ROLLER COASTER POLYNOMIALS - Mrs. R.'s Pages

!Ms. Peralta’s IM3ROLLER COASTER POLYNOMIALSNames:Purpose:In real life, polynomial functions are used to design roller coaster rides. In this project, you will applyskills acquired in Unit 4 to analyze roller coaster polynomial functions and to design your own rollercoaster ride.Project Components:Application Problems – You will have to answer questions and solve problems involving polynomialfunctions presented in real life scenarios. Through your work, you are expected to gain an in depthunderstanding of real life application of concepts such as sketching and analyzing graphs ofpolynomial functions, dividing polynomials, determining zeros of a polynomial function, determiningpolynomial function behavior, etc.Project Evaluation Criteria:Your project will be assessed based on the following general criteria: Application Problems – ONE GROUP ANSWER SHEET: will be graded on correctness andaccuracy of the answers. Provide all answers in a full sentence form. Make sure you clearly justify your answerswhere requiredPossible points: 0–3 points per question based on accuracy54 pointsProfessional appearance of the work10 pointsTOTAL POINTS 64 points

ROLLER COASTER POLYNOMIALSAPPLICATION PROBLEMS:Fred, Elena, Michael, and Diane enjoy roller Coasters. Whenever a new roller Coaster opens neartheir town, they try to be among the first to ride. One Saturday, the four friends decide to ride a newcoaster. While waiting in line, Fred notices that part of this coaster resembles the graph of apolynomial function that they have been studying in their IM3 class.1. The brochure for the coaster says that, for the first 10 seconds of the ride, the height of the coastercan be determined by h(t) 0.3t3 – 5t2 21t, where t is the time in seconds and h is the height in feet.Classify this polynomial by degree and by number of terms.This polynomial is a cubic trinomial2. Graph the polynomial function for the height of the roller coasteron the coordinate plane at the right.253. Find the height of the coaster at t 0 seconds.Explain why this answer makes sense.h(0) 0This value means that the ride starts on the ground24684. Find the height of the coaster 9 seconds after the ride begins. Explain how you found the answer.h(9) 2.7 feetthe answer is found by substituting 9 for each x in the equation5. Evaluate h(60). Does this answer make sense? Identify practical (valid real life) domain of the ridefor this model. CLEARLY EXPLAIN your reasoning. (Hint.: Mt. Everest is 29,028 feet tall.)h(60) 48,060 feetthe answer is not reasonable, because it is too high for a roller coaster ride.On the graph, after 10 seconds, ride keeps increasing in height to infinity, therefore practiceldomain is no more than D: 0 x 12

6. Next weekend, Fred, Elena, Michael, and Diane visit another roller coaster. Elena snaps a pictureof part of the coaster from the park entrance. The diagram at the right represents this part of thecoaster.Do you think quadratic, cubic, or quartic function would be thebest model for this part of the coaster? Clearly explain your choice.This model must be Quartic function,because it has 3 relative extrema.The highest degree expected would be 4.7. The part of the coaster captured by Elena on film is modeled by the function below.h(t) -0.2t4 4t3 – 24t2 48 tGraph this polynomial on the grid at the right.8. Color the graph blue where the polynomial is increasingand red where the polynomial is decreasing.Identify increasing and decreasing intervals. Increasing intervals: (0, 1.5) and (4.3, 9.2)Decreasing intervals: (1.5, 4.3) and (9.2, 10) 309. Use your graphing calculator to approximate relative maxima and minima of this function. Roundyour answers to three decimal places.MAXIMA: (1.5, 30.5) and (9.2, 92.2)MINIMUM: (4.3, 12.3)

10. Clearly describe the end behavior of this function and the reason for this behavior.It is a 4th degree function with a negative leading coefficient function will have the same behavior on both ends, Left end falls, right end rises11. Suppose that this coaster is a 2-minute ride. Do you think thatgood model for the height of the coaster throughout the ride? Clearly explain and justify yourresponse.is a2 – minute ride is NOT reasonable, because we have already seen possible Max/ Min points andafter 2 minutes the ride would continue decreasing below the ground. It needs to come back to theground to h 0.12. Elena wants to find the height of the coaster when t 8 seconds, 9 seconds, 10 seconds, and 11seconds. Use synthetic division to find the height of the coaster at these times. Show all work.SYNTHETIC DIVISION NEEDED FOR ALL 4h(8) 76.8 fth(9) 91.8 ft h(10) 80 fth(11) 19.8 ftDiane loves coasters that dip into tunnels during the ride. Her favorite coaster is modeled byh(t) -2t3 23t2 – 59t 24. This polynomial models the 8 seconds of the ride after the coaster comesout of a loop.13. Graph this polynomial on the grid at right.MAX( 4.8, 50)

14. Why do you think this model’s practical domainis only valid from t 0 to t 8?8After t 8 sec., function keeps decreasing to - which is illogical.Can not go so deep below the ground.D: 0 x 8makes sense. You end the ride on the ground.Mathematical end behavior of this function does not make sense practically.15. At what time(s) is this coaster’s height 50 feet? Clearly explain how you found your answer.H 50 is reached at about t 4.8 sec. and t 7.1 sec.Diane wants to find out when the coaster dips below the ground.16. Identify all the zeros of h(t) -2t3 23t2 – 59t 24. Clearly interpret the real-world meaning ofthese zeros.REAL ZEROES are: h( ½) 0 , h(3) 0 , h(8) 0The coaster went into the tunnel at ½ seconds and 8 seconds.At 3 seconds it came out of the underground tunnel.

Mt. Everest is 29,028 feet tall.) h(60) 48,060 feet the answer is not reasonable, because it is too high for a roller coaster ride. On the graph, after 10 seconds, ride keeps increasing in height to infinity, therefore practicel domain is no more than D: 0 x 12. 6. Next weekend, Fred, Elena, Michael, and Diane visit another roller coaster.