Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.
27 Pages
Texture_l4
Course: CSC 290, Fall 2009School: Hofstra
Rating:
Word Count: 1138
Document Preview
surfacedetail:woodgrain,stoneroughness, Texture Motivation:tomodelrealisticobjectsneed scratchesthataffectshininess,grass,wall paper. Usegeometry,modelsurfacedetailwith polygons;goodforlargescaledetail,too expensiveotherwise. Improvement:mapanimageofthedetails ontosimplegeometry HofstraUniversityCSC171A 1 06/01/09 Texturemapping Texturemapping:addingsurfacedetailby mappingtexturepatternstothesurface...
Unformatted Document Excerpt
Coursehero >> New York >> Hofstra >> CSC 290Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
surfacedetail:woodgrain,stoneroughness, Texture
Motivation:tomodelrealisticobjectsneed scratchesthataffectshininess,grass,wall paper. Usegeometry,modelsurfacedetailwith polygons;goodforlargescaledetail,too expensiveotherwise. Improvement:mapanimageofthedetails ontosimplegeometry
HofstraUniversityCSC171A 1
06/01/09
Texturemapping
Texturemapping:addingsurfacedetailby mappingtexturepatternstothesurface TechniquedevelopedbyCatmull(1974),Blinn andNewell(1976).
06/01/09
HofstraUniversityCSC171A
2
Texturemappingmethods
2Dtexturemapping:paint2Dpattern ontothesurface Environmental(reflection)mapping Bumpmapping:perturbsurfacenormals tofoolshadingalgorithms Proceduraltexturemapping,3Dtexture
06/01/09
HofstraUniversityCSC171A
3
2Dtexturemappingoverview
Texturearrayisa2Dimagepattern Withelementstexels Valueatatexelaffectssurface appearance Thetexturemapdetermineshowthe patternliesonthesrface
06/01/09
HofstraUniversityCSC171A
4
2Dtexturemappingoverview
Renderingusesthetexturemapping Findsurfacethatisfrontmostatcurrentpixel Findthethesurfacepatchcorrespondingto thepixel Findthepartofthetexturepattern correspondingtothesurfacepatch Usethatpartofthetexturepatterninsetting thepixelcolor
06/01/09 HofstraUniversityCSC171A 5
06/01/09
HofstraUniversityCSC171A
6
2Dtexturemapping
Source:2Dpatternfromdrawing,photo, procedure Destination:anysurface,easierifsurface giveninparametricform Themapfrom2Dtexturecoordto3Dobject Texturemappingtransformation:2Dscreen coord3Dobjectcoord2Dtexturecoord andback(seepreviousslide)
06/01/09
HofstraUniversityCSC171A
7
Mappingthe2Dtexturetothesurface
Themap:2Dtexture(s,t)3Dobject(x,y,z) Mappingontotriangleisnotdifficult Mappingontotriangularmeshismoredifficult (havetohandletexturediscontinuity) Mappingontoparametricsurfaceiseasier Alternative:useanintermediateparametric surface(cylinder,sphere)
06/01/09
HofstraUniversityCSC171A
8
Mappingtextureontoparametricsurface
p : x = x(u , v), y = y (u , v), z = z (u , v)
Pointontheparametricsurface
Themapfromtexturetotheparametric coord,invertible
u = as + bt + c v = ds + et + f
06/01/09 HofstraUniversityCSC171A 9
Mappingtextureontoparametricsurface
Exampleofalinearmapping Doesnottakeintoaccountcurvatureofsurface Equalsizetexturepatchesarestretchedtofitvariousareas
06/01/09
HofstraUniversityCSC171A
10
Mappingtexturetoasurfaceusingan intermediatesurface
Twostepmapping
Mapthetexturetoasimpleintermediate surface(sphere,cylinder,cube) Maptheintermediatesurface(withthe texture)ontothesurfacebeingrendered
06/01/09
HofstraUniversityCSC171A
11
Twostepmappingexample
cylinder:x=rcos(2PIu) y=rsin(2PIu) z=vh firststep:u=s,v=t sphere cube
06/01/09 HofstraUniversityCSC171A 12
Twostepmappingexample
Secondstep:mapintermediatesurfacetothe surfacebeingrendered Variousstrategies:a,b,c
06/01/09
HofstraUniversityCSC171A
13
Thetexturemappingtransformation
06/01/09
HofstraUniversityCSC171A
14
Texturemappingtransformation
Considersurfacevisibleatcurrentpixel. Findthepatchonthesurfacethatcorrespondstoit. Mapscreencoordofpixelcornersbacktoobject Findtexelsthatmaptothesurfacepatch Ifmultipletexelslieonpatchcombinethem: weightedavg;supersamplingwithpostfiltering
06/01/09 HofstraUniversityCSC171A 15
EnvironmentalMapping
Puttextureonahighlyreflectiveobject bypickinguptexturefromthe environmentinwhichtheobjectis immersed. Realizedastwostepprocess
Projecttheenvironment(excludingthe object)ontoanintermediatesurface. intermediatesurfacetoobject
HofstraUniversityCSC171A Placeobjectback,andmaptexturefrom 16
06/01/09
Bumpmapping
2DTexturemapcreatesoddlookingrough surfaces Bumpmapping:texturemapthatalters surfacenormals.
Usetexturearraytosetafunctionwhichperturbs surfacenormals Alterednormalsmatchabumpysurface Applyingilluminationmodeltothenewnormals shadesthebumpscorrectly
HofstraUniversityCSC171A 17
06/01/09
Bumpmapping
Bumpmapisintexturearray:d(s,t)<<1 ppointonthesurfacecorrespondingtotexture coordinatess,t. Nthenormalatp pthebumppointforp p=p+d(s,t)N Weactuallydonotbumpthesurface,justthe normalatp. Nthenormalatp.Thisnormalusedbythe illuminationmodelatp.
06/01/09 HofstraUniversityCSC171A 18
HowtogetN: giventwovectorstangenttothebumpy surface,Nistheircrossproduct Thetwovectorsfollowfromthepartial derivativesofthepequationwrtu,v p=p+d(s,t)N Thesepartialderivativesexpressedinterms ofthederivativesofd(s,t)ass,tchange
Bumpmapping
06/01/09
HofstraUniversityCSC171A
19
2DTexturemappinginOpenGL Pixelpipeline
Texturemapdoneatrasterizationstage
06/01/09 HofstraUniversityCSC171A 20
TextureMappinginOpenGL
Getholdoftexturearray Createtextureobjectandspecify textureforit Specifymodeforapplyingtextureto pixels Enabletexturemapping Drawthesceneprovidingbothtexture andgeometriccoordinates
HofstraUniversityCSC171A 21
06/01/09
Createtextureobjectandtextureforit
Texelvaluescouldbeupto4D (R,G,B,A) Texturingisexpensive.Textureobjects similartodisplaylists:fasterto bind(reuse)storedtexturethanload withglTexImage*D()
06/01/09 HofstraUniversityCSC171A 22
Usingtextureobjects
Createtexturenames,glGenTextures(), Returnsavailableidsinsecondparameter whichispassedbyreference Createandusetextureobjects:
staticGluinttexName; glGenTextures(1,&texName);
glBindTexture(GL_TEXTURE_2D,texName);
Infirstuse:newtextureobjectiscreatedand assignedthename;subsequentuses: activatethetextureobject
HofstraUniversityCSC171A 23
06/01/09
Usingtextureobjects
glBindTexture(),setstexturestate, subsequentcallstoglTexImage, glTe...
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Below is a small sample set of documents:
Below is a small sample set of documents:
Hofstra - CSC - 290
Discrete Techniques06/01/09 Computer Graphics, Shading (G)1OverviewWe will discuss The point: enhance visual effect of the scene rather than photorealism. Better terms: color v.s. texture mapping. One works directly in a some pi
Hofstra - CSC - 290
Illumination: Lights and Materials. Shading Readings: Review Chapter 6: "Shading" , OpenGL Guide Ch.5 "Lighting"06/01/09 AG S20011Shading Review 2a circle and a triangle or a sphere and a cone? 06/01/09 AG S20012Shading Revi
Hofstra - CSC - 290
Normal Vectors For smooth, flat surfaces, the vector normal to the surface exists at every point Plane: ax + by + cz + d = 0 Also, n (p - p0) = 0 where p0 is a point on the plane, p is any point (x,y,z) on the plane and n is the normal to the
Hofstra - CSC - 290
ScanLineInterpolationTocomputeattributepfor everypixelintherasterbuffer Normalvectorcomponents Color TextureStartwiththevaluesofpat eachvertexofapolygonin thebuffer Computepalongeachedge usinglinearinterpolation. Tofilltheinteriorpixel
Hofstra - CSC - 290
TransmittedLightIdealconditionisasurfacethattransmitsall thelightthatstrikesit Lightisbentatthesurfaceduetothe differenceinthespeedoflightinthe materials land taretheindicesofrefraction,the relativespeedoflightintwomaterials Snellslaw,sin
Hofstra - CSC - 290
%!PS-Adobe-3.0 %BoundingBox: (atend) %Pages: (atend) %PageOrder: (atend) %DocumentFonts: (atend) %Creator: Frame 4.0 %DocumentData: Clean7Bit %EndComments %BeginProlog % % Frame ps_prolog 4.0, for use with Frame 4.0 products % This ps_prolog file is
Hofstra - CSC - 290
%!PS-Adobe-3.0 %BoundingBox: (atend) %Pages: (atend) %PageOrder: (atend) %DocumentFonts: (atend) %Creator: Frame 4.0 %DocumentData: Clean7Bit %EndComments %BeginProlog % % Frame ps_prolog 4.0, for use with Frame 4.0 products % This ps_prolog file is
Hofstra - CSC - 290
%!PS-Adobe-3.0 %BoundingBox: (atend) %Pages: (atend) %PageOrder: (atend) %DocumentFonts: (atend) %Creator: Frame 4.0 %DocumentData: Clean7Bit %EndComments %BeginProlog % % Frame ps_prolog 4.0, for use with Frame 4.0 products % This ps_prolog file is
Hofstra - CSC - 290
%!PS-Adobe-3.0 %BoundingBox: (atend) %Pages: (atend) %PageOrder: (atend) %DocumentFonts: (atend) %Creator: Frame 4.0 %DocumentData: Clean7Bit %EndComments %BeginProlog % % Frame ps_prolog 4.0, for use with Frame 4.0 products % This ps_prolog file is
Hofstra - CSC - 290
Hofstra - CSC - 290
Viewing06/01/09Hofstra University CSC171A1Transformation ProcessCamera Analogy Setting up your tripod and pointing the camera at the scene (viewing transformation). Arranging the scene to be photographed into the desired composition
Hofstra - CSC - 290
TextureMappingandAntialiasingConcernedsolelywithantialiasing duringthetexturemappingphase UniqueproblemsandattributesofTM: needshapevaryingfilters,texturemaps areknownapriori,sosome precalculationscouldbedone06/01/09HofstraUniversityCSC290B
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:FINAL EXAMINATION VERSION A December 18, 2006SECTION: In which section are you enrolled? Please check one box. CRN 465 11:00-12:15 CRN 467
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:FINAL EXAMINATION VERSION B December 18, 2006SECTION: In which section are you enrolled? Please check one box. CRN 465 11:00-12:15 CRN 467
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:FINAL EXAMINATION VERSION C December 18, 2006SECTION: In which section are you enrolled? Please check one box. CRN 465 11:00-12:15 CRN 467
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #1 VERSION A "Introduction to Economics" September 20, 2006INSTRUCTIONS: This exam is closed-book, closed-notes. Simpl
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #1 VERSION C "Introduction to Economics" September 20, 2006INSTRUCTIONS: This exam is closed-book, closed-notes. Simpl
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #3 VERSION A "Long-Run Economic Growth and Inflation" November 6, 2006INSTRUCTIONS: This exam is closed-book, closed-n
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #3 VERSION C "Long-Run Economic Growth and Inflation" November 6, 2006INSTRUCTIONS: This exam is closed-book, closed-n
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #4 VERSION B "Short-Run Business Cycles" December 6, 2006INSTRUCTIONS: This exam is closed-book, closed-notes. Simple
Drake - ECON - 001
Principles of Macroeconomics (Econ 001) Drake University, Fall 2006 William M. BoalSignature: Printed name:MIDTERM EXAMINATION #4 VERSION C "Short-Run Business Cycles" December 6, 2006INSTRUCTIONS: This exam is closed-book, closed-notes. Simple
Bowling Green - A - 201
NAME: _ October 24, 1994 ASTRONOMY 201 - Second ExamINSTRUCTIONS: Choose the most nearly correct answer. Guess, if necessary, after eliminating obviously wrong answers. 1. This means measuring the brightness of as
Bowling Green - A - 201
NAME:_ April 19, 1996 Astronomy 201 - Hour Exam #3Instructions: Choose the best response to each item.1. The Coal Sack, a dark patch visible in the Milky Way from the Southern Hemisphere, is an example of a(n) _.A.
Bowling Green - A - 201
Name:_ May 7, 1996Astronomy 201 - Final Examination PLEASE SIGN YOUR NAME ABOVE AND ON THE ANSWER SHEET!Instructions: Choose the best answer from the choices given. 1. The average ti
Mississippi State - ECE - 4263
March 2007/V0.2Lab 6: Logical Effort, Gate Sizing, Transistor SizingThe learning objectives of this lab: Use different techniques for gate sizing to improve the delay of the decoder design given in Figure 1 by sizing the gates in Stage1 through S
Bowling Green - A - 201
October 25. 1996Astronomy 201 Hour Exam #2INSTRUCTIONS: Choose the best answer from those supplied.1. The best measure of overall quality of an astronomical telescope is the size of its _.A. mount B. objective C. drive motor D. eye
Bowling Green - A - 201
Astronomy 201 Final ExamDecember 1996INSTRUCTIONS: Choose the best answer from those supplied. Use a soft lead pencil. Be sure to sign both this sheet and the answer sheet, and to "bubble in" your PIN.1. At which of the following days would t
Mississippi State - ECE - 4263
February 2007/V0.3Lab 5: Logical Effort IntroductionThe learning objectives of this lab: Be able to measure the parameters of logical effort and parasitic delay for different gates and different technologies As always, read through the entire la
Cleveland State - COM - 22106
Cleveland State - C - 320
COM 320, History of the Moving Image Issues Raised with the Introduction of Sound Generally negative 1. Loss of visual artistry (according to cinematographers in Vision of Light: The Art of Cinematography (U.S./Japan, 1992)-camera immediately became
New Mexico - ECE - 447
FFT of Raw Data 45 40 35 30 Amplitude 25 20 15 10 5 0 050100150200 250 300 350 FFT Data Point Number400450500
San Jose State - ISE - 201
ISE 201 Assignment Sheet - (preliminary - subject to change)Spring 2007Classes 1 & 2Date W 1/24Material Covered/Reading Due Chap 1: Statistics Intro (skip 1.9) Chap 2 thru Sec 2.3: Probability Sec 2.3-2.7: Probability and Conditional Probabil
San Jose State - ISE - 201
Getting Started ISE 201Spring 2007-Course Detail ISE 201: Software Engineering Analysis Catalog Description: Mathematical concepts, techniques relevant to software engineering, motivation by real world examples. Probability and statistics includi
San Jose State - ISE - 201
Green Sheet - Spring 2007 ISE 201 - Software Engr Analysis (#25816) Section 1 - W 6:00-8:45pm - Engr 486Instructor: Steve Kennedy Office: Engr 485B Office Hours: W 5:00-5:45pm W 8:45-9:15pmCourse description: Mathematical concepts, techniques rel
San Jose State - ISE - 201
Walpole Ch 01: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 1 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007St
San Jose State - ISE - 201
Walpole Ch02:1Probability&Statistics forEngineers&Scientists, by Walpole,Myers,Myers& Ye ~ Chapter2NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007ProbabilityIntros Thesam
San Jose State - ISE - 201
Walpole Ch 03: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 3 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007A
San Jose State - ISE - 201
Walpole Ch 04: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 4 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007Me
San Jose State - ISE - 201
Walpole Ch 05: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 5 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007Di
San Jose State - ISE - 201
Walpole Ch06:1Probability&Statistics forEngineers&Scientists, by Walpole,Myers,Myers& Ye ~ Chapter6NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007ContinuousUniformDistribu
San Jose State - ISE - 201
Walpole Ch08:1Probability&Statistics forEngineers&Scientists, by Walpole,Myers,Myers& Ye ~ Chapter8NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007Populations&Samplesobser
San Jose State - ISE - 201
Walpole Ch 09: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 9 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007Un
San Jose State - ISE - 201
Walpole Ch10:1Probability&Statistics forEngineers&Scientists, by Walpole,Myers,Myers& Ye ~ Chapter10NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007StatisticalHypothesess
San Jose State - ISE - 201
Walpole Ch 11: 1 Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 11 NotesClass notes for ISE 201San Jose State University Industrial & Systems Engineering Dept. Steve KennedySpring 2007S
E. Kentucky - EECS - 512
Find Butterworth filter polesSolutions: 1 1 / N s = p 2m - 1 exp j + 2 2N m = 1, 2, .N 1/ N 1: Find a circle of radius 0 = p (1 / ) which is centered at s = 0.j P1/2NP2/N0 2: Find = / N which gives angular di
E. Kentucky - EECS - 512
Problem 8.56, Perform break-loop analysis for the following feedback circuits and fine expressions of their loop gains. (a) Based on the following equivalent circuit, the loop gain can be found as, (R1 / Ri ) v RL / (R2 + (R1 / Ri ) A = - r = vt RL
E. Kentucky - EECS - 512
E. Kentucky - EECS - 512
E. Kentucky - EECS - 512
Homework Solution #58.40: Fig.P8.40 (page 866) is a series-series feedback circuit. The parameters are: R2 = 10k, RL = 1k and r = 100,. The Op-amp input resistance is Rid = 10k and output resistance is r0 = 100.Rin+ A _Rout R2 R1I0 RL_ Vs +
E. Kentucky - EECS - 512
E. Kentucky - EECS - 512
E. Kentucky - EECS - 512
Addition homework problemConsider the feedback circuit shown in the following figure:I0RL2 RL1 IS RS RF REwith Rs = 10k, RL1 = 10k, RL2 = 10k, RE = 100, RF = 20k, r1 = r2 =2.5k, r01 = r02 = 20k and hfe1 = hfe2 = 100. ib+ic C hfeib r0Bvb
E. Kentucky - EECS - 512
EECS 512 Exam I Review (Spring 2009) Feedback (Ch. 8)Negative feedback: stabilizes circuit operation, reduces gain sensitivity, increases bandwidth, reduce noise Closed-loop gain: Af = 1 +A , A Loop gain: L = A Amount of feedback: 1+AIf A > 1 (an
E. Kentucky - EECS - 512
Example: Three-stage BJT amplifier with series-shunt feedbackQ3 RS Q1 R1 Vs + _ RF RE RL Q2 R2+v0_-circuitGiven: Rs = 150, RE = 100, R1 = 9k, R2 = 5k, RF = 650,and RL = 100. DC bias current of the transistors: Ic1 = 0.5mA, Ic2 = 1mA, and Ic
E. Kentucky - EECS - 512
Series-series feedback, Example 2Vcc RS R5 RD VDD+ A _ _ Vs +RF R3 R7 R4 RLFing the midband transconductance gain Small-signal equivalent circuit:RS+ + Avi _R0 Ri2 vi_+v2_R gm(v2-v1)Vs + _Ri1RF R3 R7-circuitImpedances of c
E. Kentucky - EECS - 728
Homework #7 Problem 1: Suppose an EDFA has a noise figure of Fact. However, in the presence of source spontaneous emission (SSE), the measured EDFA noise figure using an OSA is Fmea. Giving the SSE spectral density sse is 42 dBm/nm, find the measurem
New Mexico - CS - 150
CS 150 Computing for Business Students Hybrid Course Syllabus 2008 Fall Semester, Section 501, CRN 26813 Lab on Wednesdays, 9:15a 10:15a, V111 Instructor Reinaldo A. Z. Garcia, Ed.D., Professor of Information Technology & Dean of Instruction.
Laurentian - GEOG - 200901
GEOG 1010Introduction to GeographyENVIRONMENTAL ISSUESAir, Water, Land, Species Quality, Quantity, Distribution* Air/Water/Soil Quality and Quantity * Energy * Climate change * Pollution & Hazardous Waste * Land Use and Degradation * Toxics & Pe
UC Davis - CS - 145
ECS 145, Scripting Languages and Their ApplicationsNorm Matloff Spring 2009, updated May 26, 2009 for futureYOU WILL BE TESTED ON THIS SYLLABUS!1Contents1 2 Crucial Importance of This Syllabus Consultation 2.1 2.2 3 Ofce and Ofce Hours . . .
UC Davis - CS - 132
From Algorithms to Z-Scores: Probabilistic and Statistical Modeling in Computer ScienceNorm Matloff University of California, DavisfX (t) = ce0.5(t) 1 (t)library(MASS) x <- mvrnorm(mu,sgm)0.015 0.010 0.005 5 10 5 0 0 x1 5 5 10 10 10http:/h
New Mexico - MATH - 150
Math 150 - 003 Test 2 Oct 10, 2001 Name: . 1. Divide x 3 8 x 2 + 10 by x 2 + 3x 2 .2. Find all intercepts and asymptotes of R( x ) =2 x 2 16 x 40 . x 2 813. Divide without the calculator:8 5i 6 5i4. Sketch the graph of f ( x ) =x2