Crime Scene Decoders: Rational Functions

Transcription

1 Crime Scene Decoders: Rational Functions Standard Focus: Patterns, Algebra & Functions Time Range : 1-3 Days Topics of Focus : - Multiplying and Dividing Rational Expressions - Solving Equations with Rational Functions - Simplifying Complex Fractions - Asymptotes - Graphing Rational Functions Quantities Creating Equations Creating Equations Reasoning with Equations and Inequalities Interpreting Functions Interpreting Functions Procedures: N-Q A-CED A-CED A-REI F-IF F-IF 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on axes with labels and scales. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 7. Graph functions expressed symbolically and show key features of the graph, by hand and using technology for more complicated cases. 7d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. A.) Student will be given the letter from Chief Harris, the possible suspects, and crime scene puzzles. B.) Students will work in pairs or individually to solve the crime. C.) Students will determine which suspect should be arrested. D.) Students will create a portfolio of evidence proving that they have arrested the right person and will demonstrate their understanding of their mathematical content present in the problem. *Extensions.) Students can present their evidence to the class as an oral presentation.

2 CSD: The Evidence Name: 1. Clue 2. Clue 3. Clue

3 4. Clue 5. Clue 6. Clue Cryptic Text Message Suspect

4 Crime Scene Decoders Algebra 2 & Pre-Calc: Rational Functions Detectives, The boys and girls have once again come out to play. This time, the international evil genius terrorist group the Mathemagicians are running amok over Central America. Although we have apprehended a number of their associates, we have yet to recover any pieces of their ill-conceived world conquering device that they soon may deploy to conquer the world. From the intelligence we have gathered, the anonymous henchman Asymptote Swaggster is the culprit behind these robberies. As with other crimes related to the Mathemagicians, they have left behind a trail of mathematical puzzles and a cryptic text message that needs to be deciphered. When substituted correctly, the text message will calculate to the Swaggster s favorite number. So far there are six suspects that police have questioned. It is hoped that someone with a relatively strong number sense can crack some codes that have puzzled the detectives on the case so far. Your job is to bring Asymptote Swaggster to justice and save the planet. You need to be prepared to state your case and demonstrate your understanding of the following skills that Asymptote Swaggster is known to use in the notes. - Multiplying and Dividing Rational Expressions - Solving Equations with Rational Functions - Simplifying Complex Fractions - Asymptotes - Graphing Rational Functions Be sure to include: - Other examples of the concepts - Definitions - Any other relevant information. This is not a time to be sloppy. The slightest miscalculation or illegible footnote could result in a not guilty verdict. Oh, did I mention that use of a calculator might prematurely set off his world conquering device? Good luck to you, gumshoe. Chief Harris

5 Who is Asymptote Swaggster? Name: Hazel Occupation: Pharmacist Favorite Number: 8 Name: Enrique Occupation: Pro Ping Pong Player Favorite Number: -59 Name: Quinn Occupation: Cake Decorator Favorite Number: 17 ¼ Name: Waldo Occupation: Private Investigator Favorite Number: 911 Name: Stillman Occupation: ATV Salesman Favorite Number: -77 Name: Ting Fen Occupation: Speech Therapist Favorite Number: -90

6 Scene #1 Television Manufacturer - Juárez, Mexico Asymptote Swaggster broke into an international TV manufacturer and the inventory of unused cathode ray tubes was stolen. Investigators found this note in the parking lot. Hello all! It s Swaggster here and I m gonna perplex you with puzzles. While I m stealing stuff I m going to look for some rational mathematics. Just like this one I pulled off the CEO s desk. Giga Electronica Proposal: Add computer manufacturing to business. Build 650 computers as trial run. Estimated Start-Up (Fixed) Costs: 250,000 pesos Itemized Cost: each computer from scratch is 2,000 pesos. (ffffffffff cccccccc)+cccc Necessary Function: cc =, where cc is the cost of each item, xx x is the number of items produced and cc is the average total cost of each item. Implementation Restrictions: We will only do this if the average total cost for each computer is under 2,400 pesos. Should we do this? Yes No m = 1000 r = Scene #2 Instituto Tecnologica de Costa Rica - San José, Costa Rica After a campus tour, Swaggster snuck away from the tour group and into the Department of Engineering. From there, they discovered the laser cutter was taken. While I robbed the local college, I peeked in a math class. I can t help myself. Isabella and Santiago graph functions on a calculator. Isabella graphs and Santiago graphs They both have vertical asymptotes at x xx xx = 0 and horizontal asymptotes at y = 6. They are confused. What is the correct explanation? They have the same asymptotes, but are different functions They have the same asymptotes because they re the same function They are incorrect a = 0 c = 6 i = -10

7 Scene #3 Mayan Ruins - Zaculeo, Guatemala Conservationists were stunned to find out that the large stucco mask of Temple 33 was taken. It is unclear to investigators how the mask may be used in the World Conquering Device. The Mayan could build cities with rocks, mud and their hands. Can you solve five rational function questions in a row? I have my doubts. The excluded value of g(x) equals b gg(xx) = xx2 +4xx 5 (xx 4)(xx+aa) dd(xx) = bb + 1 xx 6aa + 2 xx 3 The excluded value of f(x) equals a ff(xx) = xx2 6xx+ 17 xx+1 yy(xx) = dd(xx) 3xx3 + 1 xx 4 START HERE What are the correct horizontal and vertical asymptotes of z(x) Horizontal at y = 0 Vertical at x = -8, x = 0, x = 2 e =3 Horizontal at y = 0 Vertical at x = -2, x = 0, x = 4 i =3 Horizontal at y = 1 Vertical at x = -4, x = 0, x = 8 r =3 zz(xx) = yy(xx) xx2 + 6xx 16 xx(xx + 4)

8 Scene #4 Hospital CIMA - San José, Costa Rica One of the nicest hospitals in Central America, Hospital CIMA was broken into last night and their radiology equipment was stolen. Later, nurses found this note from Asymptote Swaggster. The Doctor is in! The Dr. of Swagg, that is. While robbing the hospital, I decided to impersonate an Anesthesiologist for a few hours. I had to administer some anesthesia, did I mess up? Anesthesiology Report Local anesthesia was administered to the patient at 1:58 prior to surgery. The concentration of the drug in the bloodstream (in mg/l) is modeled by the 8tt function where t is minutes since receiving the drug: CC(tt) = 0.3tt Note: When the concentration drops below 0.1 mg/l the patient may become conscious. They will need the additional anesthesia administered one half hour prior to reaching this level. The surgery expects to last 6 hours. Based on this information, when would I need to administer more anesthesia? 5:48 5:54 6:02 a = 548 i = 554 m = 602 Scene #5 Sinkhole - Guatemala City, Guatemala One of the largest sinkholes on record, the Guatemala City Sinkhole was looted last night. It is believed that Asymptote Swaggster stole the underground piping. I ve got some bad news. The earth is about to fall out from under you. Luckily, my fortune teller has given me an equation that models its destruction. Graph it! SSSSSSSShoooooo(xx) = 3 xx 4 12xx 3 +22xx 2 +84xx How many Sinkholes are going to form? One A=1 Two R=2 Three M=3

9 Scene #6 Caribbean Sea - Belize City, Belize A figure parachuted from a helicopter and landed on a yacht in the middle of the Caribbean Sea. The passengers were sent to shore in lifeboats, but the yacht was taken. Victims claimed the figure kept yelling My name is Asymptote Swaggster! Go tell police! I m Swagg! I m floating off into the sunset on my brand-new stolen yacht. There are a lot of vessels out on the Caribbean. Jet Skis are flying around circles around me and I m just waiting for one of them to crash. The routes of three Jet Skis are modeled by a rational function. Graph the function and determine how many collisions (if any) there will be. ff(xx) = 2xx2 + 3 xx 2 9 How many yachts will get crashed into? 0 a = 0 3 c = 3 5 r = 5 CRYPTIC PUZZLE SOLVER TEXT MESSAGE Good luck. Lol. Stay Swaggin. C A + (M (E + R) i) C A Asymptote Swaggster

10 Who is Asymptote Swaggster? Name: Hazel Occupation: Pharmacist Favorite Number: 8 Name: Enrique Occupation: Pro Ping Pong Player Favorite Number: -59 Name: Quinn Occupation: Cake Decorator Favorite Number: 17 ¼ Name: Waldo Occupation: Private Investigator Favorite Number: 911 Name: Stillman Occupation: ATV Salesman Favorite Number: -77 Name: Ting Fen Occupation: Speech Therapist Favorite Number: -90

11 Scene #1 Television Manufacturer - Juárez, Mexico Asymptote Swaggster broke into an international TV manufacturer and the inventory of unused cathode ray tubes was stolen. Investigators found this note in the parking lot. Hello all! It s Swaggster here and I m gonna perplex you with puzzles. While I m stealing stuff I m going to look for some rational mathematics. Just like this one I pulled off the CEO s desk. Giga Electronica Proposal: Add computer manufacturing to business. Build 650 computers as trial run. Estimated Start-Up (Fixed) Costs: 250,000 pesos Itemized Cost: each computer from scratch is 2,000 pesos. (ffffffffff cccccccc)+cccc Necessary Function: cc =, where cc is the cost of each item, xx x is the number of items produced and cc is the average total cost of each item. Implementation Restrictions: We will only do this if the average total cost for each computer is under 2,400 pesos. Should we do this? Yes 625 No m = 1000 r = Scene #2 Instituto Tecnologica de Costa Rica - San José, Costa Rica After a campus tour, Swaggster snuck away from the tour group and into the Department of Engineering. From there they discovered the laser cutter was taken. While I robbed the local college, I peeked in a math class. I can t help myself. Isabella and Santiago graph functions on a calculator. Isabella graphs and Santiago graphs They both have vertical asymptotes at x xx xx = 0 and horizontal asymptotes at y = 6. They are confused. What is the correct explanation? They have the same asymptotes, but are different functions They have the same asymptotes because they re the same function They are incorrect a = 0 c = 6 i = -10

12 Scene #3 Mayan Ruins - Zaculeo, Guatemala Conservationists were stunned to find out that the large stucco mask of Temple 33 was taken. It is unclear to investigators how the mask may be used in the World Conquering Device. The Mayan could build cities with rocks, mud and their hands. Can you solve five rational function questions in a row? I have my doubts. The excluded value of g(x) equals b gg(xx) = xx2 + 4xx 5 (xx 4)(xx + aa) gg(xx) = (xx+5)(xx 1) B = 4 (xx 4)(xx 1) dd(xx) = bb + 1 xx 6aa + 2 xx 3 dd(xx) = 4xx3 + xx 2 6xx The excluded value of f(x) equals a a = -1 ff(xx) = xx2 6xx+ 17 xx+1 yy(xx) = dd(xx) 3xx3 + 1 xx 4 yy = 4xx3 + xx 2 6xx xx3 + 1 xx 4 yy = 4xx + 1 2xx 2 START HERE What are the correct horizontal and vertical asymptotes of z(x) Horizontal at y = 0 Vertical at x = -8, x = 0, x = 2 e =3 Horizontal at y = 0 Vertical at x = -2, x = 0, x = 4 i =3 Horizontal at y = 1 Vertical at x = -4, x = 0, x = 8 r =3 zz(xx) = yy(xx) xx2 + 6xx 16 xx(xx + 4) zz = 4xx + 1 (xx + 8)(xx 2) 2xx 2 xx(xx + 4) (4xx + 1)(xx)(xx + 4) zz = (2xx 2 )(xx + 8)(xx 2) = (4xx + 1)(xx + 4) (2xx)(xx + 8)(xx 2)

13 Scene #4 Hospital CIMA - San José, Costa Rica One of the nicest hospitals in Central America, Hospital CIMA was broken into last night and their radiology equipment was stolen. Later, nurses found this note from Asymptote Swaggster. The Doctor is in! The Dr. of Swagg, that is. While robbing the hospital, I decided to impersonate an Anesthesiologist for a few hours. I had to administer some anesthesia, did I mess up? Anesthesiology Report Local anesthesia was administered to the patient at 1:58 prior to surgery. The concentration of the drug in the bloodstream (in mg/l) is modeled by the 8tt function where t is minutes since receiving the drug: CC(tt) = 0.3tt Note: When the concentration drops below 0.1 mg/l the patient may become conscious. They will need the additional anesthesia administered one half hour prior to reaching this level. The surgery expects to last 6 hours. Based on this information, when would I need to administer more anesthesia? minutes ->4 hr 26 minutes 5:48 5:54 6:02 a = 548 i = 554 m = 602 Scene #5 Sinkhole - Guatemala City, Guatemala One of the largest sinkholes on record, the Guatemala City Sinkhole was looted last night. It is believed that Asymptote Swaggster stole the underground piping. I ve got some bad news. The earth is about to fall out from under you. Luckily, my fortune teller has given me an equation that models its destruction. Graph it! SSSSSSSShoooooo(xx) = 3 xx 4 12xx 3 +22xx 2 +84xx How many Sinkholes are going to form? One A=1 Two R=2 Three M=3

14 Scene #6 Caribbean Sea - Belize City, Belize A figure parachuted from a helicopter and landed on a yacht in the middle of the Caribbean Sea. The passengers were sent to shore in lifeboats, but the yacht was taken. Victims claimed the figure kept yelling My name is Asymptote Swaggster! Go tell police! I m Swagg! I m floating off into the sunset on my brand-new stolen yacht. There are a lot of vessels out on the Caribbean. Jet Skis are flying around circles around me and I m just waiting for one of them to crash. The routes of three Jet Skis are modeled by a rational function. Graph the function and determine how many collisions (if any) there will be. ff(xx) = 2xx2 + 3 xx 2 9 How many yachts will get crashed into? 0 a = 0 3 c = 3 5 r = 5 CRYPTIC PUZZLE SOLVER TEXT MESSAGE Good luck. Lol. Stay Swaggin. C A + (M (E + R) i) C A Asymptote Swaggster

15 From Puzzle 1 m = 1000 From Puzzle 2 c = 6 From Puzzle 3 e =3 From Puzzle 4 i = 554 From Puzzle 5 R=2 From Puzzle 6 a = 0 C A + (M (E + R) i) C A (1000 (3 + 2) 554) 6 0 = -59