{"id":2804,"date":"2025-08-13T18:43:19","date_gmt":"2025-08-13T18:43:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2804"},"modified":"2026-02-24T21:30:47","modified_gmt":"2026-02-24T21:30:47","slug":"conics-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/conics-background-youll-need-2\/","title":{"raw":"Conics: Background You'll Need 2","rendered":"Conics: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Graph circles in standard form<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>circles<\/h3>\r\nA circle with center [latex](h,k)[\/latex] and radius [latex]r&gt;0[\/latex] has equation [latex](x-h)^{2}+(y-k)^{2}=r^{2}[\/latex].\r\n\r\n<\/section>To graph, plot the center, mark points [latex]r[\/latex] units away in the four cardinal directions, and sketch the circle.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1628\" data-end=\"1682\">Graph [latex]x^{2}+(y+2)^{2}=25[\/latex].<\/p>\r\n<p data-start=\"1684\" data-end=\"1801\">Identify center and radius:\r\n[latex]\r\n\\begin{align}\r\nh&amp;=0 \\\\ k&amp;=-2\\\\ r^{2}&amp;=25\\Rightarrow r=5.\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"1803\" data-end=\"1996\">Plot center [latex](0,-2)[\/latex]. Use the center to mark points on the edge of the circle at [latex](0,3), (5,-2), (0,-7), \\text{ and } (-5,-2)[\/latex].<\/p>\r\n<p data-start=\"1803\" data-end=\"1996\"><img class=\"alignnone wp-image-4865\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM.png\" alt=\"Circle with center marked by green dot at (0,-2) and a purple line indicating radius out to touch the circle at (5,-2)\" width=\"466\" height=\"428\" \/><\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">If the equation is expanded, complete the square in [latex]x[\/latex] and in [latex]y[\/latex] to return to standard form.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313910[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313911[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Graph circles in standard form<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>circles<\/h3>\n<p>A circle with center [latex](h,k)[\/latex] and radius [latex]r>0[\/latex] has equation [latex](x-h)^{2}+(y-k)^{2}=r^{2}[\/latex].<\/p>\n<\/section>\n<p>To graph, plot the center, mark points [latex]r[\/latex] units away in the four cardinal directions, and sketch the circle.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1628\" data-end=\"1682\">Graph [latex]x^{2}+(y+2)^{2}=25[\/latex].<\/p>\n<p data-start=\"1684\" data-end=\"1801\">Identify center and radius:<br \/>\n[latex]\\begin{align}  h&=0 \\\\ k&=-2\\\\ r^{2}&=25\\Rightarrow r=5.  \\end{align}[\/latex]<\/p>\n<p data-start=\"1803\" data-end=\"1996\">Plot center [latex](0,-2)[\/latex]. Use the center to mark points on the edge of the circle at [latex](0,3), (5,-2), (0,-7), \\text{ and } (-5,-2)[\/latex].<\/p>\n<p data-start=\"1803\" data-end=\"1996\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4865\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM.png\" alt=\"Circle with center marked by green dot at (0,-2) and a purple line indicating radius out to touch the circle at (5,-2)\" width=\"466\" height=\"428\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM.png 686w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM-300x276.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM-65x60.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM-225x207.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/23175537\/Screenshot-2025-10-23-at-10.55.29%E2%80%AFAM-350x321.png 350w\" sizes=\"(max-width: 466px) 100vw, 466px\" \/><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">If the equation is expanded, complete the square in [latex]x[\/latex] and in [latex]y[\/latex] to return to standard form.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313910\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313910&theme=lumen&iframe_resize_id=ohm313910&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313911\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313911&theme=lumen&iframe_resize_id=ohm313911&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2804"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2804\/revisions"}],"predecessor-version":[{"id":5731,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2804\/revisions\/5731"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2804\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2804"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2804"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2804"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}