{"id":2726,"date":"2025-08-13T18:31:39","date_gmt":"2025-08-13T18:31:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2726"},"modified":"2025-10-14T15:45:55","modified_gmt":"2025-10-14T15:45:55","slug":"periodic-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/periodic-functions-background-youll-need-2\/","title":{"raw":"Periodic Functions: Background You'll Need 2","rendered":"Periodic Functions: 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\">Find the distance between and midpoint of two numbers<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>distance between two numbers<\/h3>\r\nThe distance between two numbers, [latex]a[\/latex] and [latex]b[\/latex] is the absolute value of their difference: [latex]|a-b|[\/latex] or [latex]|b-a|[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"2289\" data-end=\"2354\">Find the distance between [latex]3[\/latex] and [latex]-5[\/latex].<\/p>\r\n<p data-start=\"2289\" data-end=\"2354\">\r\n[reveal-answer q=\"746603\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746603\"][latex] \\begin{align} D &amp;= |x_2 - x_1| \\[4pt] &amp;= |-5 - 3| \\[4pt] &amp;= |-8| = 8 \\end{align} [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>midpoint<\/h3>\r\n<p data-start=\"2001\" data-end=\"2143\">The midpoint of two numbers [latex]a[\/latex] and [latex]b[\/latex] is the number exactly halfway between them: [latex]M=\\frac{a+b}{2}[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"2509\" data-end=\"2574\">Find the midpoint between [latex]-4[\/latex] and [latex]6[\/latex].<\/p>\r\n<p data-start=\"2509\" data-end=\"2574\">[reveal-answer q=\"725974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725974\"][latex] \\begin{align} M &amp;= \\frac{x_1 + x_2}{2} \\[4pt] &amp;= \\frac{-4 + 6}{2} \\[4pt] &amp;= \\frac{2}{2} = 1 \\end{align} [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313582[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313584[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Find the distance between and midpoint of two numbers<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>distance between two numbers<\/h3>\n<p>The distance between two numbers, [latex]a[\/latex] and [latex]b[\/latex] is the absolute value of their difference: [latex]|a-b|[\/latex] or [latex]|b-a|[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"2289\" data-end=\"2354\">Find the distance between [latex]3[\/latex] and [latex]-5[\/latex].<\/p>\n<p data-start=\"2289\" data-end=\"2354\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q746603\">Show Solution<\/button><\/p>\n<div id=\"q746603\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align} D &= |x_2 - x_1| \\[4pt] &= |-5 - 3| \\[4pt] &= |-8| = 8 \\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>midpoint<\/h3>\n<p data-start=\"2001\" data-end=\"2143\">The midpoint of two numbers [latex]a[\/latex] and [latex]b[\/latex] is the number exactly halfway between them: [latex]M=\\frac{a+b}{2}[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"2509\" data-end=\"2574\">Find the midpoint between [latex]-4[\/latex] and [latex]6[\/latex].<\/p>\n<p data-start=\"2509\" data-end=\"2574\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q725974\">Show Solution<\/button><\/p>\n<div id=\"q725974\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align} M &= \\frac{x_1 + x_2}{2} \\[4pt] &= \\frac{-4 + 6}{2} \\[4pt] &= \\frac{2}{2} = 1 \\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313582\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313582&theme=lumen&iframe_resize_id=ohm313582&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313584\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313584&theme=lumen&iframe_resize_id=ohm313584&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/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":191,"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\/2726"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2726\/revisions"}],"predecessor-version":[{"id":4652,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2726\/revisions\/4652"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2726\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2726"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2726"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2726"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2726"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}