Pre-Algebra 2. Unit 8. Rational Equations Name Period

Transcription

1 Pre-Algebra Unit 8 Rational Equations Name Period

2

3 Basic Skills (7B after test) Practice PAP Algebra Name Per NON-CALCULATOR Simplify: y 5 6 y a 3c c ( 1)

4 Solve:

5 Notes 8.1 Simplifying Rational Epressions Day 1 (Multiplying/Dividing) PAP Algebra II FACTOR EACH a 3 SIMPLIFY EACH y 5y y Multiplying/Dividing Fractions 1) 3 ) ) )

6 5) ) ) ) 6 15 (3 5 ) 4 4

7 PAP Algebra II WS 7.1 Name: For 1 6, simplify each rational epression. 1. 6c 9c 3c. z 49 z y 8y4 y 8y a 3 8. a a1 a a 9 7a y 1 6y ( f g) 3 5 f g f g 10 13) )

8 ) ) ) ) 0 n ( ) 1 19) (a.) Simplify, where is an integer and n is a positive integer. (b.) Use the results in 8a to show n 1 that the value of the given epression is always an odd integer. 19) Which of the epressions below always has a positive value? Assume 0, 1, or a. 1 ( - 1) b. 1 1 ( - 1) 1 1 c. ( 1 1) d. 1 1 ( 1) 1 1 e ) Simplify: a b

9 PAP Algebra II Notes 7. Adding/Subtracting Rational Functions To add or subtract rational functions you must get a common denominator ) Find the least common multiple (LCM) and - 8 Simplify: y y y 5 5 y 7

10 7. y y y 1 y

11 WS 7. Name: PAP Algebra II Add or subtract the following rational functions and simplify completely ) 3 4 ) ) ) ) ) - y y 9

12 4 15 7) 8) y 9) 3y 5y - y 10) ) )

13 13) ) 3a 6a - 3a 6 15) 3 8t + 1 t 16) ) ( 3) 18)

14 19) ) ) ) ) )

15 Notes 8.3 Transformational Graphing of Rationals A rational parent function or reciprocal function has the equation: f() = y Horizontal Asymptote Vertical Asymptote Domain and Range for the parent function: What can not be? What can y not be? (This is a discontinuity.) Domain Range 1 1 The general form for translating a reciprocal function (f() = ) is: f ( ) a k h Eample: 1 f() = 4 Translations: HA VA Domain Range 13

16 Graph using transformations, eplain what happens, give the domain and range, and give the asymptote equations: 3. 1 y ( 3) 4. 1 y 5. y ( 5) Domain: Domain: Domain: Range: Range: Range: VA: HA: VA: HA: VA: HA: 1 6. y 3 For each of the following, list the equation, asymptotes, domain and range. Domain: Range: VA: HA: 14

17 Graphing Rationals WS 8.3 Graph the following functions. 1 1) y 4 1 VA: HA: domain: range: ) Name: Per 3 y 3 VA: HA: domain: range: 3) y 3 1 4) y ( 3) VA: VA: HA: HA: domain: range: domain: range: 5. Determine if the table represents a direct or inverse variation. Find the constant of variation, and specific equation for each y y Architects have to consider how sound travels when designing large buildings such as theaters, auditoriums, or museums. Sound intensity, I, is inversely proportional to the square of the distance from the sound source, d. a) Write an equation that represents this situation. b) If a person in a theater moves to a seat twice as far from the speakers, compare the new sound intensity to that of the original. 15

18 Simplify each epression m3 8m4 7. m 6m 9 9 m ( y)( ) y 1 1 ( y)( ) y

19 PAP Algebra II Notes 8.4A Graphing Rational Functions A rational function is a function that can be written as a fraction of two polynomials: p( ) f ( ) q( ) The graph has zeros (i.e roots or -intercepts) where p ( ) 0 The graph (usually) has vertical asymptotes where q ( ) 0 The horizontal asymptote depends on the highest power of in the numerator and denominator. If the highest power of is in the denominator, then the horizontal asymptote is y 0 If the highest power if is the same in the numerator & denominator, then the horizontal asymptote will be the coefficients in front of those terms. For eample, if the rational function is y then the horizontal asymptote will be y 3/ If the highest power of is in the numerator, then there is no horizontal asymptote EX1: Graph f ( ) 5 VA: HA: ZEROS: Domain: 17

20 EX: Graph 6 1 f ( ) 3 9 VA: HA: ZEROS: Domain: EX3: Graph f ( ) VA: HA: ZEROS: Domain: 18

21 EX4: Graph f 4 ) ( VA: HA: ZEROS: Domain: 19

22 PAP Algebra II WS 8.4A NONCALCULATOR! Name: Find the vertical asymptotes, horizontal asymptotes, -intercepts (i.e zeros), domain and range and graph the rational function. 1) 3 f ( ) VA = 4 HA = Zeros = Domain : ) 3 6 f ( ) VA = 1 HA = Zeros = Domain: 3) 8 16 f ( ) VA = 10 HA = Zeros = Domain : 0

23 4) 6 f ( ) VA = 16 HA = Zeros = Domain : 5) f ( ) VA = 40 HA = Zeros = Domain : 6) 3 f ( ) VA = HA = Zeros = Domain : 1

24 7) f ( ) VA = 7 10 HA = Zeros = Domain: 8) f ( ) VA = 1 HA = Zeros = Domain:

25 PAP Algebra II Notes 8.4B A rational function is going to have a hole (also known as a removable discontinuity) wherever a factor that cancels from the numerator & denominator equals zero. E: ( 1)( ) f ( ) ( 1)( 4) 4 Although a rational function can never cross a vertical asymptote, SOMETIMES it can cross the horizontal asymptote. EX1: f 3 ) ( V.A.: H.A.: Zeros: Holes: Domain: Range: 3

26 EX: 3 f ( ) V.A.: H.A.: Zeros: Holes: Domain: Range: Does the graph cross the horizontal asymptote? If so, where? EX3: f ( ) V.A.: H.A.: Zeros: Holes: Domain: Range: Does the graph cross the horizontal asymptote? If so, where? 4

27 PAP Algebra II WS 8.4B NONCALCULATOR! Name: Graph the following rational functions. Be sure to label asymptotes and holes in the graph, and find the specified limits for the function. 1) y 6 9 HA = VA = ZEROS = HOLES: Domain: ) f 6 8 ) 1 ( HA = VA = ZEROS = HOLES: Domain: 5

28 3) f ( ) HA = VA = ZEROS = HOLES: Domain: 4) 3 3 f ( ) HA = VA = ZEROS = HOLES: Domain: 6

29 Graph the following transformations to 1 1 y or y. 5) 1 y 3 6) y ( 4) 3 7) y ( ) 1 8) y 5 9) What are the transformations from f() to g()? a) g( ) f ( 3) c) g( ) f ( 3) b) 1 g( ) f ( 3) d) 1 g( ) f ( 3) 7

30 Simplify: y 3y p p m Condense: 4ln 5ln 0. Epand: log 6y Solve. Remember to check for etraneous solutions.: 1. log 6(314) log 6 5 log 6( ). log log ( 4) 5 8

31 Notes 8.5 Direct and Inverse Variation Direct Variation Equation Graph Eample Direct Variation Eamples: 1. Julio s wages vary directly with the number of hours that he works. If his wages for 5 hours are $9.75, how much he be paid for 30 hours?. If y varies directly with, and y = 8 when = 7, find when y = A standard shower head uses 18 gallons of water in 3 minutes. Complete the table below that shows that gallons used, y, is a function of time in the shower,. (minutes) y (gallons) a. What is the k? k = b. Write the equation for the function c. If 70 gallons were used, how many minutes were spent in the shower? 9

32 Inverse Variation Equation Graph Eample Inverse Variation Eamples: 4. The volume V of gas varies inversely to the pressure P. The volume of a gas is 00 cm 3 under pressure of 3 kg/cm. What will be its volume under pressure of 40 kg/cm? 5. The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h? 6. The number of people, P, in a theater varies inversely with the number of empty seats, s. Is this an eample of a direct variation situation, inverse variation situation, or neither type? Why? 7. Write the inverse variation equation that represents: m is inversely proportional to the square of n. 30

33 Mied Eamples: 8. a) Is this direct, inverse or neither? How do you know? y b) Find the constant of variation, k, and c) Write an equation to model this data y a) Is this direct, inverse or neither? How do you know? b) Find the constant of variation, k, and c) Write an equation to model this data a) Is this direct, inverse or neither? How do you know? y If each point is from a model of inverse variation, find the constant of variation. (Hint: think about the equation y = k to remember how to find the constant) a) ( 3, 7 ) b) (.5, 1.5 ) c) ( 15, 3 1 ) 1. Each pair of points is from a model for inverse variation. Find the missing value. (Hint: Write the equation y = k and use it to solve for the unknown) a) ( 3, 4 1 ) and ( 1, y ) b) (10, 1 ) and (, 5 ) Classify the following graphs as a) Direct b) Inverse c) Neither 13) 14) 15) 16) 31

34 8.5 Inverse Variation Worksheet Name: Period: 1. The amount of food a dog eats varies directly with the number of days the dog is fed. If a dog is fed for 10 days, the dog eats 5 cups of food. How much food would the dog eat in 50 days?. The rent for an apartment varies inversely with the number of people sharing the cost. Four people sharing an apartment pay $10 each per month. How many people would be needed so that each would pay $80 per month? For 3-8 state whether the problem is inverse, direct or neither. If the problem is inverse or direct variation, state the constant of variation: y 3. y = 7 4. y + = y = y 8. y If and y vary inversely and = 1 when y = 11, find when y = If and y vary inversely and =.5 when y = 100 find when y = 5 3

35 11. Heart rates and life spans of most mammals are inversely related. A cat lives for about 15. years on average and has a heart rate of 16 beats per minute. a. What is the constant of variation? b. A hamster has a heart rate of about 634 beats per minute. About how long will a hamster live? c. An elephant lives for about 70 years. About how many times per minute does an elephant s heart beat? 1. Two gears are used to operate a machine. Gear A has 60 teeth and Gear B has 45 teeth. The speed at which you turn Gear A is 5400 rpm. The number of teeth and speed in rpm are inversely related. a. What is the constant of variation? b. At what speed will Gear B turn? 13. The grade you earn in math varies inversely with the number of minutes per night you watch television. If you watch 90 minutes per night, you get a 60 in math. a. What is the constant of variation? b. How much television can you watch if you want to make a 70? c. You cut back on your television to only 75 minutes a night, what grade will you make in math? d. What is the maimum amount of television you can watch and still make a 100? 14. The amount of water that has leaked from a faucet varies directly with time. In hours, 10 gallons of water leak. a. Describe what happens to the amount of water as time increases. b. What is the constant of variation? c. How much water leaks in 100 hours? d. How long does it take for 100 gallons to leak? 33

36 Simplify: m 3 m m y y m Solve. Check for etraneous solutions: log 6 3 log

37 Notes 8.6 Solving Rational Equation Word Problems PAP Algebra Nancy drove from Houston to San Antonio, a distance of 50 miles. She increases her speed by 10 miles per hour for a 360 miles drive from San Antonio to Dallas. If the trip took a total of 11 hours, how fast did she drive on both legs of the trip? 4. You take your boat on a trip 36 miles down the river. On the way to your destination (ie downstream), the 3mph current speeds you up. On the way back (ie upstream) the same current slows you down. Your total travel time is 9 hours. How fast is the boat in still water? Timeupriver + Timedownriver = Timetotal 5. The speed of a stream is 4 km/hr. A boat travels 6 km upstream in the same time it takes to travel 1 km downstream. How fast is the boat in still water? 35

38 Worksheet 8.6 Name Per 1) Natalie traveled 700 miles in her private jet in the same time Allie and Stephanie flew on a jumbo jet that traveled 3600 miles. If the jumbo jet flew 150 miles per hour faster than the private jet, what were the speeds of both jets? ) A speed skater travels 9 kilometers in the same amount of time it takes another speed skater to travel 8 kilometers. The first skater travels 4.38 kilometers per hour faster than the second skater. How fast is each skater? 3) The speed of a river s current is 3 miles per hour. You travel two miles with the current and then return to where you started in a total of 1.5 hours. What is your speed in still water? 4) A car travels 10 miles in the same amount of time that it takes a truck to travel 100 miles. The car travels 10 miles per hour faster than the truck. Find the speed of the truck. 36

39 5) David canoes upstream a distance of 8 miles and then returns. The round-trip took current of the stream was flowing at miles per hour. What was David s canoeing speed? hours. The 6) A boat can travel 8 miles an hour in still water. If it can travel 15 miles down a stream in the same time that it can travel 9 miles up the stream, what is the rate of the stream? 7) Two trains starting at the same time from stations 396 miles apart, meet in 4½ hours. How far has the faster train traveled when it meets the slower one, if the difference in their rates is 8 miles per hour? 8) Graph: f( ) HA = VA = ZEROS = HOLES: Domain: 37

40 Write the Equations of the following Graphs. 9) )

41 Notes 8.7 More Rational Equation Work Problems 1. Kyle can paint a room in 4 hours. If Cole helps, together they can paint the room in 1.5 hours. How long would it take Cole to paint the room by himself?. Ashley and Lindsay want to start a business painting fences. The figured out that they would paint a fence in 40 minutes together. Ashley can paint the fence in 70 minutes alone. How long would it take for Lindsay to paint the fence alone? 3. Mrs. Zurek is putting toys in a toy bo at the same time her son, Noah is taking them out. Mrs. Zurek can fill the toy bo in 3 minutes and her son can empty the bo in 5 min. How long will it take Mrs. Zurek to pick up the toys if her son is emptying the bo while she works? 39

42 4. For the student council to attend the national convention HHS will charter an airplane. The cost to charter a plane is $5000. Each student going on the trip is going to pay an equal share of the plane and $300 for food and lodging. a. Write an equation that shows the cost (c) per the number of students () attending the convention. b. If no more than 100 people may fly on the charter plan, what is the minimum cost per person? c. If we needed to make sure we spend no more than $400 per student, what is the minimum number of students we need to take? 5. Dr. Waldrip is so tall he decided to try professional basketball as a potential career. He made 30 out of 50 free throws to start. To earn a job with the Mavericks, he must have an 80% free throw average. How many consecutive free throws must he make to get a job with the Mavs? 40

43 Worksheet 8.7 Name Per 1. Billy can mow a lawn by himself in three hours. It takes Bobby five hours when he mows the same lawn. If the two boys worked together, how long would it take them to mow the lawn?. A basketball player has made 1 of her last 30 free throws an average of 70%. What is the number of consecutive free throws the player needs to raise her success rate to 75%? 3) You can paint a room in 8 hours. But you and a friend can paint the same room in just 5 hours. How long would it take for your friend to paint the room alone? R R R 1 together 4. HHS mathletes want to attend a competition. The school board has agreed to allow the mathletes to attend as long as the cost per mathlete is less than $0. Write an epression for the cost per student if the bus costs $40, and the entry fee per student is $1. Then, write your epression as a single fraction. 5. What is a reasonable domain and a reasonable range for #6? 41

44 6. The Pee-Wee soccer team has won 4 games and lost 6. The players have decided to throw a party if they can get the winning record to 60%. How many consecutive games must they win in order to reach this winning record? 7. You scored a hole-in-one 7 out of the last 18 times while playing put-put golf. If you continued playing put-put golf, what function would best model the percentage of hole-in-ones if the net puts dropped in on the first try? 8. Mrs. Elvington and Mrs. Poltl have agreed to contact every student at Heritage High School by phone to inform them of the spring dance. Mrs. Elvington can complete the calls in si days if she works alone. Mrs. Poltl can complete them in 4 days. How long will they take to complete the calls working together? 9. Bella can clear a lot in 5.5 hours. Rooshan can do the same job in 7.5 hours. How long would it take them to do it together? 10. For the student council to attend the national convention HHS will charter an airplane. The cost to charter a plane is $5000. Each student going on the trip is going to pay an equal share of the plane and $300 for food and lodging. a. Write an equation that shows the cost (c) per the number of students () attending the convention. b. If no more than 100 people may fly on the charter plan, what is the minimum cost per person? 4

45 11. The speed of a river s current is 3 miles per hour. You travel two miles with the current and then return to where you started in a total of 1.5 hours. What is your speed in still water? 1. The members of our math department are ordering Pi Day T-shirts. There is a one-time charge of $99 for artwork on the T-shirts and a $1 charge for each T-shirt ordered. If represents the number of T-shirts ordered, write an equation that represents the total cost, C, in dollars per T-shirt? a. What is the most epensive price shirts could be? (imagine only 1 person orders a shirt) b. What is the cheapest shirts can be? (imagine lots and lots and lots of people order shirts. They are math shirts after all.) c. What is the range for the cost of the shirts? 13. You are organizing your high school s sports banquet. The banquet rental hall is $350. In addition to this one time charge, the meal will cost $8.50 per plate. Mr. Spain has decided that you can only go if the cost per student C(), is below $15. Write an inequalities that could be used to find the number of students, needed to offer the banquet? a. What is the most epensive price the meal could cost? b. What is the cheapest meals can be? c. What is the range for the cost of the meals? 43

46 16. You have subscribed to a cable television service. The cable company charges you a one-time installation fee of $30 and a monthly fee of $50. The function gives the average cost per month as a function of the number of 30 50m months you have subscribed to the service is: C. What is a reasonable range for the average cost m of cable service per month? a. What is the most epensive price per month for cable? b. What is the cheapest price per month for cable? c. What is the range for the average monthly cost for cable? 17. Bob can paint a fence in 6 hours and Sam can paint a fence in 10 hours. How long would it take both men to paint 3 fences working together? 18. Two printing machines can complete a job in 1 hours. One machine works twice as fast as the other. How long would it take the slower machine to complete the job, working by itself? 44