One Or More 3 Xs To One Y - MR. PUNPANICHGUL
One or MoreXs to One Y3Defining Functional RelationshipsWARM UPLEARNING GOALSEvaluate each expressiongiven the set of values{1, 6, 12, 25}.1. 5x Describe a functional relationship in terms of a rule whichassigns to each input exactly one output. Determine whether a relation (represented as a mapping,set of ordered pairs, table, sequence, graph, equation, orcontext) is a function.12.x112KEY TERMS3. x 2 8 amappingsetrelationinputoutput functiondomainrangescatter plotvertical line testThroughout middle school, you have investigated different types of relationships betweenvariable quantities: additive, multiplicative, proportional, and non-proportional. What arefunctional relationships?LESSON 3: One or More Xs to One Y M2-205
Getting StartedWhat’s My Rule?Rules can be used to generate sequences of numbers. They can alsobe used to generate (x, y) ordered pairs.XDole y5112044431. Write an equation to describe the relationship between eachindependent variable x and the dependent variable y. Explainyour reasoning.a.1212 1224 VYou cansketch thegraph to helpdetermine mc.b.xI 245 125y40d.210922102154.5542012m442. Create your own table and have a partner determine theequation you used to build it.yM2-206 TOPIC 3: Introduction to Functions42x212
Functions as Mappings fromOne Set to AnotherAC T I V I T Y3.1As you learned previously, ordered pairs consist of an x-coordinateand a y-coordinate. You also learned that a series of ordered pairson a coordinate plane can represent a pattern. You can also usea mapping to show ordered pairs. A mapping represents twosets of objects or items. Arrows connect the items to represent arelationship between them.When you write the ordered pairs for a mapping, you arewriting a set of ordered pairs. A set is a collection of numbers,geometric figures, letters, or other objects that have somecharacteristic in common.Use braces, { }, todenote a set.1. Write the set of ordered pairs that represent a relationship ineach mapping.a.11233547kikc.144,3115575d.1,1 2,1 3,5342734956282075131312B75241 1b.qa.nca.aox4axo.axs.sn2,3 2,5 3,5 4,7 5,72. Create a mapping from the set of ordered pairs.O O0.0a. {(5, 8), (11, 9), (6, 8), (8, 5)}b.{(3, 4), (9, 8), (3, 7), (4, 20)}LESSON 3: One or More Xs to One Y M2-207
X3. Write the set of ordered pairs to represent each table.ya.input outputdobmaind valuesdangey Output21010 20 C 5 10Sets0a5,10 10,20group of ordered pairsThe mappings and ordered pairs shown in Questions 1 through 3form relations. A relation is any set of ordered pairs or the mappingbetween a set of inputs and a set of outputs. The first coordinate ofan ordered pair in a relation is the input, and the second coordinate isthe output. A function maps each input to one and only one output.In other words, a function has no input with more than one output.The domain of a function is the set of all inputs of the function. Therange of a function is the set of all outputs of the function.Notice theuse of setnotation whenwriting thedomain andrange.oone.cionexWORKED EXAMPLEIn each mapping shown, the domain is {1, 2, 3, 4}.11112323353474The range is {1, 3, 5, 7}.7The range is {1, 3, 7}.Each mapping represents a function because no input, or domainvalue, is mapped to more than one output, or range value.M2-208 TOPIC 3: Introduction to FunctionsI
WORKED EXAMPLENOTESIn the mapping shown, the domain is {1, 2, 3, 4, 5} and the range is{1, 3, 5, 7}.1032451357This mapping does not represent a function.4. State why the relation in the worked example shown is nota function.function becausenot one to one322,352,5Nota5. State the domain and range for each relation in Questions 2and 3. Then, determine which relations represent functions.If the relation is not a function, explain why not.235yesg6Domain5,6 205,0 5,10functionNota77one10functionLESSON 3: One or More Xs to One Y M2-209to o
Think aboutthe mappingsas orderedpairs.6. Review and analyze Emil’s work. Explain why Emil's mapping isnot an example of a function.EmilMy mapping represents a function.31247of56912Not a functionbecause4 maps to43E5357. Determine if each sequence represents a function. Explain whyor why not. If it is a function, identify its domain and range.Create a mapping to verify your answer.a. 2, 4, 6, 8, 10, b. 1, 0, 1, 0, 1, Rememberthat asequence hasa term numberand a termvalue.c. 0, 5, 10, 15, 20, M2-210 TOPIC 3: Introduction to Functions
AC T I V I T Y3.2Functions as MappingInputs to OutputsYou have determined if sets of ordered pairs represent functions.In this activity you will examine different situations and determinewhether they represent functional relationships.Read each context and decide whether it fits the definition of afunction. Explain your reasoning.YesNo1. Input:Output:Sue writes a thank-you note to her best friend.Her best friend receives the thank-you note in the mail.2. Input:A football game is being telecast.Output: It appears on televisions in millions of homes.3. Input:yesYesNo1homesThere are four puppies in a litter.Output: One puppy was adopted by the Smiths, another bythe Jacksons, and the remaining two by the Fullers.4. Input:The basketball team has numbered uniforms.Output: Each player wears a uniform with her assignednumber.5. Input:Output:0Beverly Hills, California, has the zip code 90210.There are 34,675 people living in Beverly Hills.If6. Input:NoA sneak preview of a new movie is being shown in alocal theater.Output: 65 people are in the audience.LESSON 3: One or More Xs to One Y M2-211
7. Input:Tara works at a fast food restaurant on weekdays anda card store on weekends.MOutput: Tara’s job on any one day.ToWYesRThF8. Input:Output:Janelle sends a text message to everyone in hercontact list on her cell phone.There are 41 friends and family on Janelle’scontact list.satSurNoDetermining Whethera Relation Is a FunctionAC T I V I T Y3.3Analyze the relations in each pair. Determine which relations arefunctions and which are not functions. Explain how you know.1.Mapping 000,20093000M2-212 TOPIC 3: Introduction to FunctionsMapping B10100011200001312Not3000afunctionDomain 10,11 12,13Range10092000,3000
2.Table AFunctionDomain2Table BInputOutputxy22421100010240011240O2140,5221Nota functionDomain0,52214Range4 1,944Range 0,443. Sequence A7, 10, 13, 16, 19, Sequence B10, 30, 10, 30, 10, 4. Set A{(2, 3), (2, 4), (2, 5), (2, 6), (2, 7)}Set B{(2, 1), (3, 1), (4, 1), (5, 1), (6, 1)}NotaFunctionfunctionDomain 25. Scenario3,4A 36,7Input:The morningannouncementsare read over the schoolintercom system duringhomeroom period.DomainRangeOutput:All students report tohomeroom at the start ofthe school day to listento the announcements.Not1Toa2,34 36 Range IScenario BInput:Each student goesthrough thecafeteria line.FunctionOutput:Each student selects alunch option from themenu.function35LESSON 3: One or More Xs to One Y M2-213
AC T I V I T YFunctions as Graphs3.4represented as agraph.A scatter plot is a graph of a collection of ordered pairs that allowsan exploration of the relationship between the points.1. Determine if each scatter plot represents a function.Explain your reasoning.yOutputa.yb.66554OutputA relation can be3232O10401123 4Input5600xbi 2,3 31 4,411,1Function23 4Input56x2,314,4Nota functionThe vertical line test is a visual method used to determine whethera relation represented as a graph is a function. To apply the verticalline test, consider all of the vertical lines that could be drawn on thegraph of a relation. If any of the vertical lines intersect the graph ofthe relation at more than one point, then the relation is not a function.WORKED EXAMPLEConsider the scatter plot shown. y987In this scatter plot, the relationis not a function. The inputvalue 4 can be mapped totwo different outputs, 1 and 4.Those two outputs are shownas intersections to the verticalline drawn at x 5 4.M2-214 TOPIC 3: Introduction to Functions654210Xµ3012nIion34056789x
2. Use the definition of function to explain why the vertical linetest works.NOTESvertical linetest A relation is afunction if the vertical line crossestmeeach point3. Use the vertical line test to determine if each graph representsa function. Explain your reasoning.a.b.yy665544332211001234560xNot a functionM0123456xFunction4. Use the 12 cards that you sorted in the previous lesson.Sort the graphs into two groups: functions and non-functions.Use the letter of each graph to record your findings.FunctionsH BLKGCNon-functionsEIFJDLESSON 3: One or More Xs to One Y M2-215
NOTESAC T I V I T Y3.5Functions as EquationsSo far, you have determined whether a mapping, context, or a graphrepresents a function. You can also determine whether an equationis a function.WORKED EXAMPLEThe given equation can be used to convert yards to feet. Let xrepresent the number of yards, and let y represent the numberof feet.y 5 3xTo test whether this equation isa function, first, substitute valuesfor x into the equation, and thendetermine if any x-value canbe mapped to more than oney-value. If each x-value has exactlyone y-value, then it is a function.Otherwise, it is not a function.xy 5 3x1339412824In this case, every x-value can be mapped to only one y-value.Each x-value is multiplied by 3. Some examples of ordered pairsare (2, 6), (10, 30), and (5, 15). Therefore, this equation is a function.It is not possible to test every possible input value in order todetermine whether or not the equation represents a function. You cangraph any equation to see the pattern and use the vertical line test todetermine if it represents a function.M2-216 TOPIC 3: Introduction to Functions
1. Determine whether each equation is a function. List threeordered pairs that are solutions to each. Explain yourreasoning.y mxtbLinear equationfunctionFunctionc.recognize the graphof the equation, use a2b. y 5 xOa. y 5 5x 1 3If you do notTQuadraticgraphing calculator tosee the pattern.parabolafunctiond. x2 1 y2 5 1y 5 x AbsolutevalueequationFunctionCircleNotae. y 5 4functionf. x 5 2horizontal linevertical lineeNot a functionFunction2. Explain what is wrong with Taylor's reasoning.TaylorIf two differentinputs go tothe sameoutput, itcan still be afunction.The equation y2 x represents a function.xy4293255LESSON 3: One or More Xs to One Y M2-217
NOTES0TALK the TALKFunction Organizer1. Complete the graphic organizer for the concept of function.Write a definition for function in your own words. Then,create a problem situation that can be represented using afunction. Finally, create a table of ordered pairs and sketcha graph to represent the function.DefinitionProblemSituationoneOne X tocapital toGoConeontoPalateFunctiondiscrete21 5535µGraphM2-218 TOPIC 3: Introduction to Functionscontinuous4154,570,5 3,5 4,57Table/Ordered Pairs
AssignmentWriteRememberWrite the term from the box that best completes each sentence.A relation is any set of orderedscatter plotoutputrelationinputvertical line testmappingsetdomainrangefunction1. A(n)pairs or the mapping betweena set of inputs and a set ofoutputs.is any set of ordered pairs or the mappingA relation is a function whenbetween a set of inputs and a set of outputs.each input value maps to one2. The first coordinate of an ordered pair in a relation is theand only one output value.3. The second coordinate of an ordered pair is the.4. A(n)maps each input to one and only one output.5. A(n)is a graph of a collection of ordered pairs.6. Theis a visual method of determining whethera relation represented as a graph is a function by visualizingwhether any vertical lines would intersect the graph of therelation at more than one point.7. A(n)shows objects in two sets connectedtogether to represent a relationship between the two sets.8. A(n)is a collection of numbers, geometricfigures, letters, or other objects that have some characteristicin common.9. Theof a function is the set of all inputs of thefunction.10. Theof a function is the set of all outputs ofthe function.Practice1. A history teacher asks six of her students the number of hours that they studied for a recent test. Thediagram shown maps the grades that they received on the test to the number of hours that they studied.a. Is the relation a function? If the relation is not a function, explainwhy not.185b. Write the set of ordered pairs to represent the mapping.c. What does the first value in each ordered pair in part (b) represent?What does the second value in each ordered pair represent?70953456d. Create a scatter plot. Does the graph agree with your conclusion frompart (a)? Explain your reasoning.2GradeHours StudiedLESSON 3: One or More Xs to One Y M2-219
2. The science teacher created the set of ordered pairs {(100, 6), (90, 5), (80, 3), (70, 1), (90, 4), (80, 2)} torepresent six students' grades on the midterm to the number of hours that they had studied. Create amapping from this set of ordered pairs.a. Is the relation a function? If the relation is not a function, explain why not.b. List all the inputs of the relation.c. List all the outputs of the relation.d. Instead of mapping grades to hours studied, the teacher decides to create a new diagram.This diagram maps hours studied to grades. Show the mapping that would result.e. Write the set of ordered pairs to represent the mapping in part (d).f. Is the relation in part (d) a function? If the relation is not a function, explain why not.g. Create a scatter plot. Does the graph agree with your conclusion from part (f)? Explain your reasoning.3. At the end of the year, a principal decides to create the given mapping.Input: the 82 total students in the history classOuput: the final grades they received for the classDoes this mapping fit the definition of a function? Explain your reasoning.4. Use the vertical line test to determine if each graph represents a function. Explain your tretchDescribe how you can tell from an equation whether a function is increasing, decreasing, or constant.M2-220 TOPIC 3: Introduction to Functionsx
ReviewTell whether each graph is discrete or continuous. Also, tell whether each graph is increasing, decreasing,both, or 345678910xDetermine the slope and y-intercept of the linear relationship described by each equation.x3. y 5152x4. y 54Calculate the slope of the line represented by each table.5.xy26.xy212831.5424462456.59213LESSON 3: One or More Xs to One Y M2-221
The mappings and ordered pairs shown in Questions 1 through 3 form relations . A relation is any set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate of an ordered pair in a relation is the input, and the second coordinate is the output. A function maps each input to one and only one output.
Luciferian) Luke 23:35 the chosen of God His Chosen One (NIV, Living Bible) Only One (f) (Luciferian) The one and only God only one the One (NASB) only one the one (NIV) Luciferian “One-Only One” there standeth one among you one is your Master One is your teacher (NASB) one is your Father One is your father (NASB) John 8:50 one One
on The Grey Funnel Line VERSE 4 TUNE,¶OO SDVV WKH WLPH OLNH VRPH PDFKLQH Until the waters turn to green 7KHQ ,¶OO GDQFH RQ GRZQ WKDW ZDON DVKRUH And sail The Grey Funnel Line no more And sail The Grey Funnel Line no more HIGH AND LOW HARMONIES One more day, one more day One more day, one more day *LYH PH ZLQJV 1RDK¶V GRYH Fly - - - to the .
Book One: XLIX His Relationship with King Nicomedes Book One: L His Affairs with Roman Women Book One: LI His Reputation Elsewhere Book One: LII His Royal Love Affairs Book One: LIII His Food And Drink Book One: LIV His Cupidity Book One: LV His Oratory Book One: LVI His Writings Book One: LVII His Physical Skills and Powers of Endurance
equipment is maintained in depot stocks. One set of M4T6 can be used to construct any one of the following: One 141-foot 8-inch normal bridge One 108-foot reinforced bridge One four-float normal raft One five-float normal raft One four-float reinforced and one five-float reinforced raft One
With the One & Done Workout Program Follow your One & Done workout calendar and do 1 One Minute Finishers exercise after your One & Done Workout, before doing the Flow Down. You can choose any one of the One Minute Finishers exercises you like. If you have purchased One Minute Abs, follow your One & Done One Minute Abs workout calendar.
FP bead, one 7/0 bead, one FP bead, (8 beads total). 2) Slide all the beads to the end of the thread and tie a square knot to form a circle. Then pass through the nearest FP bead. 3) String one 7/0 bead, one FP bead, one 7/0 bead, one FP bead, one 7/0 bead, one FP bead, one 7/0 bead (7 beads total) and then pass through the same FP bead.
d) One (1) "Cut & Paste" copy if the plan has more than one (1) sheet. e) One (1) copy of the Stormdrain Plan. f) One (1) copy of the Stormwater Management Plan. g) One (1) copy of the Grading Sediment & Erosion Plan. h) One (1) copy of the Forest Stand Delineation Plan, Narrative and Data Sheets. i) One (1) copy of the Forest Conservation Plan.
Total English Form 1 Heritage Studies Form 3 SECONDARY BOOK PRESS Plus One ICT Grade 1 Plus One ICT Grade 2 Plus One ICT Grade 3 Plus One ICT Grade 5 Plus One ICT Grade 6 Plus One Visual and Performing Arts Grade 6 Plus One Agriculture Grade 6 Plus One Visual and Performing Arts Grade 6 .
