{"id":2723,"date":"2025-08-13T18:31:09","date_gmt":"2025-08-13T18:31:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2723"},"modified":"2025-10-14T15:40:07","modified_gmt":"2025-10-14T15:40:07","slug":"periodic-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/periodic-functions-background-youll-need-1\/","title":{"raw":"Periodic Functions: Background You'll Need 1","rendered":"Periodic Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Apply transformations to basic functions<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>transformation of functions<\/h3>\r\n<p data-start=\"356\" data-end=\"462\">A transformation changes the appearance or position of a graph. The general form for transforming a function is [latex]y=\\pm af(b(x-h))+k[\/latex].<br data-start=\"423\" data-end=\"426\" \/>The most common transformations are:<\/p>\r\n\r\n<ul data-start=\"464\" data-end=\"1084\">\r\n \t<li data-start=\"464\" data-end=\"611\">\r\n<p data-start=\"466\" data-end=\"611\">Vertical shifts: move the graph up or down.<br data-start=\"513\" data-end=\"516\" \/>[latex]f(x) + k[\/latex] shifts up by [latex]k[\/latex]; [latex]f(x) - k[\/latex] shifts down.<\/p>\r\n<\/li>\r\n \t<li data-start=\"612\" data-end=\"767\">\r\n<p data-start=\"614\" data-end=\"767\">Horizontal shifts: move the graph left or right.<br data-start=\"666\" data-end=\"669\" \/>[latex]f(x - h)[\/latex] shifts right by [latex]h[\/latex]; [latex]f(x + h)[\/latex] shifts left.<\/p>\r\n<\/li>\r\n \t<li data-start=\"768\" data-end=\"906\">\r\n<p data-start=\"770\" data-end=\"906\">Reflections: flip the graph.<br data-start=\"802\" data-end=\"805\" \/>[latex]-f(x)[\/latex] reflects across the x-axis; [latex]f(-x)[\/latex] reflects across the y-axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"907\" data-end=\"1084\">\r\n<p data-start=\"909\" data-end=\"1084\">Stretches and compressions: make the graph taller or wider.<br data-start=\"972\" data-end=\"975\" \/>[latex]a f(x)[\/latex] vertically stretches if [latex]|a| &gt; 1[\/latex], compresses if [latex]|a| &lt; 1[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1370\" data-end=\"1458\">Given [latex]f(x) = |x|[\/latex], describe how to get [latex]g(x) = 2|x - 4| - 3[\/latex].<\/p>\r\n[reveal-answer q=\"301325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"301325\"]The function has been stretched vertically by a factor of [latex]2[\/latex], then shifted to the right [latex]4[\/latex] units and down [latex]3[\/latex] units.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313580[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Transformation inside the parentheses cause horizontal transformations.Transformations outside the parentheses cause vertical transformations.<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Apply transformations to basic functions<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>transformation of functions<\/h3>\n<p data-start=\"356\" data-end=\"462\">A transformation changes the appearance or position of a graph. The general form for transforming a function is [latex]y=\\pm af(b(x-h))+k[\/latex].<br data-start=\"423\" data-end=\"426\" \/>The most common transformations are:<\/p>\n<ul data-start=\"464\" data-end=\"1084\">\n<li data-start=\"464\" data-end=\"611\">\n<p data-start=\"466\" data-end=\"611\">Vertical shifts: move the graph up or down.<br data-start=\"513\" data-end=\"516\" \/>[latex]f(x) + k[\/latex] shifts up by [latex]k[\/latex]; [latex]f(x) - k[\/latex] shifts down.<\/p>\n<\/li>\n<li data-start=\"612\" data-end=\"767\">\n<p data-start=\"614\" data-end=\"767\">Horizontal shifts: move the graph left or right.<br data-start=\"666\" data-end=\"669\" \/>[latex]f(x - h)[\/latex] shifts right by [latex]h[\/latex]; [latex]f(x + h)[\/latex] shifts left.<\/p>\n<\/li>\n<li data-start=\"768\" data-end=\"906\">\n<p data-start=\"770\" data-end=\"906\">Reflections: flip the graph.<br data-start=\"802\" data-end=\"805\" \/>[latex]-f(x)[\/latex] reflects across the x-axis; [latex]f(-x)[\/latex] reflects across the y-axis.<\/p>\n<\/li>\n<li data-start=\"907\" data-end=\"1084\">\n<p data-start=\"909\" data-end=\"1084\">Stretches and compressions: make the graph taller or wider.<br data-start=\"972\" data-end=\"975\" \/>[latex]a f(x)[\/latex] vertically stretches if [latex]|a| > 1[\/latex], compresses if [latex]|a| < 1[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1370\" data-end=\"1458\">Given [latex]f(x) = |x|[\/latex], describe how to get [latex]g(x) = 2|x - 4| - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q301325\">Show Solution<\/button><\/p>\n<div id=\"q301325\" class=\"hidden-answer\" style=\"display: none\">The function has been stretched vertically by a factor of [latex]2[\/latex], then shifted to the right [latex]4[\/latex] units and down [latex]3[\/latex] units.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313580\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313580&theme=lumen&iframe_resize_id=ohm313580&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Transformation inside the parentheses cause horizontal transformations.Transformations outside the parentheses cause vertical transformations.<\/section>\n","protected":false},"author":67,"menu_order":2,"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\/2723"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2723\/revisions"}],"predecessor-version":[{"id":4651,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2723\/revisions\/4651"}],"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\/2723\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2723"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2723"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2723"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}