12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention 12-1 Inverse Variation 2 3 B. Vocabular inverse variation 3 5 6 3 15 20 25 Find for each ordered pair. Write the function rule. 2( ) 12 Ever -term is multiplied b () 12 to get the -term. ( ) The product is a constant so the relationship an inverse variation. Can the relationship be written in k form? So this relationship an inverse variation. Graphing an Inverse Variation Write and graph the inverse variation in which 3 when 2. Step 1 Find k. k Write the rule for constant of variation. 2( ) Substitute for and for. Step 2 Use the value k to write an inverse variation equation. k Write the rule for inverse variation. Substitute 6 for. Step 3 Use the equation to make a table of values. Complete the table. Step Plot the points and connect them with smooth curves. 3 1 3 2 3 6 Undef. 3 200 Holt Algebra 1
12A Read To Go On? Problem Solving Intervention 12-1 Inverse Variation Inverse variation implies that when one quantit increases the other quantit will. A group of college students build decks over the summer for etra mone. The time required to completel build a deck varies inversel as the number of people who are involved in the building. If it takes hours for 6 people to build the deck, how long will it take 2 people to build the same deck assuming the work at the same rate? Understand the Problem 1. In the first situation, how man people are building the deck? 2. How long does it take to completel build the first deck? 3. In the second situation, how man people are building the deck?. What are ou asked to find? Make a Plan 5. What is the relationship between the number of workers and the amount of time? 6. Write the Product Rule for Inverse Variation. 7. What do ou substitute for 1 1?. What do ou substitute for 2? Solve 9. Using the Product Rule for Inverse Variation, solve for 2. (6)() (2)( 2 ) 2 10. How long will it take 2 people to build the deck at the same rate as the first deck was built? Look Back 11. Substitute the time found for 2 in Eercise 9 into the original equation for the Product Rule for Inverse Variation. (6)() 2( ) Is our answer correct? 2 201 Holt Algebra 1
12A Find these vocabular words in Lesson 12-2 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 12-2 Rational Functions rational function ecluded value discontinuous function asmptote Identifing Ecluded Values Identif the ecluded value for each rational function. A. 9 1 B. 2 6 Set the denominator equal to 0. Factor the denominator. ( )( ) 0 Set the denominator equal to 0. The ecluded value is. ( )( ) 0 The ecluded values are and. Graphing Rational Functions Using Asmptotes Identif the ecluded values and the vertical and horizontal asmptotes for the 2 rational function 3. Then graph the function. 0 To identif the ecluded value, set the denominator of the function equal to zero. Solve for. a A rational function in the form c has a b asmptote at the ecluded value, or b, and a asmptote at c. Vertical asmptote: Horizontal asmptote On the coordinate grid, graph the asmptotes using dashed lines. Make a table of values. Choose -values on both sides of the vertical asmptote. 0 2 2.5 undefined Plot the points and connect them with smooth curves. The curves will get ver close to the asmptotes, but will not touch them. 202 Holt Algebra 1
12A Read To Go On? Skills Intervention 12-3 Simplifing Rational Epressions Find this vocabular word in Lesson 12-3 and the Multilingual Glossar. Identifing Ecluded Values Find an ecluded value of each rational epression. 11 A. k 3 k 3 Set the denominator equal to. k Solve for k. The ecluded value is. 7 B. 12 2 12 2 Set the equal to zero. ( ) 0 Factor. or Use the Zero Product Propert and solve for. The ecluded values are and. Vocabular rational epression Simplifing Rational Epressions 2 Simplif the rational epression, if possible. Identif an ecluded values. 2 2 2 2 2 _ (2 )( ) ( )( ) Factor the numerator and denominator. ( )( ) ( 2)( ) ( 2)( 2) ( 2)( 1) Determine the ecluded values. 1 The ecluded value is. Factor 1 from (2 ). Divide out the common factor. Simplif. Set the denominator equal to zero. Solve for. 203 Holt Algebra 1
12A Read To Go On? Quiz 12-1 Inverse Variation Tell whether each relationship represents an inverse variation. Eplain. 1. 2 3 12 6 2. 5 7 9 10 1 1 3. 1 2. 3 5. 2 6. 2 7. Write and graph the inverse. Write and graph the inverse variation in which 3 when 3. variation in which 2 when 3. 9. The cost of campaign buttons for the student government elections varies inversel to the number of buttons ordered. Fift buttons cost $0.90 each. How man buttons can be purchased if the cost $0.75 each? 12-2 Rational Functions Identif the ecluded values and the vertical and horizontal asmptotes for each rational function. Then graph each function. 10. 1 11. 2 1 20 Holt Algebra 1
12A 12. Read To Go On? Quiz continued 5 2 13. 3 3 2 1. Sara is joining a mail order CD club. She has $60 to spend on CD s. There is a $ shipping and handling charge. The number of CD s that Sara can bu is given b 60, where represents the cost of each CD in dollars. Describe a reasonable domain and range and graph the function. 16 16 16 16 12-3 Simplifing Rational Epressions Find an ecluded values of each rational epression. 15. 2t 16. t 3 17. _ 6 t 1 t t 2 1 1. 3 _ t 2 1 Simplif each rational epression, if possible. Identif an ecluded values. 19. n 12 n 20. n 2 3n 21. _ 3 22. _ 2 12 3 6n 2 2 15 2 2 23. Suppose the radius of a circle is equal to half the length of the side of a square. Find the ratio of the area of the circle to the area of the square. 205 Holt Algebra 1
12A Read To Go On? Enrichment Other Tpes of Variation Two tpes of variation have been discussed thus far. Direct variation is an equation of the form k while inverse variation is an equation of the form k or k. Two important laws from Chemistr arise from the idea of variation. Bole s Law states that the volume of a gas at a given temperature varies inversel with applied pressure. Mathematicall, this inverse variation can be epressed as V k P, where V is the volume, P is the applied pressure, and k is a constant. Charles Law states that the volume of a gas at a given pressure varies directl with temperature. Mathematicall, this direct variation can be stated V kt, where V is the volume, T is the temperature, and k is a constant. Combining the two laws results in what is called a joint variation: The volume of a gas varies directl with the temperature and inversel with pressure. Answer each question. 1. Write the mathematical statement for the joint variation of the two gas laws. 2. a. If the volume of a sample of gas is 3.21 L under a pressure of 0.20 atm at a temperature of 300 Kelvin, find k. b. If the pressure was adjusted to 0.50 atm and the temperature was changed to 320 Kelvin, determine the volume of the sample of gas. c. If the temperature of the sample was held constant at 320 Kelvin, what would the pressure need to be adjusted to in order to return the volume to 2 L? 3. a. Suppose a 5 L sample of gas under went the following changes: the pressure was changed from 0.1 atm to 0.07 atm, and the temperature was changed from 00 Kelvin to 320 Kelvin. Determine the volume of the sample of gas. b. Holding the pressure constant at 0.07 atm, what change in temperature would return the gas to a volume of 5 L? 206 Holt Algebra 1
Read To Go On? Skills Intervention 12- Multipling and Dividing Rational Epressions Multipling Rational Epressions. Multipl 16 b c 3 _ 15 a 3 b. Simplif our answer. 3ac b 3 c 2 16( ) a 3 (b ) c 3 3() b 3 (c ) Multipl the numerators and denominators. Arrange the epression so like variables are together. a b c 3 a b 3 c Simplif. a b 2 c 0 Divide out common factors. Use properties of eponents. a b 2 Simplif. Remember that c 0 1. Multipling Rational Epressions Containing Polnomials Multipl m 2 m 12 m 2 9m 20 m 2 10m 25. Simplif our answer. 5m 15 (m )(m 3) (m )(m ) (m 5)(m ) 5(m ) Factor. (m )(m 3) (m 5)(m ) (m )(m ) 5(m ) m Divide out common factors. Simplif. Dividing b Rational Epressions and Polnomials Divide 2 2 11 12 2 _ 2 6. Simplif our answer. 3 2 2 11 12 3 Write as multiplication b the reciprocal. (2 )( ) 3 ( ) (2 )( ) (2 )( ) 3 3 ( ) (2 )( ) 2 ( )( ) 2 Factor. Divide out common factors. Simplif. 207 Holt Algebra 1
Subtracting Rational Epressions with Like Denominators Subtract 2 7 3 5 15. Simplif our answer. 3 2 7 3 Read To Go On? Skills Intervention 12-5 Adding and Subtracting Rational Epressions 5 15 3 2 7 ( ) 3 2 7 3 2 3 ( )( ) _ 3 Add the opposite. Distribute the negative. Combine like terms. Factor. Simplif. Adding and Subtracting with Unlike Denominators Add or Subtract. Simplif our answer. A. 3 7 2 9 3 Step 1 Identif the LCD. 2 2 9 3 3 LCD 2 3 3 Step 2 Multipl each term b an appropriate form of 1. Step 3 Write each epression using the LCD. 3 7 2 9 3 2 36 3 36 3 Step Add the numerators. _ 2 36 3 Step 5, 6 No factoring is needed. The problem is in simplest form. B. 3 7 Step 1 The denominators are opposite binomials. The LCD can be either or. Step 2 Multipl the second term b 1 1. 3 7 Step 3 Write each epression using the LCD. Step Subtract the numerators. Step 5, 6 No factoring is needed. Just simplif. 3 _ ( ) 3 ( ) 20 Holt Algebra 1
Read To Go On? Skills Intervention 12-6 Dividing Polnomials Dividing a Polnomial b a Monomial Divide. (2 3 6 2 10) 2 (2 3 6 2 10) Write as a rational epression. 2 3 2 2 10 2 32 3 2 2 2 10 2 2 5 Divide each term in the polnomial b the monomial. Divide out common factors. Simplif. Polnomial Long Division Divide using long division. (2 2 23 ) ( ) Step 1 2 Step 2 2 23 2 Step 3 2 23 2 2 Step 2 23 2 2 0 Step 5 2 23 2 2 0 Step 6 2 23 2 2 (7 2)) 0 Write in long division form with epressions in standard form. Divide the first term of the dividend 2 b the first term of the divisor to get the first term of the quotient. Multipl the first term of the quotient b the binomial divisor ( ). Place the product under the dividend, aligning like terms. Subtract the product from the dividend. Bring down the net term in the dividend. Repeat Steps 2 5 as necessar. 209 Holt Algebra 1
Read To Go On? Skills Intervention 12-7 Solving Rational Equations Find this vocabular word in Lesson 12-7 and the Multilingual Glossar. Solving Rational Equations b Using Cross Products 6 Solve 7 1. Identif an etraneous solutions. 2 6( ) Multipl cross products. 6 Distribute 6 on the left side. 7 Subtract from both sides. Subtract 12 from both sides. Divide both sides b 5. Vocabular rational equation Solve Rational Equations b Using the LCD. Solve 1 n 2 n. Identif an etraneous solutions. Step 1 Find the LCD. Include ever factor of the denominators. The LCD is n 2. Step 2 Multipl both sides of the equation b the LCD. Distribute on the left side. n 2 n n 2 1 n n 2 Step 3 Simplif and solve. 1n n 2 0 n 2 1n 0 (2 n 2 n ) 0 2(2n )(n ) 0 2n or n 0 n or n Check: Verif that our solutions make the equation true. n 1 2 1 2 2 n 1 () 2 1 7 2 2 32 1 2 1 2 The solutions are, and. There are etraneous solutions. 210 Holt Algebra 1
Read To Go On? Problem Solving Intervention 12-7 Solving Rational Equations A rational equation is an equation that contains one or more rational epressions. Mort can refinish a wood table in 6 hours. It takes his business partner Rebecca 10 hours to refinish the same table. How long will it take them to refinish the table if the work together? Understand the Problem 1. What are ou being asked to determine? 2. Mort refinishes the table in hours, so he completes 1 of the table per hour. 6 3. Rebecca refinishes the table in hours, so she completes of the table per hour. Make a Plan. Mort s rate, times the number of hours worked, plus Rebecca s rate, times the number of hours worked, equals the complete time to refinish the table. Let h the number of hours worked. Mort s rate Rebecca s rate complete job 1 Solve 5. Solve the rational equation. 1 6 h 1 h 1 What is the LCD? 10 1 6 h 1 10 h 1 Multipl both sides b the LCD. 10h 60 Distribute 60 on the left side and solve the equation. 60 60 h 3 6. Mort and Rebecca, working together, can refinish the table in hours. Look Back 7. In 3 3 hours, Mort completes 15 of the table and Rebecca completes 15 1 of the table. Together, the complete 5 3 or 1 table. 211 Holt Algebra 1
Read To Go On? Quiz 12- Multipling and Dividing Rational Epressions Multipl. Simplif our answer. 1. m _ 3 m 2 (2 m 2 m) 2. 3 3 ( 2 6 9) 3. _ 12 3 2 7 3 2 _ 9 3(. a 3 a) 2 3 2 a 1 1 a 1 5. 2 2 2 3 3 2 6. z 2 z 6 z 2 2z z _ 2 7z 12 z 2 9 Divide. Simplif our answer. 7. b 5 b 3. z _ 2 z 2 3z 2 2 z 2 z 2 z 9. 2 2 2 6 3 12-5 Adding and Subtracting Rational Epressions Add or subtract. Simplif our answer. 10. 7 3 3 11. 2 3 6 3 12. 2 5 6 3 6 13. 3m 5m 1 m 2 1. 5 3 2 6 15. _ 5a 2 a 2 a 20 3 a 5 a a 212 Holt Algebra 1
Read To Go On? Quiz continued 16. A triathlon consists of a 1-mile swim, a 30-mile bike ride, and a 6-mile run. Jose averages 20 times faster on his bike than he does on the swim. He can complete the run in one-fourth the time it takes to complete the swim. Let r represent Jose swimming rate. Write and simplif an epression, in terms of r, that represents the time it takes Jose to complete the triathlon. Then determine how long it will take Jose to complete the triathlon if he swims an average of 1 mile per hour. 12-6 Dividing Polnomials Divide. 17. (20 n 2 10n) 5n 1. (12 p p 3 2 p 2 ) ( p 2 ) 19. ( 2 15) ( 3) Divide using long division. 20. ( 2 5 36) ( ) 21. ( m 2 22m 121) (m 11) 22. (3 2 7 9) ( 1) 12-7 Solving Rational Equations Solve. Identif an etraneous solutions. 23. 3 1 6 2. 2 1 2 6 25. 2 3 7 2 11 2 10 21 26. 2 3 1 3 27. _ 1 2 2. 2 3 3 5 12 29. You have a lawn-car business. You are looking to add a partner to help ou mow lawns. There is a lawn that takes ou 30 minutes to mow. The person ou are considering to hire can mow it in 5 minutes. How long will it take the two of ou to mow the lawn working together? 213 Holt Algebra 1
Snthetic Division Snthetic division is a shortcut that can be used when dividing a polnomial b a binomial. In order for snthetic division to work, the divisor must be in the form of c, that is, a variable minus a constant. Eample: ( 3 6 2 30) ( 2). Divide. The value of c is 2. Write the coefficients of the dividend and the value for c in the upper left corner. 2 1 6 1 30 2 1 6 1 30 2 1 6 1 30 1 Read to Go On? Enrichment 1 2 2 16 30 1 15 0 Bring down the first coefficient 1 and write it below the horizontal bar. The quotient is 2 15. Multipl 2 b 1 to get 2. Write the product under the net coefficient and add. Repeat the steps (multipl, write the product under the net coefficient and add) with the remaining numbers. Use snthetic division to find each quotient. 1. ( 2 19 5) ( 5) 2. (3 2 5 12) ( 3) 3. ( a 3 3 a 2 2a 3) (a 1). (5 w 3 6 w 2 3w 1) (w 1) 5. ( 3 1) ( 1) (Hint : There are missing terms, fill in the missing terms with 0.) 6. (2 5 5 3 2 6 23) ( 3) 21 Holt Algebra 1