[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572289-105":53,"doc-detail-572289-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","132-the-graphs-of-polar-equations","13.2 - The Graphs of Polar Equations","","Explains how to graph polar equations on the Cartesian xy-plane by analyzing representations of points in polar coordinates. Uses examples to show how freedom of r or theta determines the resulting curve, including circles centered at the origin, lines determined by terminal sides, and the y-axis. Develops a theorem for constant r and constant theta, then demonstrates a strategy for graphing r = 6 cos(theta), including handling negative r values and determining which quadrants the curve traces.",{"@graph":63,"@context":119},[64,80,102],{"@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/132-the-graphs-of-polar-equations/572289/",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/132-the-graphs-of-polar-equations/572289.png","ImageObject",442,249,{"name":88,"@type":89},"adawds","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How do you determine the graph from a polar equation like r = a or θ = θ̄?","Question",{"text":109,"@type":110},"For r = a, the graph is a circle centered at the origin with radius |a|. For θ = θ̄, the graph is the line containing the terminal side of θ̄ when drawn in standard position.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What does it mean when θ is free in an equation such as r = 4?",{"text":114,"@type":110},"If θ is free, the point must take the form (4, θ) for any θ, meaning all points at distance 4 from the origin are included. This produces a circle centered at the origin.",{"name":116,"@type":107,"acceptedAnswer":117},"Why does r = 6 cos(θ) require considering negative r values and quadrants?",{"text":118,"@type":110},"Negative r changes where the curve appears on the xy-plane: instead of tracing one quadrant, the same r-values can trace through another quadrant depending on how negative radii are interpreted. This affects which portions of the curve are traced and whether the graph begins to retrace itself.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572289,1792022856,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":45,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":101},3985747872983,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","13.2 The Graphs of Polar Equations  \nIn this section, we discuss how to graph equations relating the polar coordinate variables r and 􀀒 on the rectangular coordinate plane. Since every point in the plane has infinitely many different representationsin polar coordinates, in order for a point P to be on the graph of a given equation, there must be at least one representation of P (r , 􀀒) that satisfies that equation.  \nIn our first example, only one of the variables r and 􀀒 is present making the other variable free.1 This makes these graphs easier to visualize than others.  \nExample 13.2.1. Graph the following polar equations in the xy-plane.  \n1. r = 4 2. r = 􀀀3 p2 3. 􀀒 = ~~5􀀙4 ~~ 4. 􀀒 = 􀀀 ~~3􀀙2~~  \nSolution.  \n1. In the equation r = 4, 􀀒 is free. The graph of this equation is, therefore, all points which have a polar coordinate representation (4, 􀀒), for any choice of 􀀒 . In other words, we trace out all of the points 4 units away from the origin. This is exactly the definition of circle, centered at the origin, with a radius of 4 .  \nIn r = 4, 􀀒 is free The graph of r = 4  \n2. Once  again we have 􀀒 being free in the equation r = 􀀀3 p2. Plotting all of the points of the form (􀀀3 p 2, 􀀒) gives us a circle of radius 3 p 2 centered at the origin.  \n1 See the discussion in Example 13.1.3 number 2a in Section 13.1.  \n1108 Polar Coordinates and Parametric Equations  \n\n| \u003Cbr>\u003Cbr>\u003Cbr>| y\u003Cbr>|\n| --- | --- |\n| | \u003Cbr>\u003Cbr>x |\n\nIn r = 􀀀3 p2 , 􀀒 is free The graph of r = 􀀀3 p2  \n3. In the equation 􀀒 = ~~5􀀙4~~ , r is free, so we plot all of the points with polar representation 􀀀 r , ~~5􀀙4~~ 􀀁 . The result is the line containing the terminal side of 􀀒 = ~~5􀀙4~~ , when plotted in standard position.  \n\n| | y \u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr> r \u003C 0\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>|  |\n| --- | --- | --- |\n| 􀀒 = ~~5􀀙4~~ |  |  |\n|  |  |  |\n| | r = 0 | \u003Cbr>x |\n| \u003Cbr>\u003Cbr>\u003Cbr>r > 0\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>|  |  |\n|  |  |  |\n\nIn 􀀒 = ~~5􀀙4~~ , r is free  \n4. As in the previous example, the variable r is free in the equation 􀀒 = 􀀀 ~~3􀀙2~~ . Plotting 􀀀 r , 􀀀 ~~3􀀙2~~ 􀀁 for various values of r shows us that we are tracing out the y-axis.  \n13.2 The Graphs of Polar Equations 1109  \nIn 􀀒 = 􀀀 ~~3􀀙2~~ , r is free The graph of 􀀒 = 􀀀 ~~3􀀙2~~  \nHopefully, our experience in Example 13.2.1 makes the following result clear.  \nTheorem 13.2. Graphs of Constant r and 􀀒 : Suppose a and 􀀋 are constants, a  0.  \n• The graph of the polar equation r = a on the Cartesian plane is a circle centered at the origin of radius j a j .  \n• The graph of the polar equation 􀀒 = 􀀋 on the Cartesian plane is the line containing the terminal side of 􀀋 when plotted in standard position.  \nSuppose we wish to graph r = 6 cos(􀀒) . A reasonable way to start is to treat 􀀒 as the independent variable, r as the dependent variable, evaluate r = f (􀀒) at some `friendly' values of 􀀒 and plot the resulting points.2  \n\n| 􀀒  |  | r = 6 cos(􀀒) | (r , 􀀒) |\n| --- | --- | --- | --- |\n| 0 |  | 6 | (6, 0) |\n| 􀀙\u003Cbr>4 |  | 3 p2 | 􀀀3 p2 , ~~􀀙4~~ 􀀁 |\n| 􀀙\u003Cbr>2 |  | 0 | 􀀀0, ~~􀀙2~~ 􀀁 |\n| 3􀀙 4 |  | 􀀀3 p 2 | 􀀀􀀀3 p 2, ~~3􀀙4~~ 􀀁 |\n| 􀀙 |  | 􀀀6 | (􀀀6, 􀀙) |\n| 5􀀙 4 |  | 􀀀3 p2 | 􀀀􀀀3 p2 , ~~5􀀙4~~ 􀀁 |\n| 3􀀙 2 |  | 0 | 􀀀0, ~~3􀀙2~~ 􀀁 |\n| 7􀀙 4 |  | 3 p 2 | 􀀀3 p 2, ~~7􀀙4~~ 􀀁 |\n| 2􀀙 |  | 6 | (6, 2􀀙) |\n\n2 For a review of these concepts and this process, see Section 1.1.  \n1110 Polar Coordinates and Parametric Equations  \nDespite having nine ordered pairs, we get only four distinct points on the graph. For this reason, we employ a slightly different strategy. We graph one cycle of r = 6 cos(􀀒) on the 􀀒r-plane3 below on the left and use it to help graph the equation on the xy-plane below on the right.  \nWe see that as 􀀒 ranges from 0 to ~~􀀙2~~ , r ranges from 6 to 0 . In the xy-plane, this means that the curve starts 6 units from the origin on the positive x-axis (􀀒 = 0) and gradually returns to the origin by the time the curve reaches the y-axis (􀀒 = ~~􀀙2~~ ) .  \nThe arrows drawn in the figure below are meant to help you visualize this process. In the 􀀒 r-plane, ","cbCaipEBIJuCAQyU","https://ap.wps.com/l/cbCaipEBIJuCAQyU","pdf",429265,"English","# Example 13.2.1 - Graph the given polar equations in the xy-plane\n## Key cases when one variable is free (r or θ)\n# Theorem 13.2 - Graphs of Constant r and θ\n# Graphing r = 6 cos(θ) using cycles and quadrants\n# Example 13.2.2 - Graph the following polar equations in the xy-plane","[{\"question\":\"How do you determine the graph from a polar equation like r = a or θ = θ̄?\",\"answer\":\"For r = a, the graph is a circle centered at the origin with radius |a|. For θ = θ̄, the graph is the line containing the terminal side of θ̄ when drawn in standard position.\"},{\"question\":\"What does it mean when θ is free in an equation such as r = 4?\",\"answer\":\"If θ is free, the point must take the form (4, θ) for any θ, meaning all points at distance 4 from the origin are included. This produces a circle centered at the origin.\"},{\"question\":\"Why does r = 6 cos(θ) require considering negative r values and quadrants?\",\"answer\":\"Negative r changes where the curve appears on the xy-plane: instead of tracing one quadrant, the same r-values can trace through another quadrant depending on how negative radii are interpreted. This affects which portions of the curve are traced and whether the graph begins to retrace itself.\"}]","13.2 - The Graphs of Polar Equations | PDF",1791926580]