{"id":469,"date":"2025-07-10T16:41:27","date_gmt":"2025-07-10T16:41:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=469"},"modified":"2025-12-17T15:39:24","modified_gmt":"2025-12-17T15:39:24","slug":"introduction-to-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-functions-background-youll-need-2\/","title":{"raw":"Introduction to Functions: Background You'll Need 2","rendered":"Introduction to Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the slope between two points<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>slope formula<\/h3>\r\nSlope measures how steep a line is by comparing the vertical change (rise) to the horizontal change (run) between two points. The slope formula is:\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\">To find the slope between two points:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the coordinates of both points: [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute the values into the slope formula: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the numerator: [latex]y_2 - y_1[\/latex] (vertical change)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the denominator: [latex]x_2 - x_1[\/latex] (horizontal change)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the fraction if possible<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Find the slope between the points [latex](2, 3)[\/latex] and [latex](6, 11)[\/latex].<\/p>\r\n[reveal-answer q=\"150200\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"150200\"]Solution: Let [latex](x_1, y_1) = (2, 3)[\/latex] and [latex](x_2, y_2) = (6, 11)[\/latex] Using the slope formula: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{11 - 3}{6 - 2} = \\frac{8}{4} = 2[\/latex]\r\n\r\nThe slope is 2.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">It doesn't matter which point you choose as [latex](x_1, y_1)[\/latex] and which as [latex](x_2, y_2)[\/latex]. Just be consistent with your choice throughout the calculation. If you switch the order, you'll get the same slope value.<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]317454[\/ohm_question]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the slope between two points<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>slope formula<\/h3>\n<p>Slope measures how steep a line is by comparing the vertical change (rise) to the horizontal change (run) between two points. The slope formula is:<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\">To find the slope between two points:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Identify the coordinates of both points: [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute the values into the slope formula: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the numerator: [latex]y_2 - y_1[\/latex] (vertical change)<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the denominator: [latex]x_2 - x_1[\/latex] (horizontal change)<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the fraction if possible<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Find the slope between the points [latex](2, 3)[\/latex] and [latex](6, 11)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q150200\">Solution<\/button><\/p>\n<div id=\"q150200\" class=\"hidden-answer\" style=\"display: none\">Solution: Let [latex](x_1, y_1) = (2, 3)[\/latex] and [latex](x_2, y_2) = (6, 11)[\/latex] Using the slope formula: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{11 - 3}{6 - 2} = \\frac{8}{4} = 2[\/latex]<\/p>\n<p>The slope is 2.<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">It doesn&#8217;t matter which point you choose as [latex](x_1, y_1)[\/latex] and which as [latex](x_2, y_2)[\/latex]. Just be consistent with your choice throughout the calculation. If you switch the order, you&#8217;ll get the same slope value.<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm317454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317454&theme=lumen&iframe_resize_id=ohm317454&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":13,"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":36,"module-header":"- Select Header -","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\/469"}],"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\/13"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/469\/revisions"}],"predecessor-version":[{"id":5115,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/469\/revisions\/5115"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/469\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=469"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=469"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=469"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}