UNIT Rational Equations You can use a rational function to model the intensit of sound. Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
The intensit of sound is inversel proportional to the square of the distance from the sound source. For instance, if Paulina is twice as far from the stage as Ric, then she hears sound that is one-fourth the intensit Ric hears. Big Ideas The laws of arithmetic can be used to simplif algebraic epressions and equations. Epressions and equations epress relationships between different entities. A function is a correspondence between two sets, the domain and the range, that assigns to each member of the domain eactl one member of the range. Unit Topics Foundations for Unit Dividing Monomials and Polnomials Operations with Rational Epressions Compound Fractions Solving Rational Equations Reciprocal Power Functions Graphing Rational Functions RATIONAL EQUATIONS 0 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Foundations for Unit Before ou learn about rational equations, ou should know how to do the following: Add and subtract fractions. Multipl fractions. Use quotient properties of eponents to simplif epressions. Adding and Subtracting Fractions You can add or subtract two fractions with a common denominator b adding or subtracting the numerators of the fractions and keeping the denominator the same. Definition least common denominator (LCD) the least common multiple of (the smallest number that all the denominators will divide into evenl) Eample Add or subtract the following fractions. A + + B z z z + 9 7 5 C a b + d c a d b d + c b d b ad bd + bc bd ad + bc bd 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Unit Foundations Problem Set A Add or subtract the following fractions... 5 + 5.. a b + b c 5. + a b r_ t s_ t. a b c d Multipling and Dividing Fractions To multipl two fractions, multipl the numerators and multipl the denominators. Eample Multipl. A a b c d a c b d ac bd B C r_ t t s r t s t rs t To divide two fractions, ou find the reciprocal of the divisor and then multipl the numerators and multipl the denominators. Eample Divide. A 7 7 7 Problem Set B 7 B a b c d a b d c a d b c ad bc C z z z z z z Multipl. 7.. Divide. 7 7 9. 0. 7 0 5. u v w t. a b a c.. 5 5 5.. 7. z. c d e f r_ s r s_ FOUNDATIONS FOR UNIT 05 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Unit Foundations Using Properties of Eponents to Simplif an Algebraic Epression You can use properties of negative eponents and quotients to simplif epressions. PROPERTIES OF EXPONENTS Let a and b be nonzero real numbers. Let m and n be integers. Propert Statement Eamples Negative Eponent a m a m Quotient of Powers am 5 5 5 5 a a n m n 5 Power of a Quotient ( a b ) m am b ( m ) 7 Eample Simplif each epression. A B 7 5 7 5 C ( ) D 5 E z a z z b a b F ( ab ) c (ab) a b c Problem Set C Simplif each epression. 9.. ( ) 5. 0. c c c 5 5. a b. ( st r ) 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Dividing Monomials and Polnomials A ratio is a comparison of two quantities, often written as a fraction. If the quantities are represented b polnomials, the ratio is a rational epression. DEFINITION A rational epression is a ratio whose numerator and denominator are polnomials, and the denominator is nonzero. THINK ABOUT IT A rational epression has the form p q, where p and q are polnomials and q 0. The restriction q 0 ensures that p q is defined. Finding Restrictions on the Domain of a Rational Epression To find the domain of a rational epression, find the values of the variables that make the denominator equal to 0. These values are ecluded from the domain. Eample Find the domain restrictions for each epression. A 5 Solution 5 0 when 0 or 0. The domain restrictions are 0 and 0. B 7 Solution 0 The domain restriction is. C _ + 0 5 + Solution + 0 ( + )( ) 0 + 0 or 0 The domain restrictions are and. DIVIDING MONOMIALS AND POLYNOMIALS 07 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Simplifing Rational Epressions Simplifing a rational epression is similar to simplifing a fraction. Identif and divide out an factors ou find in the numerator and denominator. Be careful, though; both the numerator and denominator must be written as products of factors. You can t divide out a factor in the denominator with a term that is being added in the numerator. Eample Find the domain restrictions and simplif each epression. A 5 7 Solution The domain restrictions are 0 and 0. 5 5 ċ 7 5 ċ 5 ċ 5 ċ B a 0a + a Factor the greatest common factor from the numerator and denominator. Divide out the common factor. Simplif. Solution To find the restrictions on the domain, set a 0 and solve. The domain restriction is a 0. a 0a + a C ( + )( + ) ( + )( + ) a a 0a a + a a a 0a a + a a 5 + a Simplif. Divide each term of the polnomial b the monomial. Divide out common factors. Solution To find the restrictions on the domain, set ( + )( + ) 0 and solve. The domain restrictions are and. ( + )( + ) ( + )( + ) a ( + )( + ) ( + )( + ) + + 5 a Divide out the common factor. Simplif. THINK ABOUT IT A rational epression is in simplest form when the numerator and denominator have no common factors other than. TIP In Eample A, ou can find the greatest common factor of the numerator and denominator with the same strategies that ou use to find the greatest common monomial factor in the polnomial 5 + 7. 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
D + + Solution To find the domain restrictions, set + 0 and solve. + 0 ( )( + ) 0 0 or + 0 The domain restrictions are and. + + E 5 0 ( )( + ) ( )( + ) ( )( + ) ( )( + ) _ Factor the trinomials. Divide out the common factor. Simplif. Solution To find the restrictions on the domain, set 0 and solve. 0 The domain restriction is. 5 0 5( ) Factor the numerator. 5( ) and are opposites. Rewrite ( ) in the denominator as ( ). 5( ) ( ) 5 Divide out the common factor. Simplif. TIP To demonstrate that and are opposites, choose a value for. Let 7. Then 7 and 7. In general, binomial factors of the forms (a b) and (b a) are opposites of each other, and _ a b b a. Problem Set Find the domain restrictions for each epression.... 5 5 ab a b t _ 5 + t. 5.. v(v + ) 5v d (d 7) d + ( )( + ) ( ) 7.. 9. + + a 5a + a 7a + r r + r + r DIVIDING MONOMIALS AND POLYNOMIALS 09 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
For each epression, do the following: A. Find the domain restrictions. B. Simplif the epression. 0. 0 7 _ 5 5. 5p + 5p + 0 5p. c c + c. p q 9 _ p 5 q 9. r + r + 5r r 7. h + 9h + 0 _ h +... 5.. 7. _ a b a b 7 ( )( + 5) ( )( ) ( g)(g + ) (g + )(5 g) ( n)(5 + n) (n )(5 + n) _ + 9 t t + t 0..... 5. m + m m m f 5 + f + 7f 7f s + s + s s a + a a 9 q + q 7 q 9q + v + v 5 _ v *. Challenge z z z z * 9. Challenge c c 0 (c 5) c + Solve. * 0. Challenge The vertical asmptote of the graph of an equation is where the denominator equals zero after the equation has been simplified. The following equations are related to the epressions in Problems,, and. What are the equations of the asmptotes? A. + + B. + 9 C. + 7 9 + 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Operations with Rational Epressions As with numbers, rational epressions can be added, subtracted, multiplied, or divided. Adding or Subtracting Rational Epressions with Like Denominators To add or subtract rational epressions with like denominators, add or subtract the numerators. Keep the same common denominator. Eample Add. + + _ 5 Solution + + _ 5 ( + ) + ( 5) _ 5 ( + 5)( ) Add the numerators. Combine like terms. Factor and reduce if possible. + 5 Simplif. Adding or Subtracting Rational Epressions with Unlike Denominators To add or subtract rational epressions with unlike denominators, first rewrite the epressions with a common denominator, preferabl the least common denominator. Then add or subtract. DEFINITION The least common denominator (LCD) of two rational epressions is the least common multiple of the denominators. TIP How to Find the Least Common Denominator Step Full factor each denominator. Step Write each factor the greatest number of times it appears in an one denominator. Step Multipl the factors described in Step ; the LCD is the product of those factors. OPERATIONS WITH RATIONAL EXPRESSIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Eample Subtract. 7 + A ( + )( ) 5 + Solution 7 + ( + )( ) 5 + The denominators are in factored form. The LCD is +. Write the LCD b writing each factor the greatest number of times it appears in an denominator. 7 + ( + )( ) 5 + 7 + ( + )( ) 5 5 ( + )( ) (7 + ) (5 5) ( + )( ) + ( + )( ) ( + 9) ( + )( ) B a + 5a + 5a a + 0a + 5a Solution a + 5a + 5a a + 0a + 5a 5a (a + 5) a + a(a + 5)(a + 5) The first ratio has the LCD. Multipl the second ratio b. For the appropriate form of, decide what factor(s) would be needed to get the LCD. Epand the second numerator. Subtract the numerators. Simplif the numerator. You can factor out a, but no factors can be divided out. The last epression is in simplified form. Factor the denominators so ou can find the LCD. The LCD is 5 a (a + 5)(a + 5) Write the LCD b writing each factor the greatest number of times it appears in an denominator. 5a (a + 5) _ a + 5 a + 5 a + a(a + 5)(a + 5) 5a 5a ( + ) appears ( ) appears once in the first once in the first and second denominator. denominator. { 5 appears once in the first denominator. a + 0 5a (a + 5)(a + 5) 5a + 0a 5a (a + 5)(a + 5) _ (a + 0) (5a + 0a) 5a (a + 5)(a + 5) 5a a + 0 5a (a + 5)(a + 5) { a appears twice (nd power) in the first denominator and once in the second. (a + 5) appears twice in the second denominator. Multipl each ratio b. To determine the appropriate form of for each ratio, decide what factor(s) would be needed to get the LCD as the denominator. Epand the numerators. Subtract the numerators. Simplif the numerator to get 5a a + 0, which cannot be factored. So no factors can be divided out, and the last epression is in simplified form. UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Multipling Rational Epressions To multipl rational epressions, multipl the numerators and multipl the denominators. Divide out common factors to simplif. Eample Multipl. A h h 7 (h 7)(h ) h(h + ) Solution h (h 7)(h ) h 7 h(h + ) TIP (h )(h + ) and _ h + h are both in simplified form. Recognizing that the are equal can be helpful when ou work with rational functions. h(h 7) _ (h 7)(h )(h)(h + ) The polnomials are in factored form, so multipl the numerators and the denominators. _ h(h 7) (h 7)(h )(h)(h + ) (h )(h + ) _ h + h Divide out common factors. Simplif. Alternate simplified form B 0 + 0 Solution 0 + 0 ( 5) ( + )( ) + ( 5) ( 5)( + ) ( + )( )( )( 5) ( 5)( + ) ( + )( )( )( 5) ()( ) Factor the polnomials. Multipl the numerators and the denominators. Divide out common factors. To identif the greatest common monomial factor, write as. Simplif. Alternate simplified form Dividing Rational Epressions To divide b a rational epression, multipl b its reciprocal. Eample Divide. A 5( ) ( + ) ( + ) 5( ) ( + ) Solution Write the division problem as multipling b a reciprocal. 5( ) 5( ) ( + ) ( + ) ( + ) 5( ) ( + ) Multipl b the reciprocal of the divisor. ( + ) ( + ) 5( ) 5( )()( + ) Multipl numerators and denominators. ( + )( + )(5)( ) Divide out common factors. ( + ) or _ + 9 Simplif. OPERATIONS WITH RATIONAL EXPRESSIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B _ p p + p + _ p 7p p + p Solution Write the division problem as multipling b a reciprocal. _ p p + p + _ p 7p p + p _ p p + p + _ p + p p 7p ( p + )( p ) (p + )(p + ) p(p + ) ( p )(p + ) ( p + )( p )(p)(p + ) (p + )(p + )( p )(p + ) p( p + ) (p + ) or p + p _ p + p + Multipl b the reciprocal of the divisor. Factor. Multipl numerators and denominators. Divide out common factors. Simplif. Problem Set Add or subtract the epressions and simplif..... 5.. 7. + + 5 + _ 5t 5 t + _ t + t + q q _ q + q q q + _ 5 + + 5 5 b b b 5 b + 7b b 5 _ + + + _ 5 +. 9. 0..... 5 ( )( +) 5 ( + )( ) + ( )( ) a a + a + a a + a _ + 0 + 0 v v 5v 0v + 50v + + + _ z z + 9z 5 z z z Multipl or divide the epressions and simplif. Assume that no denominator equals zero. 5.. 7.. 9. ( ) ( + 5)( ) + 5 ( ) n(n + ) n n n 5( ) + _ + c (c + )(c ) _ c (c ) (w ) _ w w w + 0..... ( ) ( + ) ( ) _ ( ) 9( ) ( ) r r r r r + r r d 5 d d d 5 d d b b b 9b 5b + 0 (b 5b + ) UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
5.. + + g g + 7g 5g g 5g 7.. ( ) + + _ 0 5 Simplif. Assume that no denominator equals zero. * k + 9. Challenge k + k + 9 + k + k ( ) * 0. Challenge a a + a a a + a a a + a OPERATIONS WITH RATIONAL EXPRESSIONS 5 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Compound Fractions A compound event involves two or more actions. Similarl, a compoundfraction has two or more operations. DEFINITION A compound fraction is a fraction that has a fraction in the numerator and/or denominator. TIP A compound fraction can contain numerical fractions, rational epressions, or both. Eamples: b +, b 5, 5 Simplifing a Compound Fraction To simplif a compound fraction, either multipl the numerator and the denominator of the compound fraction b a common denominator of the fractions within the fraction (the LCD method) or divide the numerator of the compound fraction b the denominator (the division method). Eample Simplif each epression. A 5 Solution LCD Method 5 B _ 5 5 5 _ 5 5 5 5 a b c d Solution LCD Method a a b b bd a bd d b c c d d c bd d bd b 0 9 9 ad bc Division Method 5 5 5 Division Method a b c d a b c d a 0 9 9 b d c ad bc REMEMBER The LCD is the least common denominator of two or more fractions. UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Eample Simplif each epression. A + 5 + Solution + 5 + ( + 5 ) ( + ) Multipl the numerator and denominator b the LCD:. B + 5 + Distribute. _ + 0 + 9 _ 5 + a a Solution 5 + a a ( 5 + ( _ 5 a a a ) a + a ) a Simplif. Simplif. Simplif. Rewrite as division. Rewrite the first fraction of the dividend using the LCD: a. ( 0a a + a ) a a ) a ( 0a + 0a + a 0a + a 0a + a a Simplif. Simplif. Division is the same as multiplication b the reciprocal of the divisor. Divide out the common factor. Simplif. COMPOUND FRACTIONS 7 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Using an LCD to Simplif a Compound Algebraic Fraction To simplif a compound fraction with variables, multipl the numerator and denominator b the LCD of all fractions that appear in the numerator or denominator. Then simplif. Use this method when there is addition or subtraction in a fraction of the numerator and/or denominator. Eample Simplif. + Solution + + ( ) ( ) Multipl the numerator and denominator b the LCD: ( ). _ ( ) + ( ) _ ( ) ( ) Distribute ( ). _ ( ) ( ) + ( ) _ ( ) ( ) + ( ) _ 9 Divide out common factors. Simplif. Simplif. Problem Set Simplif... 7 7.. 5 0 5. 5.. 9 0 5. a b 9. a b b a UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
0. 9 9. 5a a 0. _ a + b _ 5 + a +.. + + 5 + 9. 0. _ 5 7 + 9 7.. 5 + + 7 + + 9 + 0 + +. + 7. + + + * 9. Challenge + 9 + 5 + 0 9. 5.. + + +... 5 5 + + + * 0. Challenge + _ + + 7. 0 + 5. + 5 + 5 ( + 5) COMPOUND FRACTIONS 9 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Solving Rational Equations A rational equation is an equation that contains one or more rational epressions. Solving Rational Equations METHODS OF SOLVING RATIONAL EQUATIONS LCD method Multipl both sides of the equation b the least common denominator (LCD) of the rational epressions. Then solve the resulting equation. This method works for an rational equation. Cross-multiplication method If the equation is in a form that looks like a b c, ou can cross-multipl, d then solve the resulting equation. The method of cross-multipling is based on the following propert: a b c if and onl if ad bc. d Because rational equations have variables in denominators, ou have to be careful with the domain. An value of the variable that makes a denominator equal zero cannot be a solution, and therefore must be ecluded. It s a good idea to start b identifing those values that must be ecluded before ou begin to solve the equation. Eample Solve and check. 5 5 Solution To find the restrictions on the domain, set the denominators equal to zero and solve. The domain restrictions are 0 and. Now solve the equation. 5 5 5 5( ) Cross-multipl. 5 5 0 Distribute the 5. 0 0 Subtract 5. Divide b 0. 5 5 TIP If ou forget to identif the values that must be ecluded, ou should discover them when ou check our solutions anwa. TIP A rational equation is a proportion if it has the form a b c a, where d b and c are rational d epressions. 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Check 5 5 5 5 Substitute for in the original equation. 5 9 5 5 5 Substituting results in a true statement. is not in the domain restrictions, so the solution set is { }. Solving Rational Equations and Checking for Etraneous Solutions The zero product propert can be used to identif domain restrictions or solve more complicated equations. ZERO PRODUCT PROPERTY For an real numbers a and b, ab 0 if and onl if a 0 or b 0. Eample Solve and check. A + 5 + + + 7 + Solution First identif values of that must be ecluded from the domain. + 5 + + + 7 + Original equation ( + )( + ) + + 7 + Factor the first denominator. If, then + 0, which makes two denominators equal zero. If, then + 0, which makes two denominators equal zero. So and must be ecluded as possible solutions. Now solve the equation. ( + )( + ) + + 7 + ( + )( + ) [ ( + )( + ) + + ] ( + )( + ) 7 ( + ) ( + )( + ) + ( + )( + ) ( + )( + ) + ( + )( + ) 7 + REMEMBER See how to appl the propert with this eample. ( )( + ) 0 ( ) 0 or ( + ) 0 So the solutions to the equation are and. Keep denominators in factored form to help identif the LCD. Multipl both sides of the equation b the LCD. Distribute the LCD to ever term. + ( + )( ) ( + ) 7 Divide out common factors. + + 7 + Multipl binomials. Distribute 7. + 7 + Simplif. 5 0 Add 7 to get 0 on one side. ( 7)( + ) 0 Factor the trinomial. 7 0 or + 0 Appl the zero product propert. 7 SOLVING RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Check Substitute 7 for : Substitute for : + 5 + + + 7 + + 5 + + + 7 + 7 + 5 7 + + 7 7 + 7 7 + ( ) + 5 ( ) + + + 7 + _ 9 + 5 + + 9 7 0 0 + 0 7 90 + 90 0 90 Here, is called an etraneous solution because it does not make the 90 90 original equation true. Recall that So 7 is a solution. was identified as a value that must be ecluded as a possible solution anwa. The solution set is {7}. B 0 n n + 9 n Solution First note that 0 and must be ecluded as possible solutions because each value makes a denominator equal zero. Solve the equation. 0 n n + 9 n n(n ) 0 n + 9 n n(n ) n + n(n )( ) n(n ) 0 n + 9 n n(n ) n + n(n )( ) (n )0 n(n + 9) n(n ) 0n 0 n + 9n n + n n n 0 0 (n + 5)(n ) 0 Appl the zero product propert: n + 5 0 or n 0 n 5 n Check Substitute 5 for n: Substitute for n: 0 n n + 9 n 0 n n + 9 n 0 5 + 9 0 5 5 + 9 5 7 0 5 5 5 0 ( 5 ) ( ) The solution set is { 5, }. UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Solving a Rate Problem Eample It takes hours for pump A to fill a tank. It takes hours for pump A and pump B together to fill the same tank. How long would it take pump B alone to fill the tank? Solution Pump A alone can fill the tank in hours, so it fills of the tank each hour. Pumps A and B together can fill the tank in hours, so the fill of the tank each hour. Let be the number of hours it takes pump B to fill the tank alone. Then pump B alone fills of the tank each hour. Now reason as follows: Pump A fills of the tank in hour, of the tank in hours, of the tank in hours, and so on. Pump B fills of the tank in hour, of the tank in hours, of the tank in hours, and so on. Write an equation: the amount of the tank that pump A fills in hours the amount of the + tank that pump B fills in hours one full tank Solve the equation: + + + + + Multipl both sides b the LCD. Divide out common factors. Solve the resulting equation. Substitute for in the original equation to verif that is the solution. It would take hours for pump B to fill the tank alone. THINK ABOUT IT For the equation +, the value 0 must be ecluded as a possible solution. Solving a Work Problem Eample Mario can finish all the house chores twice as fast as his sister Ana. Together the can do the job in hours. How long would it take each of them working alone? Solution Let be the number of hours it takes Mario to do the chores alone. Mario does the job twice as fast as Ana, so Ana would take hours to do the chores b herself. Then Mario does of the chores each hour, and Ana can do of the chores each hour. Mario and Ana together can finish the chores in hours, so together the can do of the job each hour. SOLVING RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Write an equation that combines Mario s and Ana s labor for each hour: + Solve the equation: + + Multipl both sides b the LCD. + Divide out common factors. + Solve the resulting equation. Substitute for in the original equation to verif that is the solution. It would take hours for Mario to do all the house chores alone, and it would take twice as long, or hours, for Ana to do the job alone. Problem Set Solve and check.. h + h. p p 7 5. 7. + 5. 0 a + 5a. 7.. 9. k k 7 + _ w + w w _ 7 q + q + 0..... 5.. 7.. t + 7 t 5 t + n n 7 n + 5 9 c + c + 5 + c c + 5 c + 7 + 9 + + + + r _ r + r r _ b + 5b + + b b + 7 b + _ g g + 5g g + g _ + g z z 5 z + 5 z + z + 5 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
For each problem, do the following: A. Define a variable and write an equation that models the problem. B. Solve the equation. C. Answer the question. 9. It takes 0 hours for Arturo to build a fence. It takes hours for Sun and Arturo together to build the same fence. How long would it take Sun alone to build the fence? 0. Skler can complete the shopping twice as fast as her brother Blaze. Together the can do the shopping in hours. How long would it take each of them working alone?. Julio can stain the deck three times faster than his cousin Samuel. Together the can stain the deck in hours. How long would it take each of them working alone?. It takes 5 hours for Lena to paint a room. It takes hours for Lena and Gustov together to paint the same room. How long would it take Gustov alone to paint the room?. It takes Compan A hours to install a heat pump. It takes 9 hours if Compan A and Compan B work together. How long would it take Compan B alone to install the heat pump?. Brooks can mow the lawn twice as fast as his sister Katrina. Together the can do the job in.5 hours. How long would it take each of them working alone? 5. It takes Repair Works 0 hours to fi a car. It takes 5 hours if Repair Works and Brakes & Things, Inc., work together. How long would it take Brakes & Things alone to fi the same car?. Deni can clean the pool four times as fast as her sister Lamur. Together the can do the job in hours. How long would it take each of them working alone? 7. It takes the local church hours to deliver food baskets. It takes hours if the church and scouts work together. How long would it take the scouts alone to deliver the food baskets?. Yahir can assemble a book three times faster than Malia. Together the can do the job in hours. How long would it take each of them working alone? * 9. Challenge Planet Ink Co. manufactures 500 books in 5 hours. Perr s Printers can do the same job in hours. If Perr s Printers begins production hour after Planet Ink has begun, how man hours will it take to produce 500 books? * 0. Challenge Taniah can complete a project in 5 hours. Her sister, Reanna, can complete the same project in 0 hours. If Taniah works alone for hour and then Reanna works alone for hours, how man additional hours are needed to finish the project if the sisters then work together? SOLVING RATIONAL EQUATIONS 5 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Reciprocal Power Functions A power function is a function of the form f() ( ) a n, where a 0 and n is a positive integer. The functions f (), g (), and h() are power functions. Graphing a Reciprocal Function DEFINITION OF RECIPROCAL POWER FUNCTION A reciprocal power function is a power function that has the power of in the denominator of a rational function. The functions f(), g(), and h() are reciprocal power functions. Eample Graph f (). Solution Use a table to find several ordered pairs in the function. Include some fractional values between and so that ou know what happens on either side of and near 0, where f () is undefined. f () f () 5 f () 5 5 5 5 The two distinct sections of the graph are called branches. The function f () is not defined for 0, so zero is ecluded from the domain of the function. Notice that the graph gets closer and closer to the -ais as the -values approach positive infinit and negative infinit. Also, the graph gets closer and closer to the -ais as the -values get closer to zero. 0 undefined THINK ABOUT IT f() approaches b means the values of f() get closer and closer to b. f() approaches means the values of f() increase without bound. f() approaches means the values of f() decrease without bound. UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
This graph is called a rectangular hperbola. DEFINITION OF HORIZONTAL AND VERTICAL ASYMPTOTES The line b is a horizontal asmptote of the graph of the function f if f() approaches b as approaches or. The line a is a vertical asmptote of the graph of the function f if f() approaches or as approaches a, either from the left or the right. So, for f (), the -ais (or 0) is a horizontal asmptote, and the -ais (or 0) is a vertical asmptote. Finding the Domain and Range of a Reciprocal Power Function Eample Determine the domain and range of each function. A f () B g () 5 Solution Find domain restrictions b setting the denominator equal to zero. Find the range b graphing. A Domain: 0 0 If 0, then the denominator equals 0. So the domain is the set of all real numbers ecept 0. In set notation, the domain is: { and 0}. Range: Since is squared, f () can never be negative and f () cannot equal 0. So the range is the set of all real numbers greater than 0, { f () f () > 0}. B Domain: 0 0 0 The domain is the set of all real numbers ecept 0. In set notation, the domain is: { and 0}. Range: Since is cubed, g() can be negative or positive and g() cannot equal 0. So the range is the set of all real numbers ecept 0, {g() g() and g() 0}. All reciprocal power functions of the form f () a n, where n is even, have the same domain and range. The domain is { and 0}. When a is positive, the range is { f () f () > 0}, and when a is negative, the range is { f () f () < 0}. All reciprocal power functions of the form f () a n, where n is odd, have the same domain,{ and 0}, and range, { f () f () and f () 0}. RECIPROCAL POWER FUNCTIONS 7 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
The Reciprocal Power Function Famil of Graphs Equations of the form f () a n are reciprocal power functions with a horizontal asmptote at 0 and a vertical asmptote at 0. Reciprocal Power Function Graph Famil: f () a n f () h() g() f () h () g() Graphs of the parent functions If ou change the parameter a, ou get curves that var in their distance from the origin. A negative a-value will reflect the graph across the -ais. Reciprocal power functions with odd powers have the same general shape, and reciprocal power functions with even powers have the same general shape. All reciprocal power functions have the - and -aes as asmptotes. UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Graphing a Simple Reciprocal Power Function Eample Graph both on the same coordinate sstem. Compare the two graphs. A f () B g () Solution In both graphs, the domain is the set of all real numbers ecept 0: { and 0} g() 5 5 THINK ABOUT IT Values for that make the denominator zero are ecluded from the domain of f (). Since these are places where f () is not defined, the ma indicate a vertical asmptote. In functions such as f (), the value of f () is alwas greater than zero, which ma indicate a horizontal asmptote at 0. Each graph has two branches, one for positive -values and one for negative -values. Each graph has a vertical asmptote whose equation is 0 and a horizontal asmptote whose equation is 0. ASYMPTOTES OF RECIPROCAL POWER FUNCTIONS The graph of the function f () a n has the following: Vertical asmptote: -ais ( 0) Horizontal asmptote: -ais ( 0) Finding the Equation from the Graph Eample Find the equation for the reciprocal power function of the form f () a that has the following graph. f () (, ) 5 RECIPROCAL POWER FUNCTIONS 9 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Solution f () a a a f () Start with the general form of the equation. ( Because the range is < 0, and because ) > 0, notice that the value of a must be negative. To find the value of a, Problem Set. f () substitute the coordinates of the given point and solve. Substitute this value of a in the general form of the equation. Write the equation of the function. Find the domain and range of each function.. g (a) 5. s(q) a 9 q Graph each function.. f () 5. M ( p) 5p. N (d) d 5 7. V (u) u. S (r) r 9. f () 0. g (v) 9 v. M ( ). N ( r ) r. g (t) t. P (c) 5c 5. f (a) a 5 Find the equation of the function shown in the graph, given the parent function.. f () a 5 7. g () a 7 5 (, ) 5 7 5 (, ( 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
. k () a. f () a 5 (, ) (, ) 9. T () a. z () a 7 5 5 7 ( ), 7 (, 5 ) 0. b() a. f () a (, 5) 5 ( ), 9 5 RECIPROCAL POWER FUNCTIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
. V () a. m() a (, 0) 5 ( ), 9 5. h() a 7. f () a 0 5 (, ) 0 0 (, ) 0 Solve. *. Challenge What happens to the graph as the * 9. Challenge Graph two reciprocal power value of a decreases for the reciprocal power function f () a functions that contain the point (, 0).? Compare the two graphs. 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Graphing Rational Functions An function that can be written as the quotient of two polnomials can be called a rational function. Finding the Domain of a Rational Function Man rational functions have restricted domains because an epression is undefined if its denominator equals 0. Eample Determine the domain of each rational function. A _ + 5 f () Solution Find domain restrictions b setting the denominator equal to 0. 0 ( 7)( + ) 0 7 0 or + 0 If 7 or, then the denominator equals 0. So the domain is the set of all real numbers ecept 7 and. You can state this in set notation: { R, 7 and }. B _ g () + Solution Find domain restrictions b setting the denominator equal to 0. + 0 The equation has no real solution, so there are no restrictions on the domain. The domain is the set of all real numbers. In set notation, this is { R}. graphing rational functions Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
The Rational Function Famil of Graphs a Equations of the form f () h + k are rational functions with one horizontal asmptote at k and one vertical asmptote at h. Rational Function Graph Famil: f () a h + k 0 f () 0 0 0 0 g() + 5 5 0 0 h() 0 g() 0 0 0 0 h() Graph of the parent function If ou change onl the parameter h, the vertical asmptote shifts h units to the left or right. If ou change onl the parameter a, ou get curves that var in their distance from the origin. If ou change onl the parameter k, the horizontal asmptote shifts k units up or down. A negative a-value will reflect the graph across the -ais. a Graphing Rational Functions in the Famil f () h + k Eample Graph each function. A + Solution If is close to, then + is far from 0. If is far from, then + is close to 0. The domain of + is {, }. The range of + is {, 0}. The line is a vertical asmptote. The line 0 is a horizontal asmptote. 0 0 0 0 + 0 THINK ABOUT IT little big big little UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B f () Solution The domain of f () is {, 0} because the denominator cannot equal 0. The range of f () is {, } because can never be 0, so can never be. Since is greater than, the curve of the graph is farther awa from the intersection of the asmptotes. C f () + Solution The domain is {, } because is a zero of the denominator. The range of f () + is {, } because can never be 0, so + can never be. The lines and are asmptotes. f () 0 0 0 0 0 f () 7 f () + 5 f () 7 5 5 7 5 7 TIP A zero of the denominator is just a zero of the polnomial that is in the denominator. Finding the Equation when Given a Graph of a Rational Function in the Famil f () a h + k Eample Find the equation in the famil f () a + k for the graph h of each function. A f () 7 5 (, ) 7 5 5 7 5 7 Solution Step There is a horizontal asmptote at and a vertical asmptote at. The equation of the graph so far is f () a +. Step Substitute the point (, ) into the equation. a + Substitute. a + Simplif. a Solve. The equation of the graph is f () +. GRAPHING RATIONAL FUNCTIONS 5 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B f () 0 0 0 ( 7, ) 0 Solution Step There is a horizontal asmptote at and a vertical asmptote at. So the equation of the graph so far is f () a +. Step Substitute the point ( 7, ) into the equation. a 7 + Substitute. a_ Simplif. 5 5 a Solve. The equation of the graph is f () 5 +. a Graphing Rational Functions Not in the Famil f () h + k Eample Graph each function. A _ 9 + Solution Factor and identif the zeros of the numerator and denominator. _ 9 + ( + )( ) + The zeros are and. The zero is. For this function, is a zero of both the numerator and denominator, so there is a hole at. Meanwhile, is a zero of onl the numerator, so the -intercept is, and (, 0) is on the graph. There are no zeros of onl the denominator, so there is no vertical asmptote. The degree of the numerator is greater than the degree of the denominator ( > ), so there is no horizontal asmptote. If ou divide out the common factor +, ou get the linear function f (). So, to _ graph 9 +, ou can graph f (), but ou must put an open dot at (, ). 0 -intercept (, 0) 0 0 hole: (, ) ( 9) ( + ) 0 UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
B f () ( ) Solution If is close to, then is far from 0. If is ( ) far from, then is close to ( ) 0. The domain of f () is ( ) {, }. The range is {, > 0}. Identif the zeros. Because ( ) is in the denominator, the zero is. There is one vertical asmptote, at. f () ( ) 0 The degree of the numerator is 0 and the degree of the denominator is. Because 0 <, the -ais, 0, is the horizontal asmptote. f () 7 5 7 5 5 7 5 7 Problem Set Determine the domain of each rational function.. + p() ( )( + ). _ f () 9. k () _ + + _. T () + + Identif the equations of the asmptotes of each rational function. 5. h() 5 7. f (). m() + +. g () 7 + ( ) For each problem, do the following: A. Graph the function. Label asmptotes clearl. B. Write the domain and range. 9. f () 0. w () 7 +. q() +. b() 5. 5 f () + +. h() 9 7 GRAPHING RATIONAL FUNCTIONS 7 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
Find the equation in the famil f () + k for the graph of each function. h 5. f () 7. b () 0 0 (0, ) 0 0 0 0 0 (, ) 0. h (). g () 0 0 0 (, 0) 0 (, 5) 0 0 Find the equation in the famil f () 9. 0 (, ) 0 0 0 a + k for the graph of each function. h f () 0. 0 0 0 0 C () (7, ) UNIT RATIONAL EQUATIONS Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.
. N (). q () 0 0 (, 5) 0 0 0 0 (, ) 0 0 Graph each function.. g () +. f () ( + ) 5. q() 7 7. c () ( )( + ) + 7. h() + 9 +. f () + Solve. * 9. Challenge Jimi is taking a road trip during which he will drive 500 miles. He has allowed hours for rest. As Jimi drives, his speed will var and therefore his travel time will also var. * 0. Challenge Chance the gardener is planning a flower garden in the shape of a trapezoid. He has 00 square feet to work with, and one base of the trapezoid must be 0 feet. The trapezoid area A. Write a rational function to describe Jimi s travel, where time is a function of rate and distance. B. Graph the function. C. Identif an appropriate domain and range for this situation. Eplain our reasoning. formula is A h(b + b ). A. Write a rational function for height in terms of area and base. B. Graph the function. C. Identif an appropriate domain and range for this situation. Eplain our reasoning. GRAPHING RATIONAL FUNCTIONS 9 Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.