[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572984-105":53,"doc-detail-572984-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","ch4-graphs-and-transformations-cheat-sheet","Ch.4 - Graphs and Transformations Cheat Sheet","","Graphs and Transformations Cheat Sheet summarizes key methods for sketching and modifying polynomial and reciprocal graphs. It covers cubic and quartic function forms, determining general shapes, finding roots to locate x-intercepts, and applying intersection techniques to solve simultaneous equations. It also explains how translations and stretchings/reflections affect graphs, including how asymptotes move under transformations. Worked examples illustrate plotting and computing intercepts efficiently for practical exam-style problem solving.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/ch4-graphs-and-transformations-cheat-sheet/572984/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/ch4-graphs-and-transformations-cheat-sheet/572984.png","ImageObject",442,249,{"name":88,"@type":89},"Marglet","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you sketch a cubic function using its equation?","Question",{"text":108,"@type":109},"Compare the given cubic with a general cubic form to determine the graph’s shape, then solve for the roots to find where it crosses the x-axis. Plot the resulting intercept coordinates to guide the sketch.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How are intersection points of two functions found?",{"text":113,"@type":109},"Set the two function expressions equal, forming an equation in x. Solve for x, then use the corresponding coordinates to identify where the graphs intersect.",{"name":115,"@type":106,"acceptedAnswer":116},"What does a translation, stretching, or reflection do to a graph?",{"text":117,"@type":109},"A constant outside the function shifts the graph vertically, while adding a constant inside shifts it horizontally; asymptotes translate as well when they exist. Multiplying by a constant stretches the graph vertically, while scaling inside affects horizontal stretching; negatives produce reflections about the relevant axis.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572984,1792241698,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},2336477982702,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Graphs and Transformation Cheat Sheet Cubic & Quartic Graphs  \nCubic functions come in the form of 􀝂 (􀝔) = 􀜽􀝔3 + 􀜾􀝔2 + 􀜿􀝔 + 􀝀 . Quartic functions come in the form of 􀝂 (􀝔) = 􀜽􀝔4 + 􀜾􀝔3 + 􀜿􀝔2 + 􀝀􀝔 + 􀝁 , where 􀜽, 􀜾 . 􀜿, 􀝀 & 􀝁 are all real numbers and where 􀜽 ≠ 0.  \nSimilar to sketching quadratic graphs, cubic graphs can also be represented on a  \nplot. There are two main basic shapes for cubic graphs if 􀜽 > 0: 􀜽 \u003C 0:  \nExample 1: Sketch the curve with the equation 􀝕 = 􀝔 3 + 􀝔 2 − 2􀝔 .  \nBefore we start sketching, we need to know  \n• the general shape of the cubic equation  \n• the location of the roots of the equation  \nWe can compare the function to a general function of 􀝂(􀝔) = 􀜽􀝔3 + 􀜾􀝔2 + 􀜿􀝔 +  \n􀝀 to determine the shape of the curve.  \n􀜽 = 1 and 1 > 0 hence the shape of the graph would look like  .  \nThe next step would be to find the roots of the functions so we can determine where it would cross the 􀝔 axis. To do this we need to solve the quadratic equation of  \n􀝔 3 + 􀝔 2 − 2􀝔 = 0  \n􀝔 (􀝔2 + 􀝔 − 2) = 􀝔 (􀝔 + 2)(􀝔 − 1) = 0  \n􀝔 = 0 􀜽􀝊􀝀 􀝔 − 1 = 0 􀜽􀝊􀝀 􀝔 + 2 = 0 Hence the roots of the function are at:  \n􀝔 = 0 􀜽􀝊􀝀 􀝔 = 1 􀜽􀝊􀝀 􀝔 = −2  \nAt coordinates (0,0), (1,0) & (−2,0) are where the x intercepts will be located.  \nThe same method can be applied to sketching graphs of quartic functions. The basic shapes of these graphs come in the form of:  \n.  \nReciprocal graphs  \nReciprocal graphs that come in the form of 􀝂(􀝔) = 􀯔􀯫 or 􀝂(􀝔) = 􀯔􀯫2, where 􀜽 is any real number, can also be sketched by considering their asymptotes. The graphs of 􀝕 = 􀯔􀯫 and 􀝕 = 􀯔􀯫2 both have asymptotes at 􀝔 = 0 and 􀝕 = 0. The basic shapes of these reciprocal graphs can be illustrated as:  \n􀝕 = 􀜽 􀝓ℎ􀝁􀝎􀝁 􀜽 > 0 􀝔 ,  \n| 􀝕 = − 􀜽 􀝓ℎ􀝁􀝎􀝁 􀜽 \u003C 0 􀝔 ,\u003Cbr>| 􀝕 = 􀝔􀜽2 , 􀝓ℎ􀝁􀝎􀝁 􀜽 > 0\u003Cbr>|\n| --- | --- |\n\n􀝕 = − 􀜽   \n􀝔 2 , 􀝓ℎ􀝁􀝎􀝁 􀜽 \u003C 0  \nPoints of Intersection  \nMultiple functions can be sketched on a single graph to show the points of intersection, which represent the solutions to respective equations.  \nExample 2: Sketch the following functions and find the 􀝔 coordinate of the intersection points.  \n􀝂 (􀝔) = 3􀝔 − 􀝔 2 􀝃 (􀝔) = 2􀝔3 − 􀝔 2  \nThe curves have been sketched using the methods you have learnt in this course. To find the intersection point, we must obtain the 􀝔 coordinate of the locations. To solve this, we need to find the solutions to  \n• 􀝂 (􀝔) = 􀝃 (􀝔)  \n3􀝔 − 􀝔 2 = 2􀝔 3 − 􀝔 2 2􀝔 3 − 3􀝔 = 0  \n􀝔 (2􀝔2 − 3) = 0  \n􀝔 = 0 & 2􀝔2 − 3 = 0    \n􀝔 = 0, 􀝔 = √32 & 􀝔 = −√32  \nTranslating graphs  \nAdding or subtracting a constant outside, 􀝕 = 􀝂 (􀝔) + 􀜽, or inside, 􀝕 = 􀝂 (􀝔 + 􀜽), a function can translate a graph vertically or horizontally respectively. Note that when translating functions, the asymptote of that function is also translated if it has one.  \n• The translation of 􀝕 = 􀝂 (􀝔) + 􀜽 can be represented by the vector (0􀜽)−􀜽  \n• The translation of 􀝕 = 􀝂 (􀝔 + 􀜽) can be represented by the vector ( 0 ) Example 3: Given that 􀝂(􀝔) = 􀝔 2, sketch the curve of 􀝕 = 􀝂 (􀝔 − 3) .  \nThis is the graph of 􀝕 = 􀝔 2  \nApplying the translation of 􀝕 = 􀝂 (􀝔 − 3) would shift the graph by three units to the right, forming anew graph of 􀝕 = (􀝔 − 3)2  \nEdexcel Pure Year 1  \nStretching Graphs  \nSimilar to translating graphs, multiplying a constant outside, 􀝕 = 􀜽􀝂 (􀝔), or inside, 􀝕 =􀝂 (􀜽􀝔), a function stretches the graph in the vertical direction or horizontal direction respectively.  \n• 􀝕 = 􀜽􀝂(􀝔) would stretch the graph in the vertical direction by a multiple of 􀜽 .  \n• 􀝕 = 􀝂 (􀜽􀝔) would stretch the graph in the horizontal direction by a multiple of 1  \n􀯔 .  \n• 􀝕 = −􀝂(􀝔) would be the reflection of 􀝕 = 􀝂 (􀝔) in the 􀝔-axis.  \n• 􀝕 = 􀝂 (−􀝔) would be the reflection of 􀝕 = 􀝂 (􀝔) in the 􀝕-axis.  \nExample 4: Given that 􀝂(􀝔) = 16 − 4􀝔2, sketch the curve with the equation 􀝕 = 12􀝂 (􀝔) .  \nBefore we start to sketch 􀝕 = 12􀝂 (􀝔), we need to know the 􀝔 and 􀝕 intercepts of the curve 􀝕 = 􀝂 (􀝔) and how the curve looks like.  \n􀝕 = 16 − 4􀝔2  \n􀝕 = (4 − 2􀝔)(4 + 2􀝔)  \nTo find the 􀝔 intercept we need to equate 􀝕 as 0.  \n0 = (4 − 2􀝔)(4 + 2􀝔)  \n􀝔 = 2 􀜽","cbCaieEcZiGKkade","https://ap.wps.com/l/cbCaieEcZiGKkade","pdf",571610,"English","# Cubic & Quartic Graphs\n## Cubic functions: general shape and roots\n## Quartic graphs: basic shapes\n## Reciprocal graphs and asymptotes\n# Points of Intersection\n## Example: intersection of two functions\n# Translating Graphs\n## Vertical and horizontal shifts\n## Example: translating y=f(x)\n# Stretching Graphs\n## Vertical/horizontal stretch\n## Reflections across axes\n## Example: stretching a parabola\n# Transforming Graphs\n## Using intercepts, turning points, and intersections\n## Example: reflection of a given graph","[{\"question\":\"How do you sketch a cubic function using its equation?\",\"answer\":\"Compare the given cubic with a general cubic form to determine the graph’s shape, then solve for the roots to find where it crosses the x-axis. Plot the resulting intercept coordinates to guide the sketch.\"},{\"question\":\"How are intersection points of two functions found?\",\"answer\":\"Set the two function expressions equal, forming an equation in x. Solve for x, then use the corresponding coordinates to identify where the graphs intersect.\"},{\"question\":\"What does a translation, stretching, or reflection do to a graph?\",\"answer\":\"A constant outside the function shifts the graph vertically, while adding a constant inside shifts it horizontally; asymptotes translate as well when they exist. Multiplying by a constant stretches the graph vertically, while scaling inside affects horizontal stretching; negatives produce reflections about the relevant axis.\"}]","Ch.4 - Graphs and Transformations Cheat Sheet | PDF",1791930377]