Create A Roller Coaster Ride Name

Name: Create a Roller Coaster Ride GoalUse nonlinear functionsto model a roller coasterride.A   Language ObjectiveUsing mathematicallanguage and linkingwords and phrases toconnect ideas, describethe use of nonlinearfunctions in modelingreal-world situations.?   Why Use NonlinearFunctions toModel Real-WorldSituations?Sometimes therelationship betweentwo real-worldquantities is not linear.SAMPLE PLANEssential Question How can you model real-worldsituations with nonlinear functions?y218In this task, you are creating a roller coaster ride fromgraph sections of the nonlinear functions y x2 andy x2. You will develop two plans for different rollercoaster rides.Constraints: The roller coaster ride can start at any height, but theremust be at least two hills. Sections can be shifted up and to the right in wholenumber increments. The heights of the hills should decrease from left to right.y101010888(–1, 1)y664(–2, 4) 42–4 –22(1, 1)2 4(0, 0)xy x 2: 1 x 1y–4 –2–2(–1, –1)–4(0, 0)2 4(1, –1)–4 –2242xy–4 –2–2(–2, –4)–4–4 –22y4x(2, –4)–4 –2–2(–3, –9) –4–6–6–8–8–8–10–10–10y x2: 1 x 1(0, 0)4122y –(x – 9) 1010862y (x – 7) 84y (x – 13)2024628 10 12 14 16 18 20Distance (m)xDecreasing HeightIntervals3 x 79 x 13xy x2: 3 x 3(0, 0)2(3, 9)4(0, 0)y x2: 2 x 2x6(–3, 9)14Increasing HeightIntervals0 x 37 x 90 x 3y(2, 4)y –(x – 3) 1816Height (m)APPLICATION TASK(0, 0)24x(3, –9)–6y x2: 2 x 2y x2: 3 x 3Did You Know? The 570-foot-tall Skyscraper in Orlando, Florida, will be the world’s tallest roller coaster.Create a Roller Coaster RideFunctions Grade 8Copyright Imagine Learning, Inc.1

Name: BackgroundA roller coaster car goes up and down steep hills as it travels across the coastertrack. To begin with, the car is pulled to the top of the first hill by a mechanicalwinch. As it is pulled, the car acquires energy that subsequently decreases due tofriction as the car rubs against the roller coaster track.There are different kinds of energy involved in a roller coaster ride. Potential energy isthe energy stored in the roller coaster car that has the potential to be used. Potentialenergy is often referred to as energy at rest. To illustrate this, look at the diagramshown. The roller coaster car has the maximum amount of potential energy when it isat the highest point, right before it is released to begin the ride.Energy cannot be created or destroyed, but it can be converted into different formsfor use. With this in mind, you will note that when the roller coaster car begins tomove down the hill, the car’s potential energy is being changed into kinetic energy,which is energy in motion.Maximumpotential energyACADEMIC VOCABULARYSupporting Wordsfriction: the resistance of motion when oneobject rubs against another object, slowingeach other downkinetic energy: the energy that is in motionpotential energy: the stored energy in anobject based on how it is positionedroller coaster: a ride at an amusement parkwith cars that roll on tracks with sharp curvesand steep hillssteep: describes a sharp slopeExample: The graph of the line y 3 x has a steep slope.Hill height decreases to account for the lossof energy due to frictiony108y 3x642–10 –8 –6 –4 –2–22468 10x–4Potential energyis converted tokinetic energy–6–8–10Think about ItHow is this type of model similar to and different from a real rollercoaster? Why might a model be convenient to use? Use linking words and phrases inyour answer.winch: a device used to lift or move anobject, turned by a crank or other powersourceA real roller coaster and this model are similar because Create a Roller Coaster RideFunctions Grade 8Copyright Imagine Learning, Inc.2

AName: UnderstandSolve a similar problem A continuous roller coaster ride starting at x 0 has beencreated using 4 graph sections from page 1.Analyze the graph to see how it meets each constraint. To begin with, complete theequations to label each section that was shifted. Then, complete the table to describethe height intervals for the roller coaster.ConstraintsEquation Notes There must be at least two hills.y (x h)2 k The heights of the hills must decrease from left to right.y (x h)2 k The roller coaster can start at any height.h: units shifted rightLINKING WORDS AND PHRASESMathematicians use linking wordsand phrases to link related ideas as theysolve real-world problems. Below are somepossible words and phrases you can use asyou write and discuss creating a plan for aroller coaster.k : units shifted upyIncreasing Height Intervals181614Height (m)0 x 22y –(x – 2) 15y –(x –due to the fact thatfrequentlyimportantlyon the other handsincesubsequentlyto illustratewith this in mind x 2) x 121086Decreasing Height Intervals4 x 202y (x –24) 68y (x –10 12 14 16 18 20Previously used linking words and phrases: x 2) xDistance (m)Think about ItUse the vocabulary from page 4 to complete the sentence.The roller coaster is made from nonlinear graph sections that areTalk about It Talk about how to solve thisproblem with a partner. Discuss how thesample solution in section A meets therequirements of the task. Is there anotherway that you could solve the problem?.as a resultbecauseconsequentlyfor instanceto begin withExplain your reasoning using linking words and phrases and mathematical language. Create a Roller Coaster RideFunctions Grade 8Copyright Imagine Learning, Inc.3

Name: OrganizeBMATHEMATICAL LANGUAGECreate two continuous roller coaster rides starting at x 0. Add labels to show theequation for each section after it is translated.ConstraintsMajor Wordscontinuous: going on without a gap orinterruptionEquation Notes There must be at least two hills.y (x h)2 k The heights of the hills must decrease from left to right.y (x h)2 k The roller coaster can start at any height.h: units shifted right Use the 4 graph sections from page 1 for the roller coaster ride.k: units shifted updecreasing: Describes a section of a graphor function where the y-values decreaseas the x-values increase. The graph sectionappears to be going down from left to right. Shift sections up and to the right in whole-number increments.Plan 2y1818161614141212yHeight (m)Height (m)Plan 1108442246810 12 14 16 18 20Distance (m)xnonlinear: Describes a function that does nothave a constant rate of change. The graphof a nonlinear equation is not a straight line.862linear: Describes a function that has aconstant rate of change. The graph of alinear equation is a straight line.1060increasing: Describes a section of a graph orfunction where the y-values increase as thex-values increase. The graph section appearsto be going up from left to right.0translation: a transformation that moves afigure or object a certain distance withoutchanging it in any other way246810 12 14 16 18 20Distance (m)xvertex: the minimum or maximum point of aparabolaExplain ItHow is the vertex of each graph section related to where the height isincreasing and decreasing? Use linking words and phrases to explain your reasoning. Create a Roller Coaster RideFunctions Grade 8Copyright Imagine Learning, Inc.4

Name: SolveCComplete the table for each of your plans. Fill in the blanks to describe where the heightis increasing and decreasing over time for the roller coaster ride. Then analyze the plans.Plan 1Plan 2Increasing HeightIntervalsDecreasing HeightIntervalsIncreasing HeightIntervalsDecreasing HeightIntervals x x x x x x x x x x CONNECT TO SCIENCERoller coaster cars do not have engines.A mechanical winch pulls a roller coastercar up to the maximum height of the first hill.What causes the car to move after it reachesthe maximum height of the first hill? Explain ItWhy are signs used to describe where the height is increasing ordecreasing rather than signs? Why must the heights of the hills decrease asthe ride continues?CheckDYou can check your work from sections B and C by solving equations to find thevertex coordinates for each graph section. Complete the table for the Sample Plan.Then check your own work using the same process.y182y –(x – 3) 18Equation16Height (m)14128y (x 3)2 183y (3 3)2 18 18y (x 7)2 87y ( 7)2 8 2y (x – 7) 8y (x 9)2 10y (y (x 13)2y ( 9)2 10 4202y (x – 13)246810 12 14 16 18 20 13)2 Extend Create a longer roller coaster ridestarting at x 0 to a maximum distanceof x 40. In addition to showing a greaterlength of the x-axis, what other changeswill you need to make to your display ofthe graph? Explain. x Distance (m)Explain It Solve for y2y –(x – 9) 10106Vertex(x-value) What should the result be when you solve each equation for y if the workis correct? Create a Roller Coaster RideFunctions Grade 8 Copyright Imagine Learning, Inc.5

friction as the car rubs against the roller coaster track. There are different kinds of energy involved in a roller coaster ride. Potential energy is the energy stored in the roller coaster car that has the potential to be used. Potential energy is often referred to as energy