0001000101000eLinearFunc.ACT0001020017LinearFunctionGraph.EAC010000008137ɸDkäûx³[Linear Function And [ Its Graph\Authoráà\PMun Chou, Fong Mathsroof Consultancy Malaysia. e-mail: mcfong@mathsroof.comÿ[[ OBJECTIVE[I. Explore the roles of[coefficient and constant[play in linear function of[ y=Aí¸+B, and[[II. Use what we have[observed to help graph[the function of y=Aí¸+B.[[Exploration 1:[The exercise explore the[relationships between the[graph of the function of[y=í¸+B and the constant[B.[We begin by assigning the[initial value of 1 to B.[R1îBR1R[Perform the following:[- Tap on GraphStrip 1.[- Select y=í¸+B and drag[ drop it to the window.[A line which slants from [left to right is drawn.[\ GraphStrip 1ÌÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~ÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁÙÁ a]xB]¨°Ron systemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁsystemä]listsystemä^]=systemÙÁ  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`pl`pll ™pl`plll(1…0qy`l#Y‡uYƒl ˜`Y™ ™`cx”t ™O[Now scroll up and draw[the lines again but with[these values for B: -5,[-1, 4.[Remember to tap [EXE][after each change of B.[[(1) State the points[where the line passes[y-axis for each value[of B.[(2) What can you say[about B based on the[ exercise? \Exploration Notesáà8)(1) The points where the line passes y-axis for each value of B: when B=1, (0,1); when B=-5, (0,-5); when B=-1, (0,-1) and when B=4, (0,4). (2) Changes in B cause the line to shift vertically. In fact, as can be deduced from (1) above, B forms the y-coordinate of the line's y-intercept. ÿ[[In this next exercise we[explore the relationships[between the graph of the[function y=Aí¸+2 and its[coefficient A.[[Exploration 2:[We begin by modifying the[coefficient A to study the[changes to the line.[First we assign 2 to A:R2îAR2R[Now[- Tap on GraphStrip 2.[- Select y=Aí¸+2 and drag[ -drop it into the graph[ window.[The line graph of y=2í¸+2[is drawn.(Recall A=2)[\ GraphStrip 2ÌÊÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á A]R!AxbT systemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™`(1…0qy`#Y‡uYƒ ˜ÿÿÿÿÿÿÿÿÿÿÿÿ``„!&1X ™O[- Now tap on [Analysis][ î [Modify]. [The line becomes bolder,[and 'Aîx+2' is display in[the Graph message box.[We only want to modify[the coefficient A, so[- select(or highlight) the[ 'A' in this message box[- Tap on the left/right[ Graph control button[ of îˆ and î† to modify[ value of A.[Explore the line graph of[y=Aí¸+2 with different values[of A.[ [(1) What do you observe[when A= 0 and -2?[(2) What can you say[about A based on this[ exploration?\Exploration NotesáàD7(1) When A=0, the line is horizontal. When A=-2, the line is perpendicular to the original line graph of y=2í¸+2. (2) When A>0, the line slants from left to right and it slants from right to left when A<0. Also as the magnitude of A becomes greater, the slopiness of the line becomes greater too. ÿ[[ Example 1:[In this example we want[to determine x-intercept[for y=Cx+D by analysing[ the function.[[First we express y=Cx+D[ in terms  of y, and we[have x=y-DC.\ In term of yËߤR clear_a_zRdoneR y=Cx+Dîeq1Ry=Cx+DR eq1-Dîeq2Ry-D=CxR eq2î(1/C)Ry-DC=CxCR exchange(y-DC=x)Rx=y-DCRÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á©Á aseq_histbNewFolde systemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Ásystemä]listsystemä^]=system©Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[As x-intercept is where[the line meets the x-axis,[its y-coordinate is equal[ 0, and the x-cordinate[is y-DC=0-DC=-DC[î+ x-intercept of y=Cx+D[is (-DC ,0). î~[[In this final example, we[use concepts and method[learned in Exploration 1[and 2, and Example 1 to[construct the line graph[rather by using graphing[capabilities of Classpad.[[ Example 2:[The task is to construct[the graph of the function[ 2í¸+3í¹=5.[[Step 1[First we solve 2x+3y=5[for y.\ Solve for yËß°R clear_a_zRdoneRsolve(2x+3y=5,y)Ry=-2î’x3+53RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á aseq_histbNewFolde systemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[[Step 2[Next we need to find[two(2)  points from the[ function y=-2î’x3+53.[One is the y-intercept,[which is deducible from[the function as (0,53).[Another easy-to-find point[is the x-intercept. Its[x-coordinate is calculated[asR-53/-23R52R[So x-intercept is (52,0).[[Step 3[Here we plot P(52,0) and[Q(0,53) in GeometryStrip.\ GeometryStripÎ Î`P”3–"p`P”3–"`vrALCuxy ÈÈ  P@Ð6–°P ÉÈ  Q@Ð6–°ffp [If we draw an infinite[line across both points,[we would have graphed[the line of 2í¸+3í¹=5. [[We leave the drawing of[this infinite line as an[exercise. You can draw[this line through [Draw][î [Infinite line], then[tap on both points. î~[[[ Exercise 1:[Construct the line graph[of 35(7x+2)-11y=19 using[similar steps discussed in[ Example 2.[\ Solve for yË ß4R clear_a_zRdoneRÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]listsystemä^]=systemä_ä^systemä`ä_systemäaä`systemäb~¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á¬Á aseq_histbNewFolde systemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Ásystemä]listsystemä^]=system¬Á  0@PSheet1úº™^Sheet2úº™^Sheet3úº™^Sheet4úº™^Sheet5úº™^SheetSheet3D 0@PSheet1躙^Sheet2躙^Sheet3躙^Sheet4躙^Sheet5躙^SheetSheet3D i™ ™¸Dkäûx $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[Deduce y-intercept and[ x-intercept.[\Plot and GraphÎ Î` `62%€ vrALCuxy[[eActA ÿÿB dûCOEFF F/CONS ‡cM TÿN TÿS £‡cT €ÀSa b ÿÿeq1,,Fy DCxeq288F  € DyCxeq3((F 1à x€eq5S)/(eqn1PPFy eqn1 €xeqn2Fy eqn3PPFà eq X² €“ȳxeqn4DDF€ à  € x³