# Unit 4 Rational And Reciprocal Functions And Equations

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1Unit 4 Rational and ReciprocalFunctions and EquationsGeneral Outcome: Develop algebraic reasoning and number sense.Develop algebraic and graphical reasoning through the study of relations.Specific Outcomes:4.1 Determine equivalent forms of rational expressions (limited to numerators anddenominators that are monomials, binomials, or trinomials).4.2 Perform operations on rational expressions (limited to numerators and denominators that aremonomials binomials, or trinomials).4.3 Solve problems that involve rational equations (limited to numerators and denominators thatare monomials, binomials, or trinomials).4.4 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadraticfunctions).Topics Simplifying Rational Expressions(Outcome 4.1)Page 2 Multiplying & Dividing RationalExpressions(Outcome 4.2)Page 11 Adding & Subtracting RationalExpressions(Outcome 4.2)Page 19 Solving Rational Equations(Outcome 4.3)Page 31 Word Problems Involving RationalEquations(Outcome 4.3)Page 38 Reciprocal Functions(Outcome 4.4)Page 45

2Unit 4 Rational and ReciprocalFunctions and EquationsRational Expressions:Rational expressions are fractions that have a polynomialnumerator and denominator.Ex)2x 7x2 93x 22 y 4x5 y 3xAny time a variable is found in a denominator the possibility ofrestrictions exist.Ex) For each rational expression determine all non-permissiblevalues.a)5t4 st 2b)3xx (2 x 3)c)2 p 1p 2 p 12

3Equivalent Rational Expressions:Remember when the numerator and denominator of a fraction ismultiplied or divided the same value the resulting fraction isequivalent.Ex)7xx 2

4Simplifying Rational Expressions:To simplify a rational expression you must first factor thenumerator and denominator. Only like factors of the numeratorand denominator will cancel each other out.Ex) Simplify the following.x3 ya) 2 4x yb)3x 62 x 2 x 10x 2 4 x 32c)x 2 9 x 20d)1 tt2 1

516 x 2 9 y 2Ex) Consider the expression8x 6 ya) What expression represents the non-permissible valuesof x?a) Simplify the rational expression.c) Evaluate the expression for x 2.6 and y 1.2

6Simplifying Rational Expressions Assignment:1) In each case below determine what the first rational expression must bemultiplied by to obtain the second rational expression.a)3p,q 4c),m 33 p 2qpq 2 4 ( m 3 )m2 9b)2,x 42x 8x 2 16d)1,y 1y2 yy3 y2) Determine the non-permissible values for each of the following.a)3a4 ab)2b 8bc)3( y 7 )( y 4 )( y 2 )d) 7 ( x 1)( x 1)( x 3)e)2k 8k2f)6x 8( 3x 4 )( 2 x 5 )

73) Simplify the following rational expressions. State any restrictions.a)2a ( a 5 )3a ( a 5 )b)3x ( 2 x 3)2 x ( 3x 2 )c)( x 7 )( x 7 )( 2 x 1)( x 7 )d)5 ( a 3)( a 2 )10 ( 3 a )( a 2 )f)3x 610 5 x6r 2 p 3e)4rp 4

8a 2 2a 24g)2a 2 72x 4i)4 x10 x 2 55 x 75h)20 x 2 10 x 150j)5( x2 y2 )x 2 2 xy y 24) Identify the error in the simplification shown below and provide the correctsimplification.x 2 4 ( x 2 )( x 2 ) 2x 42( x 2) x 22 x 1

95) Create a rational expression that has non-permissible values of 1 and 2 .6) The area of right PQR is ( x 2 x 6 ) square units, and the length of side PQis ( x 3) units. Side PR is the hypotenuse.a) Draw a diagram of this triangle.b) Write and expression for the length of side QR. Express your answer insimplest form.c) Identify the restrictions.

107) Express the following in simplest form. Identify any restrictions.a)c)( x 2)(x22 ( x 2 ) 20x2 9 x ) 8 ( x 2 x ) 12(x2b)22 4) ( x 2)224 ( x 2 9 ) ( x 3)d)2x2 6 x 9(x2 4 x 4 ) 10 ( x 2 4 x 4 ) 92( 2 x 1) ( x 2)22

11Multiplying and Dividing Rational Expressions:Multiplying: Factor all numerators and denominators Simplify by eliminating common factors (1 from anumerator and 1 from a denominator) State all restrictionsEx) Simplify the following.x 2 3x 10 x 7a) 2x x 56 x 53 a2 b)a b4 x2 9 y 2c) 3 y3 8x4

12a 2 5a 6 a 2 a 30d) 2 2a 6a 5 a 9a 18x 2 x 12 x 2 4 x 3e) 2x2 9x 4xDividing: Factor all numerators and denominators Convert into a multiplication questionEx)2 14 2 5 7 5 7 14 Simplify by eliminating common factors State all restrictions

13Ex) Simplify the following.x 2 x 20 x 2 9 x 20a) 2x2 6 xx 12 x 36a 2 3a 28 a 2 4a 21b) 2 a 14a 45 a 2 2a 15

14(xc)2 4)( 2x 4)(x( x 3)2 2 x 3)2m2 7m 15 4m2 9d) ( 3 2m )22m 10m6

15Multiplying & Dividing Rational Expressions Assignment:1) Simplify the following expressions. Identify any restrictions.12 x 2 y 15 z a)5 zy4xc)( y 7 )( y 3) 4 ( 2 y 3)( 2 y 3)( 2 y 3) ( y 3)( y 1)d 2 10036 e)144d 10b)3( a b )( a 5 )( a 5 ) ( a 1)( a 5) 15 ( a b )d)2x 6 x 3 x 32a 3 a2 1 f)a 1 a2 9

164 x 2 25x 4 g)2 x 2 13 x 20 4 x 10h)2a 2 5a 3 a 2 12a 3 22a 36a 3 a 2a 32) Simplify the following rational expressions. Identify any restrictions.4x2 2x a)3y y2t 2 7t 3t 2 b) 2t 49 t 7

17c)510 2 ( n 1)n 1 n 1d)y2y y2 9 y 32 w2 w 6 2 w 3 e)3w 6w 2a 5 a 2 2a 15 f)aa29 x 2 1 3x 2 5 x 2 g)x 52 x8 y2 2 y 3 2 y2 3y 2 3 4 y h)y2 12y 2y 1

183) Identify the error in the simplification shown below and correct thesimplification.x2 3 x 62xx 6 2x8 x 2 ( x 6 )( x 6 ) ( 2 x )( 4 x ) 2xx 6 ( x 6 ) ( x 6 ) ( 2x ) ( 4 x ) 14x ( x 6)4) Write an expression to represent the length of the rectangle shown below.Simplify your answer.A x2 9x2 2x 3x 15) Write an expression to represent the area of the triangle shown below. Simplifyyour answer.x 2x 8x2 7 x 8x2 4

19Adding and Subtracting Rational Expressions:When adding or subtracting fractions we must have a commondenominator.Ex)1 35 3 2 10 10 108 104 5* 10 is the lowest commonmultiple of 2 & 10.When finding the lowest common multiple (or lowest commondenominator) we must consider the coefficients and thevariables involved.Ex) Find the lowest common multiple of the following.a) 2, 6b) 5, 3c) a 2 , a5d) 8t 2 , 12t 7

20Ex) Simplify the following.24a) 3y 3y4t 2 6t 2 1b) 33 3c)2 a 5a 34d)e)5 2a 2a 3f)g)6x 12x 4 1 57y 5 y 2 963x5 8x4 6x2

21Remember when adding or subtracting rational expressions wemust have a common denominator.To find the lowest common denominator we must first factoreach denominator.x3 2 x 4 3x 6Ex) Determine the lowest common denominator for each of thefollowing.a)12,xy 3x 2 yb)5x34x,,2 x 2 6 x 5 x 2 20 x 2 x 12

22Ex) Simplify the following.a)x 12 2 x 4x 4 x 1c)45 y 2 5 y 6 y 2 y 12b)d)x5 6 x 6 4 x 12x 5x 1 x 2 3 x 10 x 2 9 x 20

23e)y 20 y 2 y2 4 y 2g)52x x 2 16 x 2 2 x 247x3 x 2 10 x 24 x 421xf)1x x1

24Adding & Subtracting Rational Expressions Assignment:1) Simplify the following expressions. Express all answers in simplest form andidentify any non-permissible values.a)7 3 x xb)5t 3 3t 5 1010c)m2m m 1 m 1d)a2aa 12 a 4 a 4 a 4e)12 3a 5 af)3 1 2x 6

25g) 4 i)k)65x2x16 25 y 10 y 15 y 385 x2 4 x 2h)4z 9x xy yzj)6 xy2 2 12a b ab yl)13 x 2 x 12 x 3

26m)o)3xx x 2 2 x2hh3 2 h 9 h 6h 9 h 323 x 154 x2 1 q) 2x 25 2 x 2 9 x 5n)51 y 4 2y 1 y y yp)23 3x x 6 x 2 x 2 3xr)2xx 8 x 3 x 2 6 x x 2 5 x 242

27s)n 36 2n 5n 6 n 7 n 122t)2ww 6 2w 5w 6 w 6 w 822) Identify the error in the simplification shown below and provide the correctsimplification.6( x 2) 4 7 ( x 2)647 2 x 2 x 4 x 2( x 2 )( x 2 ) 6 x 12 4 7 x 14( x 2 )( x 2 ) x 2( x 2 )( x 2 )

283) Simplify the following. State and restrictions on the variable.6xa)91 2x2 c)33 m 2m 331 2m2m 33 3 2tb)t1 t 6 td)11 x 4 x 4x1 2x 16 x 4

29xx 1and. If all measurements are in the42same units, determine a simplified expression for the length of the hypotenuse.4) A right triangle has legs of length5) Simplify the following. Identify all non-permissible values.a)x 2 x2 2x 3 x2 2 x x 5 x2 x 6 x2 4 x2 x 2 x x 2 x 12 x 1 b)x 2 3x 2 x 2 3x 1 x 2

30c)x 2 x2 2 x 3 x2 2 x x 5 x2 x 6 x2 4 xd)x 1 x2 4 2 x2 7 x 3 x 6 x2 2 x2 x2 x

31Solving Rational Equations: Factor all denominators Determine the lowest common denominator Multiply both sides of the equation by the lowest commondenominator (this will eliminate all fractions) Solve as normal Check for extraneous roots (your answer cannot be arestriction)Ex) Solve the following.a)x3x 3 2 24b)t 53 t 82c)x 1 1 22xxd)41 2x 1 x 2

32e)15 m 2 m 4f)4 6 2xg)5 11 2 2 y 12 3 yh)2101 x 2 4 6 x 12 x 2i)9418 2y 3 y 6 y 9 y 18

33j)341 x 2 x 2 x 2 4 x 2 3x 24k 1 k 1 k 2 4k 24k) k 2 k 2k2 4

34Solving Rational Equations Assignment:1) Solve the following. Check your answers for extraneous roots.a)f 3 f 2 223b)3 y 1 1 3y4 2yc)9418 2w 3 w 6 w 9 w 18d)6 t 4t 2

35e)6c 3 2 5c 3 c 9f)d2 d1 2 d 4 d 3d 4 d 1g)x2 x 2x2 5 x 2x 1x 1h) 3 y6y 9 6 y 1y 1

36i)263 1 b 5b 2j)c 6 3 2c 2c 42) Experts claim that the golden rectangle is most pleasing to the eye. It hasl l wdimensions that satisfy the equation , where w is the width and l iswlthe length. According to this relationship, how long should a rectangularpicture frame be if its width is 30 cm?

373) A rectangle has the dimensions shown in the diagram below.2x3 xx2a) Determine the difference between the length and the width of the rectangle.b) Determine an expression that represents the area of the rectangle.c) If the perimeter of the rectangle is 28 cm, determine the value of x.

38Word Problems Involving Rational Equations: Identify the variable being used. This can be done with a letstatement, a table, or a diagram Create an equation that describes the situation Solve for the variable Check to see that you have answered the question and thatyour answer makes sense Answer the question with a sentenceEx) Find two consecutive numbers where half of the smaller isequal to 4 more than one third the larger.Ex) The average life span of a woodland caribou is 5 yearslonger than half the average life span of a moose. The sumof their life spans is 35 years. What is the life span of amoose?

39Ex) Two friends share a paper route. Sheena can deliver thepapers in 40 min. Jeff can cover the same route in 50 min.How long, to the nearest minute, does the paper route takeif they work together?Ex) In a particular dog race from Pas to Flin Flon and back thetotal distance covered was 140 miles. Conditions wereexcellent on the way to Flin Flon. However, bad weathercaused the winner’s average speed to decrease by 6 mphon the return trip. The total time for the trip was 8 1 2hours. What was the winning dog team’s average speed onthe way to Flin Flon?

40Word Problems Involving Rational Equations Assignment:1) The sum of two numbers is 25. The sum of their reciprocals is1. Determine4the two numbers.2) Two consecutive numbers are represented by x and x 1 . If 6 is added to thefirst number and two is subtracted from the second number, the quotient of the9new numbers is . Determine the value of the original numbers.2

413) A French club collected the same amount from each student going on a trip toLe Cercle Moliere in Winnipeg. When six students could not go, each of theremaining students was charged an extra 3. If the total cost was 540, howmany students went on the trip.4) A tub can be filled in 2 min. if only the cold tap is turned on. It fills in 3 min.if only the hot tap is turned on. How long will it take to fill the tub if both tapsare turned on?

425) Two hoses together fill a pool in 2 hours. If only hose A is used, the pool fillsin 3 hours. How long would it take to fill the pool if only hose B were used?6) Two kayakers paddle 18 km downstream with the current in the same time ittakes them to go 8 km upstream against the current. The rate of the current is 3km/h. Determine the rate at which the kayakers are paddling.

437) Nikita lives in Kindersley, Saskatchewan. With her old combine, she canharvest her entire wheat crop in 72 hours. Her neighbor offers to help. His newcombine can do the same job in 48 hours. How long would it take to harvestthe wheat crop with both combines working together?8) Two friends can paddles a canoe at a rate of 6 km/h in still water. It takes them1 hour to paddle 2 km up a river and back again. Determine the speed of thecurrent.

449) Suppose you have 21 days to read a 518-page novel. After finishing half of thebook, you realize that you must read 12 pages more per day to finish the novelon time. What is your reading rate for the first half of the book? Use the tablebelow to help solve the problem.Reading Rate inPages per DayFirstHalfSecondHalf1 1 1ab 4 , solve for a.10) If b and1 1 5a a bNumber ofPages ReadNumberof Days

45Reciprocal Functions:A reciprocal of a value is the “flip” of the fraction:Ex)23reciprocal325 reciprocal15 1reciprocal xxA reciprocal of a function is found by dividing ‘1’ by thefunction.If y f ( x) , then y 1is its reciprocal.f ( x)

46Ex) Complete the table of values given below, then use this to1sketch the graphs of y x and y .xxy xy 10 5 2 1 12 150151212510Characteristics:Non-Permissible ValuesAsymptotesInvarient Points1x

47Properties / Rules:y f ( x)(original)If f ( x) 0If f ( x) is undefined and hasa vertical asymptote.If f ( x) 1If f ( x) 1If f ( x) is positiveIf f ( x) is negativeIf f ( x) is increasingIf f ( x) is decreasingIf f ( x) approaches 0If f ( x) approaches ( means infinity)1f ( x)(reciprocal)y 1isf ( x)and amay exist.1 .thenf ( x)1 .thenf ( x)1 .thenf ( x)1thenis .f ( x)1thenis .f ( x)1thenis .f ( x)1thenis .f ( x)1thenapproachesf ( x)then.Then1approachesf ( x).

48Ex) Given the graph of y f ( x) x 3 , sketch the1graph of y .f ( x)Steps:Ex) For each of the following, sketc

4.4 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). Topics Simplifying Rational Expressions (Outcome 4.1) Page 2 Multiplying & Dividing Rational (Outcome 4.2) Page 11 Expressions Adding &

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