The document describes how equations and limits were used in FXgraph to plot the Batman logo and the word "Batman". Various types of equations including linear, quadratic, and circle equations were manipulated to create the desired lines and curves. Horizontal and vertical lines were created using y=c and x=c equations. Linear equations with one variable and quadratic equations with two variables were used to form diagonal lines and curves. Circle and arc equations were employed to draw the circular elements. Through combining these different equation types with limits, the full Batman logo and word were accurately replicated on the graph.

1. Batman Logo and Word Investigation
Introduction
The aim of thisinvestigationis tograph a picture of the Batman logoand the word “Batman” using
equationsandlimits inFXgraph.The mathematicsinvolvedincludedquadraticequations, functions,
limitationsandfactorising.Todrawthe Batman logoand word, linear,quadraticandothersortsof
equationswillbe putintoFXgraph and manipulated usingdifferentfunctionstocreate the desired
linesinthe desiredpositions.Additionally,the limits willbe addedtothe equationtolimithow long
the line isinorderto create the image.Itis predictedthat Batmanlogoand wordwill be plottedon
the graph accuratelyto replicate whattheywouldnormallylooklike.
Graphs andEquations
y=20 [-55,-15]
y=0.5(x+10.5)2+10 [-10.5,-15]
y=10 [-21/2,-7.5]
y=4x+40 [-7.5,-5]
y=-4x [-5,-2.5]
y=0.5x+11 [-2.5,0]
y=20 [55,15]
y=-0.5x+11 [2.5,0]
y=sqrt(50^2-2(x+35)2)-57.5 [-35,0]
y=sqrt(50^2-2(x-35)2)-57.5 [35,0]
x2+y2=3481
y=4x [5,2.5]
y=-4x+40 [7.5,5]
y=10 [21/2,7.5]
y=0.5(x-10.5)2+10 [10.5,15]
y=sqrt(50^2-5(x-55)2)-30 [35,55]
y=sqrt(50^2-5(x+55)2)-30 [-35,-55]

2. Discussion
Circles
To form the outerring/frame of the Batmanlogothe equationfora circle was used.Thisequation
takesthe form 𝑥2 + 𝑦2 = 𝑟2, where ris the radiusof the circle.The desiredsize of the circle hada
radiusof 59 as that wasthe perfectsize forthe wingstojusttouch the outercircle.Hence the
equationof the circle was 𝑥2
+ 𝑦2
= 3481. Picturedhere isa comparisonbetweendifference
sizedcirclestodemonstrate howthe functionbehaves.
y=20 [25,40]
x=32.5 [19,1]
x=32.5 [19.5,1]
y=-(x-45)2+20 [40.6,49.4]
y=-(x-54)2+20 [49.45,58.5]
y=-(x-45)2+20 [40.6,49.51]
y=-(x-54)2+20 [49.5,58.5]
y=3x-30 -150 [60,66.5]
y=3x-33 -150 [64.55,67]
y=-3x+70 +150 [67.35,69.8]
y=10.5 [65,69.7]
y=-3x+73 +150 [67.8,74.3]
y=19.7 [66.6,67.6]
y=9.5 [63.5,71]
x=76 [1,20]
x=76 [1,20]
y=-2x+172 [76,86]
y=-2x+172 [76,86]
x=86 [1,20]
x=86[1,20]
x=2 [19.3,11]
x=1 [20.5,-0.5]
x=-0.3(y-15)2+10 [10,20.5]
x=-0.4(y-15)2+9 [10.9,19.1]
x=-0.3(y-5)2+10 [10.1,-0.48]
x=-0.4(y-5)2+9 [9.1,0.85]
x=2 [9,1]
y=3x-30 [10,16.5]
y=3x-33 [14.5,17]
y=-3x+70 [17.3,19.8]
y=10.5 [15,19.7]
y=-3x+73 [17.9,24.3]
y=19.5 [17.8,16.6]
y=9.5 [13.3,21]
y=20
x=32.5 [19,1]
x=32.5 [19.5,1]
y=-(x-45)2+20 [40.6,49.4]
y=-(x-54)2+20 [49.45,58.5]
y=-(x-45)2+20 [40.6,49.51]
y=-(x-54)2+20 [49.5,58.5]
y=3x-30 -150 [60,66.5]
y=3x-33 -150 [64.55,67]
y=-3x+70 +150 [67.35,69.8]
y=10.5 [65,69.7]
y=-3x+73 +150 [67.8,74.3]
y=19.7 [66.6,67.6]
y=9.5 [63.5,71]
x=76 [1,20]
x=76 [1,20]
y=-2x+172 [76,86]
y=-2x+172 [76,86]
x=86 [1,20]
x=86[1,20]
x=76 [1,20]
x=76 [1,20]
y=-2x+172 [76,86]
y=-2x+172 [76,86]
x=86 [1,20]
x=86[1,20]
𝑥2
+ 𝑦2
= 20
𝑟 = √20
𝑟 = 4.47
𝑥2
+ 𝑦2
= 4
𝑟 = √4
𝑟 = 2

3. Horizontal andVerticalLines
On the logothe wingswere horizontal lines,meaningtheytookthe form 𝑦 = 𝑐, thiscreateda line
that wasflat andwentthroughthe y axisat c unitshigh.Forthe wingsitwas 𝑦 = 20. On the word
“BATMAN” itwas usedonthe lettersA andT, howeverthe B,T, M, and N vertical lineswere needed
to theytookthe form 𝑥 = 𝑐 as the line needtopassthrough the x axis.The same propertiesapply
excepteverythingisrotated90o
bychangingit from 𝑦 = 𝑐 to 𝑥 = 𝑐. Picturedbelow isavisual
explanation of howhorizontal andvertical lineswork. Tolimithow fara line will gosquare brackets
are placedat the endof the equationtolimitwassectionof the x or y axistheyare limitedto.
Linear Equations
To form the earsof the bat on the logolinesona slantwere needed.Tochange themfromjustflat
linesthatwere constantanothervariable wasaddedtoc to formthe equation 𝑦 = 𝑥 + 𝑐.Insteadof
c settingwere the line wasnowcsethow highor low the line wouldbe.If c=20 the y intercept
wouldstill be 20 howeverthe line wouldbe angledtoitwouldbe higher. Thisformedadiagonal line
witha 45o
angle to eitheraxis.Thisangle wastooflat,tomake it steeperacoefficientwasaddedto
x to change the steepnessof the line.A highercoefficientmeantthe linewouldbe steeperanda
lowermeantitwouldbe flatter.Thisequationtookthe form 𝑦 = 𝑚𝑥 + 𝑐 The earsrequireda
steeperline thena45o
line sotheytookthe equation 𝑦 = 4𝑥 + 40.The line wasonlyrequiredfora
small segmentandthatwas betweenx=-7.5and-5.To setthese limitsthe twox coordinateswere
put insquare bracketsat the end of the equation.The final equationforone side of one eartookthe
form 𝑦 = 4𝑥 + 40 [−7.5,−5]. To create the otherside of the ear the line hadto be goingdown
insteadof up.To achieve thisthe coefficientof x wasmade negative makingis −4𝑥 insteadof 4𝑥.
Thisprocesswas repeatedonall the letterswithdiagonallinesandthe twolinesbetweenthe bats
ears. Belowisa visual representationof how the cvalue changedthe heightof the graphand how
the coefficientof x changedthe slantof the line.

4. Quadratic equations
To connectthe earsto the top of the wingsa straightline wouldnothave createdthe ideal shape
and look.A curvedline wasnecessary. Tocreate a curvedline a 𝑥2 neededtobe addedtothe
existingequation.Hence formingthe base equation 𝑦 = 𝑥2 + 𝑚𝑥 + 𝑐.Itwasfoundthat this
equationalwayshadthe same widthof the curve so a coefficientwasnecessarycreatinganew base
equationfora quadraticbeing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐.This graphwas harderto manipulate soiswas
factorisedintothe form 𝑦 = (𝑥 + ℎ)2 + 𝑘. Inthisequationthe vertex of the quadraticfunctionwas
(– ℎ, 𝑘). Thisis because hmovesthe equationleftandrightandk movesitup anddown.It is –h as
have positive hmovesthe line leftandnegativehmovesitright. The line connectingthe earstothe
wingshadthe equation 𝑦 = 0.5(𝑥 + 10.5)2 + 10 the coefficientof x outthe front (0.5) made the
line wider.Thisseemscounterintuitive asa lowernumbermakesitwiderhoweverif the number
gestlowenoughthatis becomes0 a straightline will be formedandthatislogical asif x has a
coefficientof 0 the line will alwaysbe straight.
Visual representationof howthe coefficient
of x2
changedthe widthof the graph
Thisgraph showshow the coordinatesof
the vertex are (−ℎ, 𝑘) as the h value is -2
and k value is6 and the vertex coordinates
are (2,6)

5. Relations
Whenformingthe B on the word“BATMAN” none of the curvesthat tookthe form 𝑦 = (𝑥 + ℎ)2 +
𝑘 wouldsatisfythe line neededasthe curve wasgoingthe wrongway.To change to direction
insteadof using 𝑦 = (𝑥 + ℎ)2 + 𝑘.The x andy were swapped.Thistookthe form 𝑥 = (𝑦 + ℎ)2 + 𝑘.
Thisallowedthe quadraticequationtogohorizontallyinsteadof vertically. These are calledrelations
not functionsbecause foreachinputof x there can be more thenone value fory. A functiononlyhas
only1 yvalue foreach x input.
Arcs
Sometimesacurve wasnot a goodoptionto use for roundlinesasitwasn’tconsistentenough.For
the bottomof the wings,a segmentof asemicircle wasusedbyusinglimitationsandanew equation
that takesthe form 𝑦 = √𝑟2 − 𝑥2 thisgraph couldbe manipulatedinasimilarwaytothe quadratic
equationbychangedthe coefficientof x. Toformthe bottompart of the wingIusedthe equation
𝑦 = √50^2 − 5(𝑥 − 55)2 − 30 [35,55]. The radius need was 50 units so 𝑟 was replaced
with 50. The location of the line need to be moved 55 units to the right and down 30. Hence
55 was taken away from x and 30 was taken away from the whole equation. The segment
needed lied between x= 35 and 55 so the limits were added. This was the line created and
this is where it goes on the batman logo.

6. Conclusion
Throughthe developmentof the two images,itwasdiscoveredthatlinescanbe manipulatedby
changingspecificelementstoform the desiredline.Limitationscanthenbe setto selectthe
segmentof the line thatisneeded. Horizontalandvertical linesare the simplest formsof linesand
have no variables,linearequationshave one variable thatcanbe manipulatedintwowaysand
quadraticequationshave twovariablesinx andx2
.Quadraticequationscanbe manipulatedinfour
ways.Circlesandsemicirclesdon’ttake the traditional formof anequation astheyare not
consideredfunctions,similartothe relationswhenx isonthe LHS insteadof y.Usinglinearand
quadraticequationsin combination withsemi circlesitwaspossible tocreate the Batmanlogoand
the word “BATMAN”on FX Graph 5.