[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-315367-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-315367-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","mathematical-methods-34-summary-sheets-maths-key-formulas-and-transformations","Mathematical Methods 3,4 Summary sheets - Maths - Key formulas and transformations","","Mathematical Methods 3,4 summary sheets covering core algebra and calculus-adjacent function skills. Includes distance and midpoint formulas, identification of relations vs functions using vertical and horizontal line tests, inverse definitions via reflection in y = x, and polynomial factorisation methods such as difference of two squares, trinomials, sum/product patterns, completing the square, factor theorem, and remainder theorem with long division. Also covers quadratic formula, power/exponential/logarithmic/trigonometric functions, graph transformations, trig equation solving, and index/logarithm laws.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/mathematical-methods-34-summary-sheets-maths-key-formulas-and-transformations/315367/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/mathematical-methods-34-summary-sheets-maths-key-formulas-and-transformations/315367.png","ImageObject",300,407,{"name":42,"@type":43},"Rowan","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-05","2026-09-20",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",21,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you determine whether a relation is a function?","Question",{"text":63,"@type":64},"A relation is a set of ordered pairs. If no two ordered pairs have the same first element, then it is a function.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What are the key methods for factorising polynomials in these sheets?",{"text":68,"@type":64},"The sheets list common-factor checking, difference of two squares, trial-and-error for trinomials, sum of two cubes, grouping, completing the square, and the factor theorem.",{"name":70,"@type":61,"acceptedAnswer":71},"How are graphs transformed (and in what order should translations be applied)?",{"text":72,"@type":64},"They cover vertical/horizontal dilation, reflection in the x- and y-axes, vertical and horizontal translation. The sheets state to always carry out translations last when sketching graphs.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},315367,1789983462,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":124,"preview_page_count":4},1099514067415,"https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502","􀂔 Copyright [itute.com 2006](itute.com 2006)  \nFree download & print [from www.itute.com](from www.itute.com)[ ](from www.itute.com)Do not reproduce by other means  \nMathematical Methods 3,4 Summary sheets  \nDistance between two points  \n=  (x2 − x1 )2 + (y 2 − y 1)2 Mid-point = 􀂧􀂨 x1~~ ~~+~~ ~~x2~~ ~~,~~ ~~y~~ ~~1~~ ~~+~~ ~~y~~ ~~2~~ ~~􀂷􀂸  \n􀂩 2 2 􀂹 Parallel lines, m1 = m2 Perpendicular lines, m1m2 = −1 or m2 = − ~~ ~~1m1  \nGraphs of polynomial functions infactorised form:  \nQuadratics e.g. y = (x +1)(x − 3)  \nCubics e.g. y = 3(x + 1)(x − 1)(x − 2)  \n-1 0 1 2 e.g. y = (x +1)2 (x −1)  \n–1 0 1  \ne.g. y = (x + 1)3  \n–1 0  \nQuartics e.g. y = (x + 3)(x +1)(x −1)(x − 2)  \ne.g. y = (x + 3)2 (x − 1)(x − 2)  \n–3 0 1 2  \ne.g. y = (x + 3)2 (x −1)2  \ne.g. y = (x + 2)3 (x −1)  \n–2 0 1 e.g. y = (x + 2)4  \n–2 0  \nExamples of power functions:  \n−1 y = x  \n−2 y = x  \n􀂨􀂧􀂩y = ~~ ~~x12~~ ~~􀂷􀂹􀂸  1  \n2  \nx x )  \n􀂧 1 􀂷  \n􀂨 y = ~~ ~~ 􀂸  \n􀂩 x 􀂹  \n0  \n0  \nExponential functions:  \ny = ax where a = 2, e,10  \n1  \n10x ex 2x  \na~~symptotic ~~  \nLogarithmic functions:  \ny = loga x where a = 2, e,10  \n2x  \nx  \ne  \n10x  \n0 1 asymptotic  \nTrigonometric functions:  \n0 π 2π–1  \ny = cos x  \ny = tan x  \n0 π 2π  \nModulus functions  \n􀂭x, xy = x = 􀂮  \n􀂯 − x,  \n≥ 0  \nx \u003C 0  \nTransformations of  \ny = f (x)  \n(1) Vertical dilation (dilation away from the x-axis, dilation parallel to the y-axis) by factor k. y = kf (x)  \n(2) Horizontal dilation (dilation away from the y-axis, dilation parallel to the x-axis) by factor 1n . y = f (nx)  \n(3) Reflection in the x-axis. y = − f (x)  \n(4) Reflection in the y-axis. y = f (−x)  \n(5) Vertical translation (translation parallel to the y-axis) by c units.  \ny = f (x) ± c , + up,– down.  \n(6) Horizontal translation (translation paratofth(xxb-)a,x) blefyt, b–unrigitsht ..  \n*Always carry out translations last in sketching graphs.  \nExample 1 Sketch y = − 2(x −1) + 2 2  \nExample 2 Sketch y = 2 1 − x .  \nRewrite as y = 2 − (x −1) .  \n2  \n0 1  \nRelations and functions:  \nA relation is a set of ordered pairs (points) . If no two ordered pairs have the same first element, then the relation is a function.  \n*Use the vertical line test to determine whether a relation is a function.  \n*Use the horizontal line test to determine whether a function is one-to-one or many-toone.  \n*The inverse of a relation is given by its reflection in the line y = x .  \n*The inverse of a one-to-one function is a function and is denoted by f −1 . The inverse of a many-to-one function is not a function and therefore cannot be called inverse function, and f −1 cannot be used to denote the inverse.  \nFactorisation of polynomials:  \n(1) Check for common factors first.  \n(2) Difference of two squares,  \ne.g. x4 − 9 = (x2 )2 − 32 = (x2 − 3)(x2 + 3)  \n= (x −  3 )(x +  3 )(x2 + 3)  \n(3) Trinomials, by trial and error, e.g. 2x2 − x −1 = (2x +1)(x −1)  \n(4x3) ife(nxoyf)(twx2 cuxysy,2e).g.  \n(5) Sum of two cubes, e.g. 8 + a3 = 23 + a3 = (2 + a)(4 − 2a + a2 )  \ne(6g) . Grx3p32gwo3x n1dw(xo3, +1)+ (3x2 + 3x)  \n= (x +1)(x2 − x +1)+ 3x(x +1)  \n= (x +1)(x2 − x +1+ 3x)  \n= (x +1)(x2 + 2x +1) = (x +1)3  \n(7) Grouping three and one, .g(.x2x2)2 +y2  (x −1)2 − y 2 = (x − 1 − y )(x − 1 + y)  \n(8) Completing the square, e.g. x 2 + x −1 = x 2 + x + 􀂧􀂨  1 􀂷􀂸 2 − 􀂧􀂨  1 􀂷􀂸 2 −1  \n􀂩 2 􀂹 􀂩 2 􀂹  \n= 􀂧􀂨 x2 + x +  1 􀂷􀂸 −  5 = 􀂨􀂧x +  1 􀂷􀂸 2 − 􀂧􀂨  5 􀂷􀂸 2 􀂩􀂨 4 􀂹􀂸 4 􀂩􀂨 2 􀂹􀂸 􀂨􀂩 2 􀂹􀂸= 􀂨 x + ~~ ~~12 − ~~ ~~25~~ ~~􀂷􀂹􀂨􀂨􀂧􀂩x + ~~ ~~12 + ~~ ~~25~~ ~~􀂷􀂹  \n(9) Factor theorem, e.g. P(x) = x3 − 3x2 + 3x −1 P(−1) = (−1)3 − 3(−1)2 + 3(−1)−1 ≠ 0 P(1) = 13 − 3(1)2 + 3(1)−1 = 0 ∴ (x −1) is a factor.  \nLong division:  \nx2 − 2x +1 x −1)x3 − 3x2 + 3x −1  \n− (x3 − x2 )  \n− (− 2x2x22 3x2x) x −1  \n− (x −1)  \n∴ P(x) = (x −1)(x2 − 2x +01) = (x −1)3  \nRemainder theorem:  \ne.g. when P(x) = x3 − 3x2 + 3x −1 is divided by x + 2 , the remainder is P(− 2) = (− 2)3 − 3(− 2)2 + 3(− 2) −1 = −11 When it is divided by 2x − 3 , the remainder is P􀂧􀂨 3  􀂷􀂸 =  1  .  \n􀂩 2 􀂹 8  \nQuadratic formula:  \nSolutions of ax2 + bx + c = 0 are  \n− b ± ","cbCaigjQBDiVA47M","https://ap.wps.com/l/cbCaigjQBDiVA47M","pdf",178491,"English","# Distance between two points\n# Mid-point formula\n# Relations and functions\n## Vertical/horizontal line tests\n## Inverse relations\n# Factorisation of polynomials\n## Difference of two squares\n## Trinomials and cubes\n## Completing the square\n## Factor theorem and remainder theorem\n# Long division and quadratic formula\n# Power, exponential, logarithmic, and trigonometric functions\n# Transformations of graphs\n## Vertical/horizontal dilation and reflection/translation\n# Solving trigonometric equations\n# Exact trig values\n# Index laws and logarithm laws\n# Change of base and exponential equations","[{\"question\":\"How do you determine whether a relation is a function?\",\"answer\":\"A relation is a set of ordered pairs. If no two ordered pairs have the same first element, then it is a function.\"},{\"question\":\"What are the key methods for factorising polynomials in these sheets?\",\"answer\":\"The sheets list common-factor checking, difference of two squares, trial-and-error for trinomials, sum of two cubes, grouping, completing the square, and the factor theorem.\"},{\"question\":\"How are graphs transformed (and in what order should translations be applied)?\",\"answer\":\"They cover vertical/horizontal dilation, reflection in the x- and y-axes, vertical and horizontal translation. The sheets state to always carry out translations last when sketching graphs.\"}]","Mathematical Methods 3,4 Summary sheets - Maths - Key formulas and transformations | PDF",1789904606]