field goals during the season. Write a rational epression to represent the ratio of the number of career field goals made to the number of career field goals attempted at the end of the season. Real-World Link Elton Brand became the first Duke Blue Devil ever to be picked number one overall in the NBA Draft in 999. Source: NBA AIRPLANES For Eercises 56 58, use the formula d = rt and the following information. An airplane is traveling at the rate r of 500 miles per hour for a time t of (6 + ) hours. A second airplane travels at the rate r of ( ) miles per hour for a time t of 6 hours. 56. Write a rational epression to represent the ratio of the distance d traveled by the first airplane to the distance d traveled by the second airplane. 57. Simplify the rational epression. What does this epression tell you about the distances traveled of the two airplanes? 58. Under what condition is the rational epression undefined? Describe what this condition would tell you about the two airplanes and g() = Graphing For Eercises 59 6, consider f() = Calculator 59. Simplify. What do you observe about the epression? Graph f() and g() on a graphing calculator. How do the graphs appear? 6. Use the table feature to eamine the function values for f() and g(). How do the tables compare? 6. How can you use what you have observed with f() and g() to verify that epressions are equivalent or to identify ecluded values? H..T. Problems 63. PEN ENDED Write two rational epressions that are equivalent. a + b -a + b 64. CHALLENGE Rewrite so it has a numerator of. 65. Which ne Doesn t Belong? Identify the epression that does not belong with the other three. Eplain your reasoning REASNING Determine whether = is sometimes, always, or never true. 3 3d + 5 Eplain. d Writing in Math Use the information about rational epressions on page 44 to eplain how rational epressions are used in mitures. Include an eample of a miture problem that could be represented by y 448 Chapter 8 Rational Epressions and Equations Reuters/CRBIS

12 68. ACT/SAT For what value(s) of is 4 - undefined? A -, B -, 0, C 0, D REVIEW Which is the simplified form of 4 3 y z - ( - y 3 z )? 4 F 7 y 4 z 5 H 4 y 3 z 5 G 4y z 5 J 4 y 4 z 5 Graph each function. State the domain and range. (Lesson 7-3) 70. y = - 7. y = - 7. y = Determine whether f () = - and g () = are inverse functions. (Lesson 7-) Determine whether each graph represents an odd-degree or an even-degree polynomial function. Then state how many real zeros each function has. (Lesson 6-3) 74. f() 75. f() 76. f() 77. ASTRNMY Earth is an average of kilometers from the Sun. If light travels kilometers per second, how long does it take sunlight to reach Earth? (Lesson 6-) Solve each equation by factoring. (Lesson 5-3) 78. r - 3r = u - 3u = 80. d - 5d = 0 Solve each equation. (Lesson -4) = = + PREREQUISITE SKILL Solve each equation. (Lesson -3) = = = = = = Lesson 8- Multiplying and Dividing Rational Epressions 449

13 8- Adding and Subtracting Rational Epressions Main Ideas Determine the LCM of polynomials. Add and subtract rational epressions. Standard 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. (Key) In order to produce a picture that is in focus, the distance between the camera lens and the film q must be controlled so that it satisfies a particular relationship. If the distance from the subject to the lens is p and the focal length of the lens is f, then the formula q = f - p can be used to determine the correct distance between the lens and the film. LCM of Polynomials To find or f - p, you must first find the least common denominator (LCD). The LCD is the least common multiple (LCM) of the denominators. To find the LCM of two or more numbers or polynomials, factor each number or polynomial. The LCM contains each factor the greatest number of times it appears as a factor. LCM of 6 and 4 LCM of a - 6a + 9 and a + a - 6 = 3 a - 6a + 9 = (a - 3) 4 = a + a - = (a - 3) (a + 4) LCM = 3 or LCM = (a - 3) (a + 4) EXAMPLE LCM of Monomials Find the LCM of 8 r s 5, 4 r 3 st, and 5 s 3 t. 8 r s 5 = 3 r s 5 Factor the first monomial. 4 r 3 st = 3 3 r 3 s t Factor the second monomial. 5 s 3 t = 3 5 s 3 t Factor the third monomial. LCM = r 3 s 5 t Use each factor the greatest number of times = 360 r 3 s 5 t it appears as a factor and simplify. Find the LCM of each set of monomials. A. a b 4 3, 7ac, 8 a 5 b c B. 6 m 3 n 5, 4mnp, 36 m 3 n 4 p 450 Chapter 8 Rational Epressions and Equations

14 EXAMPLE LCM of Polynomials Find the LCM of p p + 6p and p + 6p + 9. p p + 6p = p (p + ) (p + 3) Factor the first polynomial. p + 6p + 9 = (p + 3) Factor the second polynomial. LCM = p (p + ) (p + 3) Use each factor the greatest number of times it appears as a factor. Find the LCM of each set of polynomials. A. q - 4q + 4 and q 3-3 q + q B. k 3-5 k - k and k 3-8 k + 6k Add and Subtract Rational Epressions As with fractions, to add or subtract rational epressions, you must have common denominators. Specific Case 5 = = = 9 5 Find equivalent fractions that have a common denominator. Simplify each numerator and denominator. Add the numerators. General Case a c + b d = a d c d + b c d c = ad cd + bc cd = ad + bc cd As with fractions, you can use the least common multiple of the denominators to find the least common denominator for two rational epressions. EXAMPLE Simplify y + y Monomial Denominators y 5 y + 8y. 8y = y 6 + y 5y 8y 5y = 4 90y + 5 y 90 y = y 90y The LCD is 90y. Find the equivalent fractions that have this denominator. Simplify each numerator and denominator. Add the numerators. Simplify each epression. 3A. 8a 9b - 7 ab 3C. 3y - 3 5y 3B. 8 m n + m n 3D. 6c 7 b + d 3ab Etra Eamples at ca.algebra.com Lesson 8- Adding and Subtracting Rational Epressions 45

15 Common Factors Sometimes when you simplify the numerator, the polynomial contains a factor common to the denominator. Thus the rational epression can be further simplified. EXAMPLE Polynomial Denominators Simplify w + 4w w + 4 w - 8. w + 4w w + 4 w - 8 = w + 4A (w - 4) - w + 4 (w - 4) = w + 4 (w - 4) - (w + 4) () (w - 4) () = (w + ) - ()(w + 4) 4 (w - 4) w + - w - 8 = 4 (w - 4) = -w (w - 4) = - (w - 4) or - 4 (w - 4) 4 Factor the denominators. The LCD is 4 (w - 4). Subtract the numerators. Distributive Property Combine like terms. Simplify. Simplify each epression. - 4B Personal Tutor at ca.algebra.com ne way to simplify a comple fraction is to simplify the numerator and the denominator separately, and then simplify the resulting epressions. EXAMPLE - y Simplify Comple Fractions Simplify. + - y y y - + = y + The LCD of the numerator is y. The LCD of the denominator is. y - y = Simplify the numerator and denominator. + = y - y + Write as a division epression. = y - y Multiply by the reciprocal of the divisor. + y - = or y - Simplify. y ( + ) y + y 5A. Simplify each epression. y + a y - 5B. b + - b a 45 Chapter 8 Rational Epressions and Equations

16 EXAMPLE Use a Comple Fraction to Solve a Problem Check Your Solution You can check your answer by letting p equal any nonzero number, say. Use the definition of slope to find the slope of the line through the points. CRDINATE GEMETRY Find the slope of the line that passes through A ( p, ) and B ( 3 3, m = y - y - p ). Definition of slope 3 p - = 3 - y = 3 p, y =, = 3, and = p p = 6 - p p p - 6 3p = 6 - p p - 6 p 3p = p 3p p p - 6 or - 3 The LCD of the numerator is p. The LCD of the denominator is 3p. Write as a division epression. The slope is - 3. Find the slope of the line that passes through each pair of points. w, 7, w 7 ) 6A. C ( 4, 4 q ) and D ( 5 q, 5) 6B. E ( 7 7) and F ( Eamples, (pp ) Eample 3 (p. 45) Eample 4 (p. 45) Eample 5 (p. 45) Eample 6 (p. 453) Find the LCM of each set of polynomials.. y, 6. 6ab 3, 5 b a, 0ac 3. -, , Simplify each epression. 5. y - y m - 7m - m 9. 6 d + 4d d a 5 b - b 8ab a a - a a GEMETRY An epression for the area of a rectangle is Find the width of the rectangle. Epress in simplest form. 4 Lesson 8- Adding and Subtracting Rational Epressions 453

17 HELP HMEWRK For See Eercises Eamples 8, 9 0, , , 35 6 Find the LCM of each set of polynomials s, 35 s t y, 0yz 0. 4w -, w y, 3 + y Simplify each epression.. 6 ab + 8 a v + 7 4v y - y a b - 7a 5 a 6. 7 y a 8 - y a a 8. m m y 3m + 6 y b + + b b - b - 3 b - 3b y y - 9 d - 4 d + d d + d - 6 ( + y) ( - y ) ( - y) ( + y ) 34. GEMETRY An epression for the length of one rectangle is The length of a similar rectangle is epressed as + 3. What is the scale - 4 factor of the two rectangles? Write in simplest form. 35. GEMETRY Find the slope of a line that contains the points A ( p, q ) and B ( q, p ). Write in simplest form. Find the LCM of each set of polynomials a 3, 5bc 3, b p q 3, 6pq 4,4p t + t - 3, t + 5t n - 7n +, n - n - 8 Simplify each epression r q - 5q - q 44. h - 9h h - 0h m + n m - n + m n - m + n m + n 4. 3y w - 6 5w y + y - + y + y Write ( s s + - ) s ( + in simplest form. - s) 49. What is the simplest form of ( a + ) ( 3-0? a + 7) 50. GEMETRY Find the perimeter of the quadrilateral. Epress in simplest form. y y - 3y Chapter 8 Rational Epressions and Equations

18 ELECTRICITY For Eercises 5 and 5, use the following information. In an electrical circuit, if two resistors with R resistance R and R are connected in parallel as shown, the relationship between these resistances and the resulting combination resistance R = +. is R R R 5. If R is ohms and R is 4 ohms less than R. twice ohms, write an epression for Real-World Link The Tour de France is the most popular bicycle road race. It lasts days and covers 500 miles. Source: World Book Encyclopedia R 5. A circuit with two resistors connected in parallel has an effective resistance of 5 ohms. ne of the resistors has a resistance of 30 ohms. Find the resistance of the other resistor. BICYCLING For Eercises 53 55, use the following information. Jalisa is competing in a 48-mile bicycle race. She travels half the distance at one rate. The rest of the distance, she travels 4 miles per hour slower. 53. If represents the faster pace in miles per hour, write an epression that represents the time spent at that pace. 54. Write an epression for the time spent at the slower pace. 55. Write an epression for the time Jalisa needed to complete the race. 56. MAGNETS For a bar magnet, the magnetic field strength H at a point P m m -. Write a simpler along the ais of the magnet is H = L(d - L) L(d + L) epression for H. EXTRA PRACTICE L Point See pages 908, 933. P d L Self-Check Quiz at ca.algebra.com H..T. Problems d L d 57. PEN ENDED Write two polynomials that have a LCM of d 3 - d. 58. FIND THE ERRR Lorena and Yong-Chan are simplifying a - b. Who is correct? Eplain your reasoning. Lorena b a = a b ab ab b a = ab Yong-Chan = a b a b CHALLENGE Find two rational epressions whose sum is. ( + )( - ) 60. REASNING In the epression a + b + c, a, b, and c are nonzero real numbers. Determine whether each statement is sometimes, always, or never true. Eplain your answer. a. abc is a common denominator. b. abc is the LCD. c. ab is the LCD. d. b is the LCD. bc + ac + ab e. The sum is. abc Lesson 8- Adding and Subtracting Rational Epressions Pascal Rondeau/Allsport/Getty Images 455

19 6. Writing in Math Use the information on page 450 to eplain how subtraction of rational epressions is used in photography. Include an equation that could be used to find the distance between the lens and the film if the focal length of the lens is 50 millimeters and the distance between the lens and the object is 000 millimeters. 6. ACT/SAT What is the sum of - y 5 and + y 4? A + 9y 0 B 9 + y 0 C 9 - y 0 D - 9y REVIEW Given: Two angles are complementary. The measure of one angle is 5 more than the measure of the other angle. Conclusion: The measures of the angles are 30 and 45. This conclusion F is contradicted by the first statement given. G is verified by the first statement given. H invalidates itself because a 45 angle cannot be complementary to another. J verifies itself because 30 is 5 less than 45. Simplify each epression. (Lesson 8-) y 3 (5yz) (3y) 3 0 y 66. Graph y +. (Lesson 7-7) a - 0 a + 4a 0a - 0 Find all of the zeros of each function. (Lesson 6-9) 67. g () = h () = GARDENS Helene Jonson has a rectangular garden 5 feet by 50 feet. She wants to increase the garden on all sides by an equal amount. If the area of the garden is to be increased by 400 square feet, by how much should each dimension be increased? (Lesson 5-5) 70. Three times a number added to four times a second number is. The second number is two more than the first number. Find the numbers. (Lesson 3-) PREREQUISITE SKILL Factor each polynomial. (Lesson 5-3) Chapter 8 Rational Epressions and Equations

20 8-3 Graphing Rational Functions Main Ideas Determine the limitations on the domains and ranges of the graphs of rational functions. Graph rational functions. Standard 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. (Key) New Vocabulary rational function continuity asymptote point discontinuity A group of students want to get their favorite teacher, Mr. Salgado, a retirement gift. They plan to get him a gift certificate for a weekend package at a lodge in a state park. The certificate costs $50. If c represents the cost for each student and s represents the number of students, then c = 50 s. Cost (dollars) C student could pay $50. 5 students could each pay $6. 50 students could each pay $ s Students Domain and Range The function c = 50 s is a rational function. A rational function has an equation of the form f () = p(), where p() and q() q() are polynomial functions and q() 0. Here are other rational functions. f () = + 3 g() = 5-6 h() = + 4 ( - )( + 4) No denominator in a rational function can be zero because division by zero is not defined. The functions above are not defined at = -3, = 6, and = and = -4, respectively. The domain of a rational function is limited to values for which the function is defined. The graphs of rational functions may have breaks in continuity. This means that, unlike polynomial functions, which can be traced with a pencil never leaving the paper, not all rational functions are traceable. Breaks in continuity occur at values that are ecluded from the domain. They can appear as vertical asymptotes or as point discontinuity. An asymptote is a line that the graph of the function approaches, but never touches. Point discontinuity is like a hole in a graph. Vertical Asymptote If the rational epression of a function is written in simplest form and the function is undefined for = a, then the line = a is a vertical asymptote. Vertical Asymptotes Property Words Eample Model For f () = - 3, the line = 3 is a vertical asymptote. f() f() 3 3 Lesson 8-3 Graphing Rational Functions 457

21 Point Discontinuity Property Point Discontinuity Animation ca.algebra.com EXAMPLE Words Eample If the original function is undefined for = a but the rational epression of the function in simplest form is defined for = a, then there is a hole in the graph at = a. ( + )( - ) f () = + can be simplified to f () = -. So, = - represents a hole in the graph. Model f ( ) ( )( ) f ( ) Limitations on Domain Determine the equations of any vertical asymptotes and the values of for any holes in the graph of f () = First factor the numerator and denominator of the rational epression. ( - )( + ) - = ( - )( - 5) ( - )( + ) + The function is undefined for = and = 5. Since =, ( - )( - 5) -5 = 5 is a vertical asymptote, and = represents a hole in the graph. Parent Function The parent function for the family of rational, or reciprocal, functions is y =. CHECK You can use a graphing calculator to check this solution. The graphing calculator screen at the right shows the graph of f(). The graph shows the vertical asymptote at = 5. It is not clear from the graph that the function is not defined at =. However, if you use the value function of the CALC menu and enter at the X = prompt, you will see that no value is returned for Y=. This shows that f() is not defined at =. Q Ó]ÊnRÊÃV \Ê ÊLÞÊQ ä]ê ärêãv \Ê. Determine the equations of any vertical asymptotes and the values of for any holes in the graph of f () =. For some rational functions, the values of the range are limited. ften a horizontal asymptote occurs where a value is ecluded from the range. For eample, is ecluded from the range of f () =. + The graph of f () gets increasingly close to a horizontal asymptote as increases or decreases. 458 Chapter 8 Rational Epressions and Equations f ( ) 6 8 y horizontal asymptote 4 6

22 Graph Rational Functions You can use what you know about vertical asymptotes and point discontinuity to graph rational functions. Graphing Rational Functions Finding the - and y-intercepts is often useful when graphing rational functions. EXAMPLE Graph f () = Graph with Vertical and Horizontal Asymptotes -. The function is undefined for =. Since is in simplest form, = is a - vertical asymptote. Draw the vertical asymptote. Make a table of values. Plot the points and draw the graph. As increases, it appears that the y-values of the function get closer and closer to. The line with the equation f () = is a horizontal asymptote of the function.. Graph f () = + -. f () f() f() Personal Tutor at ca.algebra.com As you have learned, graphs of rational functions may have point discontinuity rather than vertical asymptotes. The graphs of these functions appear to have holes. These holes are usually shown as circles on graphs. EXAMPLE Graph with Point Discontinuity Graph f () = Notice that - 9 = ( + 3)( - 3) or Therefore, the graph of f () = - 9 is the + 3 graph of f () = - 3 with a hole at = -3. f() f() Graph f () = In the real world, sometimes values on the graph of a rational function are not meaningful. Etra Eamples at ca.algebra.com Lesson 8-3 Graphing Rational Functions 459

23 Use Graphs of Rational Functions AVERAGE SPEED A boat traveled upstream at r miles per hour. During the return trip to its original starting point, the boat traveled at r miles per hour. The average speed for the entire trip R is given by the formula R = r r r + r. a. Let r be the independent variable and let R be the dependent variable. Draw the graph if r = 0 miles per hour. The function is R = r (0) r + 0 or R = 0 r r + 0. The vertical asymptote is r = -0. Graph the vertical asymptote and the function. Notice that the horizontal asymptote is R = 0. b. What is the R-intercept of the graph? The R-intercept is 0. c. What domain and range values are meaningful in the contet of the problem? In the problem contet, the speeds are nonnegative values. Therefore, only values of r greater than or equal to 0 and values of R between 0 and 0 are meaningful SALARIES A company uses the formula S() = to determine the + salary in thousands of dollars of an employee during his th year. Draw the graph of S(). What domain and range values are meaningful in the contet of the problem? What is the meaning of the horizontal asymptote for the graph? Eample (p. 458) Determine the equations of any vertical asymptotes and the values of for any holes in the graph of each rational function.. f () = 3. f () = Eample (p. 459) Eample 3 (p. 459) Graph each rational function. 3. f () = + 5. f () = 4 ( - ) 7. f () = f () = 6. f () = f () = 6 ( - )( + 3) Chapter 8 Rational Epressions and Equations

24 Eample 4 (p. 460) ELECTRICITY For Eercises 9, use the following information. The current I in amperes in an electrical circuit with three resistors in series is V given by the equation I =, where V is the voltage in volts in the R + R + R 3 circuit and R, R, and R 3 are the resistances in ohms of the three resistors. 9. Let R be the independent variable, and let I be the dependent variable. Graph the function if V = 0 volts, R = 5 ohms, and R 3 = 75 ohms. 0. Give the equation of the vertical asymptote and the R - and I-intercepts of the graph.. Find the value of I when the value of R is 40 ohms.. What domain and range values are meaningful in the contet of the problem? HELP HMEWRK For See Eercises Eamples , Determine the equations of any vertical asymptotes and the values of for any holes in the graph of each rational function. 3. f () = 4. f () = f () = f () = Graph each rational function. 7. f () = f () = f () = ( - ) 6. f () = f () = 3 9. f () = + 5. f () = f () = ( + 3 ) 7. f () = f () = + 5. f () = f () = - - PHYSICS For Eercises 9 3, use the following information. Under certain conditions, when two objects collide, the objects are repelled from each other with velocity given by the equation V f = m v + v ( m - m ). m + m In this equation m and m are the masses of the two objects, v and v are the initial speeds of the two objects, and V f is the final speed of the second object. Before Collision After Collision m v v V f m m m 9. Let m be the independent variable, and let V f be the dependent variable. Graph the function if m = 5 kilograms and v = 5 meters per second, and v = 0 meters per second. 30. Use the equation and the values in Eercise 9 to determine the final speed if m = 0 kilograms. 3. Give the equation of any asymptotes and the m - and V f -intercepts of the graph. 3. What domain and range values are meaningful in the contet of the problem? Lesson 8-3 Graphing Rational Functions 46

25 BASKETBALL For Eercises 33 36, use the following information. Zonta plays basketball for Centerville High School. So far this season, she has made 6 out of 0 free throws. She is determined to improve her free-throw percentage. If she can make consecutive free throws, her free-throw percentage can be determined using P() = Graph the function. 34. What part of the graph is meaningful in the contet of the problem? 35. Describe the meaning of the y-intercept. 36. What is the equation of the horizontal asymptote? Eplain its meaning with respect to Zonta s shooting percentage. Determine the equations of any vertical asymptotes and the values of for any holes in the graph of each rational function. 37. f () = f () = Graph each rational function f () = ( - )( + 5) 4. f () = f () = 6 ( - 6 ) 45. f () = f () = ( + )( - 3) 4. f () = f () = ( + ) 46. f () = PRACTICE EXTRA See pages 908, 933. Self-Check Quiz at ca.algebra.com H..T. Problems HISTRY For Eercises 47 49, use the following information. In Maria Gaetana Agnesi s book Analytical Institutions, Agnesi discussed the characteristics of the equation y = a (a - y), the graph of which is called the a curve of Agnesi. This equation can be epressed as y = 3 + a. a 47. Graph f () = 3 if a = 4. + a 48. Describe the graph. What are the limitations on the domain and range? a 49. Make a conjecture about the shape of the graph of f () = 3 if a = a Eplain your reasoning. 50. PEN ENDED Write a function the graph of which has vertical asymptotes located at = -5 and =. ( - )( + 5) 5. REASNING Compare and contrast the graphs of f () = and - g() = CHALLENGE Write a rational function for the graph at the right. y 46 Chapter 8 Rational Epressions and Equations

26 53. CHALLENGE Write three rational functions that have a vertical asymptote at = 3 and a hole at = Writing in Math Use the information on page 457 to eplain how rational functions can be used when buying a group gift. Eplain why only part of the graph of the rational function is meaningful in the contet of the problem. 55. ACT/SAT Which set is the domain of the function graphed below? A { 0, } y B { -, 0} C { < 4} D { > -4} 56. REVIEW = F G H J Simplify each epression. (Lessons 8- and 8-) 57. 3m + m + n m + n w - 4 w + 3 w Find all of the rational zeros for each function. (Lesson 6-8) 60. f () = g() = ART Joyce Jackson purchases works of art for an art gallery. Two years ago she bought a painting for $0,000, and last year she bought one for $35,000. If paintings appreciate 4% per year, how much are the two paintings worth now? (Lesson 6-5) Solve each equation by completing the square. (Lesson 5-5) = = = Write the slope-intercept form of the equation for the line that passes through (, -) and is perpendicular to the line with equation y = - +. (Lesson -4) 5 PREREQUISITE SKILL Solve each proportion v = = a = 8 s 70. b 9 = Lesson 8-3 Graphing Rational Functions 463

27 EXTEND 8-3 Graphing Calculator Lab Graphing Rational Functions Preparation for CA Mathematical Analysis Standard 6.0 Students find the roots and poles of a rational function and can graph the function and locate its asymptotes. A graphing calculator can be used to eplore the graphs of rational functions. These graphs have some features that never appear in the graphs of polynomial functions. ACTIVITY Graph y = 8-5 in the standard viewing window. Find the equations of any asymptotes. Enter the equation in the Y = list. KEYSTRKES: Y= ( 8 X,T,,n 5 ( X,T,,n ZM 6 [ 0, 0] scl: by [ 0, 0] scl: By looking at the equation, we can determine that if = 0, the function is undefined. The equation of the vertical asymptote is = 0. Notice what happens to the y-values as grows larger and as gets smaller. The y-values approach 4. So, the equation for the horizontal asymptote is y = 4. ACTIVITY Graph y = scale factors of. in the window [-5, 4.4] by [-0, ] with Because the function is not continuous, put the calculator in dot mode. KEYSTRKES: MDE This graph looks like a line with a break in continuity at = -4. This happens because the denominator is 0 when = -4. Therefore, the function is undefined when = -4. Notice the hole at 4. [ 5, 4.4] scl: by [ 0, ] scl: If you TRACE along the graph, when you come to = -4, you will see that there is no corresponding y-value. EXERCISES Use a graphing calculator to graph each function. Be sure to show a complete graph. Draw the graph on a sheet of paper. Write the -coordinates of any points of discontinuity and/or the equations of any asymptotes.. f () =. f () = 3. f () = f () = 5. f () = f () = Which graph(s) has point discontinuity? 8. Describe functions that have point discontinuity. 464 Chapter 8 Rational Epressions and Equations ther Calculator Keystrokes at ca.algebra.com

28 8-4 Direct, Joint, and Inverse Variation Main Ideas Recognize and solve direct and joint variation problems. Recognize and solve inverse variation problems. New Vocabulary direct variation constant of variation joint variation inverse variation The total high-tech spending t of an average public college can be found by using the equation t = 03s, where s is the number of students. Direct Variation and Joint Variation The relationship given by t = 03s is an eample of direct variation. A direct variation can be epressed in the form y = k. The k in this equation is called the constant of variation. Notice that the graph of t = 03s is a straight line t through the origin. An equation of a direct variation is a special case of an equation written 600 in slope-intercept form, y = m + b. When m = k 400 and b = 0, y = m + b becomes y = k. So the t 03s 00 slope of a direct variation equation is its constant of variation. 4 6 s To epress a direct variation, we say that y varies directly as. In other words, as increases, y increases or decreases at a constant rate. Direct Variation y varies directly as if there is some nonzero constant k such that y = k. k is called the constant of variation. If you know that y varies directly as and one set of values, you can use a proportion to find the other set of corresponding values. y = k and y = k y = k y = k Therefore, y = y. Using the properties of equality, you can find many other proportions that relate these same - and y-values. Etra Eamples at ca.algebra.com Lesson 8-4 Direct, Joint, and Inverse Variation 465

29 EXAMPLE Direct Variation If y varies directly as and y = when = -3, find y when = 6. Use a proportion that relates the values. y = y Direct proportion -3 = y 6 6 () = -3 ( y ) Cross multiply. 9 = -3 y Simplify. y =, = -3, and = 6-64 = y Divide each side by -3. When = 6, the value of y is If r varies directly as s and r = -0 when s = 4, find r when s = -6. Another type of variation is joint variation. Joint variation occurs when one quantity varies directly as the product of two or more other quantities. Joint Variation y varies jointly as and z if there is some nonzero constant k such that y = kz. If you know that y varies jointly as and z and one set of values, you can use a proportion to find the other set of corresponding values. y = k z and y = k z y z = k y z = k Therefore, y z = y z. EXAMPLE Joint Variation Suppose y varies jointly as and z. Find y when = 8 and z = 3, if y = 6 when z = and = 5. Use a proportion that relates the values. y z = y z Joint variation 6 5 () = y 8 (3) y = 6, = 5, z =, = 8, and z = 3 8 (3) (6) = 5 () ( y ) Cross multiply. 384 = 0 y Simplify = y Divide each side by 0. When = 8 and z = 3, the value of y is Suppose r varies jointly as s and t. Find r when s = and t = 8, if r = 70 when s = 0 and t = Chapter 8 Rational Epressions and Equations

30 Inverse Variation Another type of variation is inverse variation. For two quantities with inverse variation, as one quantity increases, the other quantity decreases. For eample, speed and time for a fied distance vary inversely with each other. When you travel to a particular location, as your speed increases, the time it takes to arrive at that location decreases. Direct Variation y varies inversely as if there is some nonzero constant k such that y = k or y =, where 0 and y 0. k Suppose y varies inversely as such that y = 6 or 6 y=. The graph of this equation is shown at the right. Since k is a positive value, as the values of increase, the values of y decrease. A proportion can be used with inverse variation to solve problems where some quantities are known. The following proportion is only one of several that can be formed. y y 6 or 6 y y = k and y = k y = y = y Divide each side by y y. y EXAMPLE Substitution Property of Equality Inverse Variation If r varies inversely as t and r = 8 when t = -3, find r when t = -. r r = t t r 8 = (-3) = -(r ) Real-World Link Mercury is about 36 million miles from the Sun, making it the closest planet to the Sun. Its proimity to the Sun causes its temperature to be as high as 800 F. Use a proportion that relates the values. r = 8, t = -3, and t = - Cross multiply. -54 = -r Simplify. 0 4 = r Divide each side by If varies inversely as y and = 4 when y = 4, find when y =. Source: World Book Encyclopedia SPACE The apparent length of an object is inversely proportional to one s distance from the object. Earth is about 93 million miles from the Sun. Use the information at the left to find how many times as large the diameter of the Sun would appear on Mercury than on Earth. Eplore The apparent diameter of the Sun varies inversely with the distance from the Sun. You know Mercury s distance from the Sun and Earth s distance from the Sun. You want to know how much larger the diameter of the Sun appears on Mercury than on Earth. Lesson 8-4 Direct, Joint, and Inverse Variation JPL/TSAD/Tom Stack & Associates 467

31 Plan Let the apparent diameter of the Sun from Earth equal unit and the apparent diameter of the Sun from Mercury equal m. Then use a proportion that relates the values. Solve distance from Mercury apparent diameter from Earth = 36 million miles unit distance from Earth apparent diameter from Mercury 93 million miles = m units Inverse variation Substitution (36 million miles) (m units) = (93 million miles) ( unit) Cross multiply. m = ( 93 million miles) ( unit) 36 million miles m.58 units Divide each side by 36 million miles. Simplify. Check Since the distance between the Sun and Earth is between and 3 times the distance between the Sun and Mercury, the answer seems reasonable. From Mercury, the diameter of the Sun will appear about.58 times as large as it appears from Earth. 4. SPACE Jupiter is about million miles from the Sun. Use the information above to find how many times as large the diameter of the Sun would appear on Earth as on Jupiter. Personal Tutor at ca.algebra.com Eamples 3 (pp ). If y varies directly as and y = 8 when = 5, find y when = 0.. Suppose y varies jointly as and z. Find y when = 9 and z = -5, if y = -90 when z = 5 and = If y varies inversely as and y = -4 when =, find when y =. Eample 4 (pp ) SWIMMING For Eercises 4 7, use the following information. When a person swims underwater, the pressure in his or her ears varies directly with the depth at which he or she is swimming. 4. Write a direct variation equation that represents this situation. 5. Find the pressure at 60 feet. 4.3 pounds per square inch (psi) 6. It is unsafe for amateur divers to swim where the water pressure is more than 65 pounds per square inch. How deep can an amateur diver safely swim? 0 ft 7. Make a table showing the number of pounds of pressure at various depths of water. Use the data to draw a graph of pressure versus depth. 468 Chapter 8 Rational Epressions and Equations

32 HELP HMEWRK For See Eercises Eamples 8, 9 0,, 3 3 4, If y varies directly as and y = 5 when = 3, find y when =. 9. If y varies directly as and y = 8 when = 6, find y when = Suppose y varies jointly as and z. Find y when = and z = 7, if y = 9 when = 8 and z = 6.. If y varies jointly as and z and y = 80 when = 5 and z = 8, find y when = 6 and z =.. If y varies inversely as and y = 5 when = 0, find y when =. 3. If y varies inversely as and y = 6 when = 5, find y when = GEMETRY How does the circumference of a circle vary with respect to its radius? What is the constant of variation? 5. TRAVEL A map of Alaska is scaled so that 3 inches represents 93 miles. How far apart are Anchorage and Fairbanks if they are.6 inches apart on the map? State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. 6. m n = = a 8. a = 5bc b 9. V = 3 Bh 0. p =.5 q. = s t. vw = y = V = π r h Real-World Career Travel Agent Travel agents give advice and make arrangements for transportation, accommodations, and recreation. For international travel, they also provide information on customs and currency echange. For more information, go to ca.algebra.com. 5. If y varies directly as and y = 9 when is -5, find y when =. 6. If y varies directly as and = 6 when y = 0.5, find y when = Suppose y varies jointly as and z. Find y when = and z = 6, if y = 45 when = 6 and z = If y varies jointly as and z and y = 8 when = and z = 3, find y when = 6 and z = If y varies inversely as and y = when = 5, find when y = If y varies inversely as and y = 4 when =, find y when = CHEMISTRY Boyle s Law states that when a sample of gas is kept at a constant temperature, the volume varies inversely with the pressure eerted on it. Write an equation for Boyle s Law that epresses the variation in volume V as a function of pressure P. 3. CHEMISTRY Charles Law states that when a sample of gas is kept at a constant pressure, its volume V will increase directly as the temperature t. Write an equation for Charles Law that epresses volume as a function. LAUGHTER For Eercises 33 35, use the following information. A newspaper reported that the average American laughs 5 times per day. 33. Write an equation to represent the average number of laughs produced by m household members during a period of d days. 34. Is your equation in Eercise 33 a direct, joint, or inverse variation? 35. Assume that members of your household laugh the same number of times each day as the average American. How many times would the members of your household laugh in a week? Geoff Butler Lesson 8-4 Direct, Joint, and Inverse Variation 469

33 BILGY For Eercises 36 38, use the information at the left. 36. Write an equation to represent the amount of meat needed to sustain s Siberian tigers for d days. 37. Is your equation in Eercise 36 a direct, joint, or inverse variation? 38. How much meat do three Siberian tigers need for the month of January? 39. WRK Paul drove from his house to work at an average speed of 40 miles per hour. The drive took him 5 minutes. If the drive home took him 0 minutes and he used the same route in reverse, what was his average speed going home? Real-World Link In order to sustain itself in its cold habitat, a Siberian tiger requires 0 pounds of meat per day. Source: Wildlife Fact File 40. WATER SUPPLY Many areas of Northern California depend on the snowpack of the Sierra Nevada Mountains for their water supply. If 50 cubic centimeters of snow will melt to 8 cubic centimeters of water, how much water does 900 cubic centimeters of snow produce? 4. RESEARCH According to Johannes Kepler s third law of planetary motion, the ratio of the square of a planet s period of revolution around the Sun to the cube of its mean distance from the Sun is constant for all planets. Verify that this is true for at least three planets. ASTRNMY For Eercises 4 44, use the following information. Astronomers can use the brightness of two light sources, such as stars, to compare the distances from the light sources. The intensity, or brightness, of light I is inversely proportional to the square of the distance from the light source d. 4. Write an equation that represents this situation. 43. If d is the independent variable and I is the dependent variable, graph the equation from Eercise 4 when k = If two people are viewing the same light source, and one person is three times the distance from the light source as the other person, compare the light intensities that the two people observe. GRAVITY For Eercises 45 47, use the following information. According to the Law of Universal Gravitation, the attractive force F in Newtons between any two bodies in the universe is directly proportional to the product of the masses m and m in kilograms of the two bodies and inversely proportional to the square of the distance d in meters between the m m bodies. That is, F = G. G is the universal gravitational constant. Its d value is Nm /kg. 45. The distance between Earth and the Moon is about meters. The mass of the Moon is kilograms. The mass of Earth is kilograms. What is the gravitational force that the Moon and Earth eert upon each other? EXTRA PRACTICE See pages 908, 933. Self-Check Quiz at ca.algebra.com 46. The distance between Earth and the Sun is about.5 0 meters. The mass of the Sun is about kilograms. What is the gravitational force that the Sun and Earth eert upon each other? 47. Find the gravitational force eerted on each other by two 000-kilogram iron balls a distance of 0. meter apart. 470 Chapter 8 Rational Epressions and Equations Lynn M. Stone/Bruce Coleman, Inc.

34 H..T. Problems 48. PEN ENDED Describe two real life quantities that vary directly with each other and two quantities that vary inversely with each other. 49. CHALLENGE Write a real-world problem that involves a joint variation. Solve the problem. 50. Writing in Math Use the information about variation on page 465 to eplain how variation is used to determine the total cost if you know the unit cost. 5. ACT/SAT Suppose b varies inversely as the square of a. If a is multiplied by 9, which of the following is true for the value of b? A It is multiplied by 3. B It is multiplied by 9. C It is multiplied by 8. D It is multiplied by REVIEW If ab = and a is less than 0, which of the following statements cannot be true? F b is negative. G b is less than a. H As a increases, b decreases. J As a increases, b increases. Determine the equations of any vertical asymptotes and the values of for any holes in the graph of each rational function. (Lesson 8-3) 53. f () = f () = f () = Simplify each epression. (Lesson 8-) y + 4 y t t + - m m t m - 5 m 59. BILGY ne estimate for the number of cells in the human body is 00,000,000,000,000. Write this number in scientific notation. (Lesson 6-) State the slope and the y-intercept of the graph of each equation. (Lesson -4) 60. y = y = y = 5 PREREQUISITE SKILL Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. (Lesson -6) 63. h () = 64. g () = f () = f () = if > 0 - if h () = g () = -3 Lesson 8-4 Direct, Joint, and Inverse Variation 47

35 CHAPTER 8 Mid-Chapter Quiz Lessons 8- through 8-4 Simplify each epression. (Lesson 8-). t - t -6 t -6t y 5. w + 5w ab 3 8 a b 4ac 9 b 4 4. w + 8w a MULTIPLE CHICE For all t 5, t - 5 = (Lesson 8-) 3t - 5 A t t + 5 B 3. C t - 5. D t a Simplify each epression. (Lesson 8-) 8. 4a + a + b b - a 5ab + 4y 3 3 a b 0. 5 n n For Eercises 4, use the following information. Lucita is going to a beach 00 miles away. She travels half the distance at one rate. The rest of the distance, she travels 5 miles per hour slower. (Lesson 8-). If represents the faster pace in miles per hour, write an epression that represents the time spent at that pace. 3. Write an epression for the amount of time spent at the slower pace. 4. Write an epression for the amount of time Lucita needed to complete the trip. Graph each rational function. (Lesson 8-3) 5. f() = - 6. f() = MULTIPLE CHICE What is the range of the function y = + 8? (Lesson 8-3) F y y ± G {y y 4} H {y y 0} J {y y 0} WRK For Eercises 8 and 9, use the following information. (Lesson 8-3) Andy is a new employee at Quick il Change. The company s goal is to change every customer s oil in 0 minutes. So far, he has changed 3 out of 0 customers oil in 0 minutes. Suppose Andy changes the net customers oil in 0 minutes. His 0-minute oil changing percentage can be determined using P() = Graph the function. 9. What domain and range values are meaningful in the contet of the problem? Find each value. (Lesson 8-4) 0. If y varies inversely as and = 4 when y = 7, find when y =.. If y varies directly as and y = when = 5, find y when =.. If y varies jointly as and z and y = 80 when = 5 and z = 4, find y when = 0 and z = 7. For Eercises 3 5, use the following information. In order to remain healthy, a horse requires 0 pounds of hay a day. (Lesson 8-4) 3. Write an equation to represent the amount of hay needed to sustain horses for d days. 4. Is your equation a direct, joint, or inverse variation? Eplain. 5. How much hay do three horses need for the month of July? 47 Chapter 8 Mid-Chapter Quiz

36 8-5 Classes of Functions Main Ideas Identify graphs as different types of functions. Identify equations as different types of functions. The purpose of the Mars Eploration Program is to study conditions on Mars. The findings will help NASA prepare for a possible mission with human eplorers. The graph at the right compares a person s weight on Earth with his or her weight on Mars. This graph represents a direct variation, which you studied in the previous lesson. Identify Graphs In this book, you have studied several types of graphs representing special functions. The following is a summary of these graphs. Constant Function y y Direct Variation Function y y Special Functions Identity Function y y The general equation of a constant function is y = a, where a is any number. Its graph is a horizontal line that crosses the y-ais at a. Greatest Integer Function y If an equation includes an epression inside the greatest integer symbol, the function is a greatest integer function. Its graph looks like steps. y The general equation of a direct variation function is y = a, where a is a nonzero constant. Its graph is a line that passes through the origin and is neither horizontal nor vertical. Absolute Value Function y y An equation with the independent variable inside absolute value symbols is an absolute value function. Its graph is in the shape of a V. The identity function y = is a special case of the direct variation function in which the constant is. Its graph passes through all points with coordinates (a, a). Quadratic Function h(t ) t 4 6 The general equation of a quadratic function is y = a + b + c, where a 0. Its graph is a parabola. Lesson 8-5 Classes of Functions 473

37 Square Root Function y y Rational Function y y Special Functions Inverse Variation Function y y If an equation includes the independent variable inside the radical sign, the function is a square root function. Its graph is a curve that starts at a point and continues in only one direction. The general equation for a rational function is y = p() q(), where p() and q() are polynomial functions. Its graph may have one or more asymptotes and/or holes. The inverse variation function y = a is a special case of the rational function where p() is a constant and q() =. Its graph has two asymptotes, = 0 and y = 0. You can use the shape of the graphs of each type of function to identify the type of function that is represented by a given graph. To do so, keep in mind the graph of the parent function of each function type. EXAMPLE Identify a Function Given the Graph Identify the type of function represented by each graph. a. b. The graph has a starting point and curves in one direction. The graph represents a square root function. The graph appears to be a direct variation since it is a straight line passing through the origin. However, the hole indicates that it represents a rational function. A. y B. y Personal Tutor at ca.algebra.com 474 Chapter 8 Rational Epressions and Equations

38 Identify Equations If you can identify an equation as a type of function, you can determine the shape of the graph. EXAMPLE Match Equation with Graph RCKETRY Emily launched a toy rocket from ground level. The height above the ground level h, in feet, after t seconds is given by the formula h(t) = -6 t + 80t. Which graph depicts the height of the rocket during its flight? a. b. c. h (t ) 0 h(t ) h (t ) t 4 6 t 4 6 t 4 6 The function includes a second-degree polynomial. Therefore, it is a quadratic function, and its graph is a parabola. Graph b is on the only parabola. Therefore, the answer is graph b.. Which graph above could represent an elevator moving from a height of 80 feet to ground level in 5 seconds? Sometimes recognizing an equation as a specific type of function can help you graph the function. EXAMPLE Identify a Function Given its Equation Identify the type of function represented by each equation. Then graph the equation. a. y = - Since the equation includes an epression inside absolute value symbols, it is an absolute value function. Therefore, the graph will be in the shape of a V. Plot some points and graph the absolute value function. y b. y = - 3 The function is in the form y = a, where a = - 3. Therefore, it is a direct variation function. The graph passes through the origin and has a slope of - y 3. 3A. y = [[ - ]] 3B. y = - + Etra Eamples at ca.algebra.com Lesson 8-5 Classes of Functions 475

39 Eample (p. 474) Identify the type of function represented by each graph.. y. y 3. y Eample (p. 475) Match each graph with an equation at the right. 4. y 5. y a. y = b. y = + c. y = + + d. y = 6. GEMETRY Write the equation for the area of a circle. Identify the equation as a type of function. Describe the graph of the function. Eample 3 (p. 475) Identify the type of function represented by each equation. Then graph the equation. 7. y = 8. y = y = + HELP HMEWRK For See Eercises Eamples Identify the function represented by each graph. 0. y. y. y 3. y 4. y 5. y Identify the type of function represented by each equation. Then graph the equation. 6. y = y =.5 8. y = 9 0. y = Chapter 8 Rational Epressions and Equations 9. y = 4. y = 3. y = 3. y =

40 Match each graph with an equation at the right. 4. y 5. y 6. y 7. y a. y = - b. y = c. y = d. y = -3 e. y = 0.5 f. y = g. y = - 3 Real-World Link When the Hope Diamond was shipped from New York to the Smithsonian Institution in Washington, D.C., it was mailed in a plain brown paper package. Source: usps.com PRACTICE EXTRA See pages 909, 933. Self-Check Quiz at ca.algebra.com H..T. Problems HEALTH For Eercises 8 30, use the following information. A woman painting a room will burn an average of 4.5 Calories per minute. 8. Write an equation for the number of Calories burned in m minutes. 9. Identify the equation in Eercise 8 as a type of function. 30. Describe the graph of the function. 3. ARCHITECTURE The shape of the Gateway Arch of the Jefferson National Epansion Memorial in St. Louis, Missouri, resembles the graph of the function f () = , where is in feet. Describe the shape of the Gateway Arch. MAIL For Eercises 3 and 33, use the following information. In 006, the cost to mail a first-class letter was 39 for any weight up to and including ounce. Each additional ounce or part of an ounce added 4 to the cost. 3. Make a graph showing the postal rates to mail any letter from 0 to 8 ounces. 33. Compare your graph in Eercise 3 to the graph of the greatest integer function. 34. PEN ENDED Find a countereample to the statement All functions are continuous. Describe your function. 35. CHALLENGE Identify each table of values as a type of function. a. f() b. f() c. f() d. f() undefined undefined undefined Terry Smith Images/Alamy Images Lesson 8-5 Classes of Functions 477

41 36. CHALLENGE Without graphing either function, eplain how the graph of y = is related to the graph of y = Writing in Math Use the information on page 473 to eplain how the graph of a function can be used to determine the type of relationship that eists between the quantities represented by the domain and the range. 38. ACT/SAT The curve below could be part of the graph of which function? A y = y B y = C y = 4 D y = REVIEW A paper plate with a -inch diameter is divided into 3 sections. 7 5 What is the approimate length of the arc of the largest section? F 0.3 inches H 4. inches G.5 inches J 6.5 inches 40. If varies directly as y and y = 5 when =, find when y =. (Lesson 8-4) 5 Graph each rational function. (Lesson 8-3) 4. f () = 3 4. f () = 8 + ( - )( + 3) 43. f () = Solve each equation by factoring. (Lesson 5-) = q + q = HT-AIR BALLNS For Eercises 46 and 47, use the table. (Lesson 3-) 46. If both balloons are launched at the same time, how long will it take for them to be the same distance from the ground? 47. What is the distance of the balloons from the ground at that time? PREREQUISITE SKILL Find the LCM of each set of polynomials. (Lesson 8-) 48. 5ab c, 6 a 3, 4bc , 5y, 5 y d - 0, 3d y, 3 + 3y 5. a - a - 3, a - a t - 9t - 5, t + t Chapter 8 Rational Epressions and Equations

42 8-6 Solving Rational Equations and Inequalities Main Ideas Solve rational equations. Solve rational inequalities. Standard 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simplify complicated rational epressions, including those with negative eponents in the denominator. (Key) New Vocabulary rational equation rational inequality A music download service advertises downloads for $ per song. The service also charges a monthly access fee of $5. If a customer downloads songs in one month, the bill in dollars will be 5 +. The actual cost per song is 5 +. To find how many songs a person would need to download to make the actual cost per song $.5, you would need to solve the equation 5 + =.5. Solve Rational Equations The equation 5 + = 6 is an eample of a rational equation. In general, any equation that contains one or more rational epressions is called a rational equation. Rational equations are easier to solve if the fractions are eliminated. You can eliminate the fractions by multiplying each side of the equation by the least common denominator (LCD). Remember that when you multiply each side by the LCD, each term on each side must be multiplied by the LCD. EXAMPLE Solve a Rational Equation 9 Solve z + =. Check your solution. 4 The LCD for the terms is 8(z + ) z + = 3 4 8(z + ) ( z + ) = 8(z + ) ( 3 8(z + ) ( 8) 9 + 8(z + ) ( 3 7 = 8(z + ) z + ) ( 3 riginal equation 4) Multiply each side by 8(z + ). Distributive Property 4) (9z + 8) + 84 = z + 4 Simplify. 9z + 0 = z + 4 Simplify. 60 = z Subtract 9z and 4 from each side. 5 = z Divide each side by. (continued on the net page) Hugh Threlfall/Alamy Images Lesson 8-6 Solving Rational Equations and Inequalities 479

43 CHECK z + = riginal equation z = 5 Simplify Simplify = 3 The solution is correct. 4 Solve each equation. Check your solution. A = B = When solving a rational equation, any possible solution that results in a zero in the denominator must be ecluded from your list of solutions. Etraneous Solutions Multiplying each side of an equation by the LCD of rational epressions can yield results that are not solutions of the original equation. These solutions are called etraneous solutions. EXAMPLE Elimination of a Possible Solution Solve r + r - 5. Check your solution. r - = r + r + r + r + r - 5 ( r - ) ( r + r - 5 ( r - )r + ( r - ) ( r - 5 r - = r + r + r + r - ) = ( r - ) ( r + r + r + r - ) = ( r - ) ( r + r + r + ( r 3 - r) + ( r - 5) = (r - )( r + r + ) Simplify. r 3 + r - r - 5 = r 3 + r - r - r - 3 = 0 riginal equation; the LCD is (r - ). ) Multiply each side by the LCD. ) Distributive Property Simplify. (r - 3)(r + ) = 0 Factor. r - 3 = 0 or r + = 0 r = 3 r = - CHECK r + r - 5 r - = r + r + r Subtract ( r 3 + r - ) from each side. Zero Product Property r + r - 5 r - = r + r + r (-) - 5 (-) - (-) + (-) = Since r = - results in a zero in the denominator, eliminate - from the list of solutions. The solution is Chapter 8 Rational Epressions and Equations

44 Solve each equation. Check your solution. A. r + - r - = - r B. 7n - 3n n - 4 = 3n n + Real-World Link The Loetschberg Tunnel is miles long. It was created for train travel and cut travel time between the locations in half. Source: usatoday.com TUNNELS The Loetschberg tunnel was built to connect Bern, Switzerland, with the ski resorts in the southern Swiss Alps. The Swiss used one company that started at the north end and another company that started at the south end. Suppose the company at the north end could drill the entire tunnel in. years and the south company could do it in.8 years. How long would it have taken the two companies to drill the tunnel? In year, the north company could complete In years, the north company could complete the tunnel. of the tunnel... or. of In t years, the north company could complete. t or t of the tunnel.. Likewise, in t years, the south company could complete.8 t or t of the tunnel..8 Together, they completed the whole tunnel. Part completed by part completed by entire the north company plus the south company equals tunnel. t + t =..8 t. + t = riginal equation ( t. + t.8 ) = () Multiply each side by t +.t = Simplify. 44t = Simplify. t Divide each side by 44. It would have taken about years to build the tunnel. This answer is reasonable. Working alone, either company could have drilled the tunnel in about years. Working together, they must be able to do it in about half that time. 3. WRK Breanne and wen paint houses together. If Breanne can paint a particular house in 6 days and wen can paint the same house in 5 days, how long would it take the two of them if they work together? Personal Tutor at ca.algebra.com Etra Eamples at ca.algebra.com Lesson 8-6 Solving Rational Equations and Inequalities 48 Yoshiko Kusano/AP/Wide World Photos

45 Rate problems frequently involve rational equations. NAVIGATIN The speed of the current in the Puget sound is 5 miles per hour. A barge travels 6 miles with the current and returns in 0 hours. What is the speed of the barge in still water? 3 Words Variables Equation The formula that relates distance, time, and rate is d = rt or d r = t. Let r = the speed of the barge in still water. Then the speed of the barge with the current is r + 5, and the speed of the barge against current is r - 5. Time going with the current 6 r + 5 plus + time going against the current 6 r - 5 equals total time. = 0 3 3( r - 5) ( 6 6 r r - 5 = 0 3 3( r - 5) ( 6 r r - 5) = 3( r - 5) ( r r + 5) - 5) ( 6 = 3( r r - 5) - 5) ( 3 riginal equation Multiply each side by 3( r - 5). 3 ) Distributive Property (78r - 390) + (78r + 390) = 3 r Simplify. 56r = 3 r Simplify. Look Back To review the Quadratic Formula, see Lesson 5-6. Use the Quadratic Formula to solve for r. = -b ± b - 4ac a r = -(-39) ± (-39 ) - 4(8)(-00) (8) 39 ± r = ± 89 r = 6 0 = 3 r - 56r Subtract 56r from each side. 0 = 8 r - 39r - 00 Divide each side by 4. Quadratic Formula = r, a = 8, b = -39, and c = -00 Simplify. Simplify. r = 8 or -3.5 Simplify. Since speed must be positive, it is 8 miles per hour. Is this answer reasonable? Interactive Lab ca.algebra.com 4. SWIMMING The speed of the current in a body of water is mile per hour. Juan swims miles against the current and miles with the current in a total time of hours. How fast can Juan swim in still 3 water? 48 Chapter 8 Rational Epressions and Equations

46 Solve Rational Inequalities Inequalities that contain one or more rational epressions are called rational inequalities. To solve rational inequalities, complete the following steps. Step State the ecluded values. Step Solve the related equation. Step 3 Use the values determined in Steps and to divide a number line into intervals. Test a value in each interval to determine which intervals contain values that satisfy the original inequality. EXAMPLE Solve 5 4a + 8a > Solve a Rational Inequality. Step Values that make a denominator equal to 0 are ecluded from the domain. For this inequality, the ecluded value is 0. Step Solve the related equation. 4a + 5 8a = Related equation 8a ( 4a + 5 8a) = 8a ( ) Multiply each side by 8a. + 5 = 4a Simplify. 7 = 4a Add. 3 = a Divide each side by 4. 4 Step 3 Draw vertical lines at the ecluded value and at the solution to separate the number line into intervals. ecluded value solution of related equation Now test a sample value in each interval to determine if the values in the interval satisfy the inequality. Test a = -. Test a =. Test a =. 4(-) + 5 8(-) 4() + 5 8() 4() + 5 8() > 7 6 a < 0 is not a solution. 0 < a < 3 4 is a solution. a > 3 is not a solution. 4 The solution is 0 < a < 3 4. Solve each inequality. 5A. 3b - 5b < 5 5B Lesson 8-6 Solving Rational Equations and Inequalities 483

47 Eample (pp ) Eample (p. 480) Eamples 3, 4 (pp. 48, 48) Eample 5 (p. 483) Solve each equation. Check your solutions.. d + 4 =. t = 0 t = 0 4. v v - 4 = 3 5. w w - + w = 4w - 3 w n n n n + 3 = 3 n WRK A worker can powerwash a wall of a certain size in 5 hours. Another worker can do the same job in 4 hours. If the workers work together, how long would it take to do the job? Determine whether your answer is reasonable. Solve each inequality c + > 9. 3v + 4v < HELP HMEWRK For See Eercises Eamples 0 5 6, 7 8 5, 3 3, 4 Solve each equation or inequality. Check your solutions. 0. y y + =. p 3 p - = 5. s + 5 = 6 s 3. a + = 6 a 4. 9 t - 3 = t - 4 t = y y y = y + 4 y a + > m + > t > 6 t d + 4 = d + 3d d. 7 - b < 5 b. NUMBER THERY The ratio of 6 more than a number to less than that number is to 3. What is the number? 3. NUMBER THERY The sum of a number and 8 times its reciprocal is 6. Find the number(s). Solve each equation or inequality. Check your solutions. 4. b - 4 b - = b - b b - n - = n + n + n n + 4 q 6. q q q - 3 = 7. 4 z - - z + 6 z + = 8. 3y + 5 6y > p + 3 4p < 484 Chapter 8 Rational Epressions and Equations 30. ACTIVITIES The band has 30 more members than the school chorale. If each group had 0 more members, the ratio of their membership would be 3:. How many members are in each group?

48 PHYSICS For Eercises 3 and 3, use the following information. The distance a spring stretches is related to the mass attached to the spring. This is represented Spring by d = km, where d is the distance, m is the mass, k cm/g and k is the spring constant. When two springs Spring with spring constants k and k are attached in a k 8 cm/g series, the resulting spring constant k is found by = +. the equation k Real-World Career Chemist Many chemists work for manufacturers developing products or doing quality control to ensure the products meet industry and government standards. For more information, go to ca.algebra.com. k k d Spring Spring 3. If one spring with constant of centimeters per gram is attached in a series with another 5g spring with constant of 8 centimeters per gram, find the resultant spring constant. 3. If a 5-gram object is hung from the series of springs, how far will the springs stretch? Is this answer reasonable in this contet? 33. CYCLING n a particular day, the wind added 3 kilometers per hour to Alfonso s rate when he was cycling with the wind and subtracted 3 kilometers per hour from his rate on his return trip. Alfonso found that in the same amount of time he could cycle 36 kilometers with the wind, he could go only 4 kilometers against the wind. What is his normal bicycling speed with no wind? Determine whether your answer is reasonable. 34. CHEMISTRY Kiara adds an 80% acid solution to 5 milliliters of solution that is 0% acid. The function that represents the percent of acid in the resulting 5(0.0) + (0.80) 5+ solution is f() =, where is the amount of 80% solution added. How much 80% solution should be added to create a solution that is 50% acid? 35. NUMBER THERY The ratio of 3 more than a number to the square of more than that number is less than. Find the numbers which satisfy this statement. EXTRA PRACTICE See pages 909, 933. Self-Check Quiz at ca.algebra.com H..T. Problems STATISTICS For Eercises 36 and 37, use the following information. A number is the harmonic mean of y and z if is the average of y and z. 36. Eight is the harmonic mean of 0 and what number? 37. What is the harmonic mean of 5 and 8? 38. PEN ENDED Write a rational equation that can be solved by first multiplying each side by 5(a + ) FIND THE ERRR Jeff and Dustin are solving - a = 3. Who is correct? Eplain your reasoning. Jeff 3 - a= 3 6a - 9 = a 4a = 9 a =.5 Dustin 3 - a = 3-9 = a - 7 = a -3.5 = a 40. CHALLENGE Solve for a if a - = c. b Lesson 8-6 Solving Rational Equations and Inequalities Keith Wood/Getty Images 485

49 4. Writing in Math Use the information about music downloads on page 479 to eplain how rational equations are used to solve problems involving unit price. Include an eplanation of why the actual price per download could never be $ ACT/SAT Amanda wanted to determine the average of her 6 test scores. She added the scores correctly to get T, but divided by 7 instead of 6. The result was less than her actual average. Which equation could be used to determine the value of T? A 6T + = 7T B T 7 = T - 6 C T 7 + = T 6 D T 6 = T REVIEW What is 0a-3 9b 5a-5 4 5b F 3 3a 8 G 5 3a b 3 H 3b3 9a 8 J 3a 9b 6b -7? Identify the type of function represented by each equation. Then graph the equation. (Lesson 8-5) 44. y = y = 46. y = If y varies inversely as and y = 4 when = 9, find y when = 6. (Lesson 8-4) Solve each inequality. (Lesson 5-8) 48. ( + )( - 3) > b - b < 6 Find each product, if possible. (Lesson 4-3) HEALTH The prediction equation y = relates a person s maimum heart rate for eercise y and age. Use the equation to find the maimum heart rate for an 8-year-old. (Lesson -5) Determine the value of r so that a line through the points with the given coordinates has the given slope. (Lesson -3) 54. (r, ), (4, -6); slope = Evaluate [(-7 + 4) 5 - ] 6. (Lesson -) 486 Chapter 8 Rational Epressions and Equations 55. (r, 6), (8, 4); slope =

50 EXTEND 8-6 Graphing Calculator Lab Solving Rational Equations and Inequalities with Graphs and Tables You can use a TI-83/84 graphing calculator to solve rational equations by graphing or by using the table feature. Graph both sides of the equation and locate the point(s) of intersection. ACTIVITY Solve 4 + = 3. Step Graph each side of the equation. Graph each side of the equation as a separate function. Enter 4 + as Y and 3 as Y. Then graph the two equations. KEYSTRKES: Y= 4 ( X,T,,n 3 ZM 6 Step Use the intersect feature. The intersect feature on the [CALC] menu allows you to approimate the ordered pair of the point at which the graphs cross. KEYSTRKES: nd [CALC] 5 Select one graph and press. Select the other graph, press, and press again. [ 0, 0] scl: by [ 0, 0] scl: Because the calculator is in connected mode, a vertical line may appear connecting the two branches of the graph. This is not part of the graph. The solution is 3. Step 3 Use the table feature. Verify the solution using the table feature. Set up the table to show -values in increments of 3. KEYSTRKES: nd [TBLSET] 0 3 nd [TABLE] The table displays -values and corresponding y-values for each graph. At =, both functions have a y-value of.5. Thus, the 3 solution of the equation is 3. ther Calculator Keystrokes at ca.algebra.com Etend 8-6 Graphing Calculator Lab 487

51 You can use a similar procedure to solve rational inequalities using a graphing calculator. ACTIVITY Solve > 9. Step Enter the inequalities. Rewrite the problem as a system of inequalities. The first inequality is > y or y < Since this inequality includes the less than symbol, shade below the curve. First, enter the boundary and then use the arrow and keys to choose the shade below icon,. The second inequality is y > 9. Shade above the curve since this inequality contains greater than. KEYSTRKES: 3 X,T,,n 7 X,T,,n 9 GRAPH Step Graph the system. KEYSTRKES: GRAPH Step 3 Use the table feature. Verify using the table feature. Set up the table to show -values in increments of 9. KEYSTRKES: nd [TBLSET] 0 9 nd [TABLE] [ 0, 0] scl: by [ 0, 0] scl: The solution set of the original inequality is the set of -values of the points in the region where the shadings overlap. Using the calculator s intersect feature, you can conclude that the solution set is 0 > > 9. Scroll through the table. Notice that for -values greater than 0 and less than 9, Y > Y. This confirms that the solution of the inequality is 0 > > 9. EXERCISES Solve each equation or inequality.. + =. - 4 = = Chapter 8 Rational Epressions and Equations = = > <

52 CHAPTER 8 Study Guide and Review Download Vocabulary Review from ca.algebra.com Be sure the following Key Concepts are noted in your Foldable. Key Concepts Rational Epressions (Lessons 8- and 8-) Multiplying and dividing rational epressions is similar to multiplying and dividing fractions. To simplify comple fractions, simplify the numerator and the denominator separately, and then simplify the resulting epression. Direct, Joint, and Inverse Variation (Lesson 8-4) Direct Variation: There is a nonzero constant k such that y = k. Joint Variation: There is a number k such that y = kz, where 0 and z 0. Inverse Variation: There is a nonzero constant k such that y = k or y = k. Classes of Functions (Lesson 8-5) The following functions can be classified as special functions: constant function, direct variation function, identity function, greatest integer function, absolute value function, quadratic function, square root function, rational function, inverse variation function. Rational Equations and Inequalities (Lesson 8-6) Eliminate fractions in rational equations by multiplying each side of the equation by the LCD. Possible solutions of a rational equation must eclude values that result in zero in the denominator. Key Vocabulary asymptote (p. 457) comple fraction (p. 445) constant of variation (p. 465) continuity (p. 457) direct variation (p. 465) inverse variation (p. 467) joint variation (p. 466) point discontinuity (p. 457) rational equation (p. 479) rational epression (p. 44) rational function (p. 457) rational inequality (p. 483) Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.. The equation y = - has a(n) + asymptote at = -.. The equation y = 3 is an eample of a(n) direct variation equation. 3. The equation y = + is a(n) polynomial equation. 4. The graph of y = 4 has a(n) - 4 variation at = The equation b = a is a(n) inverse variation equation. 6. n the graph of y = - 5, there is a break + in continuity at =. 7. The epression + - is an eample of a comple fraction. 8. In the direct variation y = 6, 6 is the degree. Vocabulary Review at ca.algebra.com Chapter 8 Study Guide and Review 489

53 CHAPTER 8 Study Guide and Review Lesson-by-Lesson Review 8 Multiplying and Dividing Rational Epressions (pp ) Simplify each epression ab c 4 c 0. a - b a + b a 6b 36 b n - 6n n y - y - y - 4 y + y - 4y n GEMETRY A triangle has an area of square meters. If the base is - meters, find the height. Eample Simplify 3 y 8 y y 8 y 3 6 = 3 y y y y 3 = y Eample Simplify p + 7p 49 - p 3p 3p -. p + 7p 49 - p 3p 3p - = p + 7p 3p - 3p 49 - p p(7 + p) = -3(7 - p) 3p (7 + p)(7 - p) = - 8 Adding and Subtracting Rational Epressions (pp ) 4 Eample 3 Simplify + y y. 4 + y y = 4 + y - 9 ( + y)( - y) Simplify each epression y y y y b - 5b - b. m + 3 m - 6m + 9 8m m BILGY For Eercises and 3, use the following information. After a person eats something, the ph or acid level A of their mouth can be determined by the formula A = - 0.4t t , where t is the number of minutes that have elapsed since the food was eaten.. Simplify the equation. 3. What would the acid level be after 30 minutes? 4( - y) = ( + y)( - y) - 9 ( + y)( - y) = = = 4( - y) - 9 ( + y)( - y) 4-4y - 9 ( + y)( - y) 5-4y ( + y)( - y) Subtract the numerators. Distributive Property Simplify. 490 Chapter 8 Rational Epressions and Equations

54 Mied Problem Solving For mied problem-solving practice, see page Graphing Rational Functions (pp ) Graph each rational function. 4. f() = 4-5. f() = f() = 7. f() = f() = f() = ( + )( - 3) 30. SANDWICHES A group makes 45 sandwiches to take on a picnic. The number of sandwiches a person can eat depends on how many people go on the trip. Write and graph a function to illustrate this situation. Eample 4 Graph f() = 5 ( + 4). The function is undefined for = 0 and =-4. Since 5 is in simplest form, ( + 4) = 0 and = -4 are vertical asymptotes. Draw the two asymptotes and sketch the graph. f() 5 ( 4) f() 8 4 Direct, Joint, and Inverse Variation (pp ) 3. If y varies directly as and y = when = 7, find when y = If y varies inversely as and y = 9 when =.5, find y when = If y varies inversely as and y = -4 when = 8, find y when = If y varies jointly as and z and = and z = 4 when y = 6, find y when = 5 and z = If y varies jointly as and z and y = 4 when = 0 and z = 7, find y when = and z = EMPLYMENT Chris s pay varies directly with how many lawns he mows. If his pay is $65 for 5 yards, find his pay after he has mowed 3 yards. Eample 5 If y varies inversely as and = 4 when y = -6, find when y = -. y = y 4 - = -6 Inverse variation 4(-6) = -( ) Cross multiply. = 4, y = -6, y = = - Simplify. 7 7 = Divide each side by -. When y = -, the value of is 7 7. Chapter 8 Study Guide and Review 49

55 CHAPTER 8 Study Guide and Review 8 5 Classes of Functions (pp ) Identify the type of function represented by each graph. 37. y Eample 6 Identify the type of function represented by each graph. a. y The graph has a parabolic shape; therefore, it is a quadratic function. 38. y b. y The graph has a stair-step pattern; therefore, it is a greatest integer function. 8 6 Solving Rational Equations and Inequalities (pp ) Solve each equation or inequality. Eample 7 Solve - + = 0. Check your solutions y + 7 y = 9 The LCD is ( - ) = = 0 6 ( - ) ( - + = ( - )(0) ) 4. r - = r + r - ( - ) ( + ( - ) - ) ( ) = ( - )(0) = + () + ( - ) = b - 3 4b > + - = = PUZZLES Danielle can put a puzzle together in three hours. Aidan can put the same puzzle together in five hours. How long will it take them if they work together? 3 = = 3 49 Chapter 8 Rational Epressions and Equations

56 CHAPTER 8 Practice Test Simplify each epression.. a - ab 3a a - b 5 b. - y y 3 y y y - 5 y Identify the type of function represented by each graph. 7. y 8. y 7. If y varies inversely as and y = 9 when = -, find when y = If g varies directly as w and g = 0 when w = -3, find w when g = Suppose y varies jointly as and z. If = 0 when y = 50 and z = 5, find when y =.5 and z = AUT MAINTENANCE When air is pumped into a tire, the pressure required varies inversely as the volume of the air. If the pressure is 30 pounds per square inch when the volume is 40 cubic inches, find the pressure when the volume is 00 cubic inches.. WRK Sofia and Julie must pick up all of the apples in the yard so the lawn can be mowed. Working alone, Julie could complete the job in.7 hours. Sofia could complete it alone in.3 hours. How long will it take them to complete the job when they work together? Graph each rational function f() = f() = ( - )( + ) Solve each equation or inequality.. - = z + = t > - t = m m + 3 = = ELECTRICITY For Eercises and 3, use the following information. The current I in a circuit varies inversely with the resistance R.. Use the table below to write an equation relating the current and the resistance. I R What is the constant of variation? 4. MULTIPLE CHICE If m =, n = 7m, p = n, q = 4p, and r =, find. q A r B q C p D r Chapter Test at ca.algebra.com Chapter 8 Practice Test 493

57 CHAPTER 8 California Standards Practice Cumulative, Chapters 8 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 5 Which graph below is a representation of the equation y = + 4? F = A - B C 4( + 3) - + D G What is 7b F 9 5a 5 c 4 G 8b9 a 5 c 4 H 6c 5ab 5 J 7c 0ab c-3 0a 3 b - c 3a b -7? H 3 If 3-8 = 5-3, what is the value of? 4 If is a real number, for what values of is the equation = + 5 true? 4 A all values of B some values of C no values of D impossible to determine J 494 Chapter 8 Rational Epressions and Equations California Standards Practice at ca.algebra.com

58 More California Standards Practice For practice by standard, see pages CA CA43. 6 Which is a simplified form of 75 y 7? A 6 y 3 75y B 75 4 y 7 C 5 6 y 3 3y D y 7 0 What are the solutions to the equation = 0? A = 6 + 4i, = 6-4i B = 3 + i, = 3 - i C = 0, = 6 D = 0, = 6i GEMETRY In the figure below, l m. Find b. 7 What are the solutions to the equation = +? F =, = - 60 a b c 0 m G = -, = H = + 3, = - 3 J = + 3, = ( ) + 3( ) = A B C D Pre-AP/Anchor Problem A gear that is 8 inches in diameter turns a smaller gear that is 3 inches in diameter. a. Does this situation represent a direct or inverse variation? Eplain your reasoning. b. If the larger gear makes 36 revolutions, how many revolutions does the smaller gear make in that time? c. If a third gear is added that is 4 inches in diameter, and the 4 inch gear makes 0 revolutions does the 8 inch gear make in that time? y 3 = F 8( - y)( + y)( - y) G 8( - y)(4 - y + y ) H 8( + y)(4 + y + y ) J 8( - y)(4 + y + y ) Question To prepare for a standardized test, make flash cards of key mathematical terms, such as direct, inverse, and joint variation. Use the glossary in this tet to determine the important terms and their correct definitions. NEED EXTRA HELP? If You Missed Question Go to Lesson Prior Course 8-4 For Help with Standard G.0 A6.0 Chapter 8 California Standards Practice 495