{"id":1699,"date":"2024-04-24T16:58:15","date_gmt":"2024-04-24T16:58:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1699"},"modified":"2025-08-17T23:04:34","modified_gmt":"2025-08-17T23:04:34","slug":"analytical-applications-of-derivatives-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/analytical-applications-of-derivatives-background-youll-need-1\/","title":{"raw":"Analytical Applications of Derivatives: Background You'll Need 1","rendered":"Analytical Applications of Derivatives: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize key features of a parabola's graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Key Features of a Parabola's Graph<\/h2>\r\n<p>The graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. Some key features of a parabola are:<\/p>\r\n<ul>\r\n\t<li><strong>Vertex<\/strong>: The extreme point of the parabola.\r\n\r\n<ul>\r\n\t<li><strong>Minimum Value<\/strong>: Vertex is the lowest point (parabola opens upwards).<\/li>\r\n\t<li><strong>Maximum Value<\/strong>: Vertex is the highest point (parabola opens downwards).<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Axis of Symmetry<\/strong>: A vertical line through the vertex dividing the parabola into two mirror images.<\/li>\r\n\t<li><strong>Intercepts<\/strong>:\r\n\r\n<ul>\r\n\t<li><strong>Y-intercept<\/strong>: Point where the graph crosses the [latex]y[\/latex]-axis.\u00a0<\/li>\r\n\t<li><strong>X-intercepts<\/strong>: Points where the graph crosses the [latex]x[\/latex]-axis (zeros or roots of the quadratic function).\u00a0<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/> Graph of a parabola with key features labeled[\/caption]\r\n<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>key features of a parabola<\/h3>\r\n<ul>\r\n\t<li><strong>Vertex:<\/strong> The highest or lowest point, indicating a maximum or minimum value.<\/li>\r\n\t<li><strong>Axis of Symmetry:<\/strong> A vertical line that divides the parabola into two symmetric parts.<\/li>\r\n\t<li><strong>Intercepts:<\/strong>\r\n<ul>\r\n\t<li><strong>Y-intercept:<\/strong> The point where the parabola crosses the [latex]y[\/latex]-axis.<\/li>\r\n\t<li><strong>X-intercepts:<\/strong> The points where the parabola crosses the [latex]x[\/latex]-axis, representing the roots of the quadratic equation.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>The places where a function's graph crosses the horizontal axis are the places where the function value equals zero. These values are called meaning names such as <em>horizontal intercepts<\/em>, [latex]x[\/latex]<em>-intercepts<\/em>, or the <em>zeros<\/em> of the graph. They can also be referred to as the <em>roots<\/em> of a function.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/> Graph of a parabola[\/caption]\r\n<\/center>\r\n<p>[reveal-answer q=\"366804\"]Show Solution[\/reveal-answer]<\/p>\r\n<p>[hidden-answer a=\"366804\"]<\/p>\r\n<p>The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287786[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize key features of a parabola&#8217;s graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\n<\/ul>\n<\/section>\n<h2>Key Features of a Parabola&#8217;s Graph<\/h2>\n<p>The graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. Some key features of a parabola are:<\/p>\n<ul>\n<li><strong>Vertex<\/strong>: The extreme point of the parabola.\n<ul>\n<li><strong>Minimum Value<\/strong>: Vertex is the lowest point (parabola opens upwards).<\/li>\n<li><strong>Maximum Value<\/strong>: Vertex is the highest point (parabola opens downwards).<\/li>\n<\/ul>\n<\/li>\n<li><strong>Axis of Symmetry<\/strong>: A vertical line through the vertex dividing the parabola into two mirror images.<\/li>\n<li><strong>Intercepts<\/strong>:\n<ul>\n<li><strong>Y-intercept<\/strong>: Point where the graph crosses the [latex]y[\/latex]-axis.\u00a0<\/li>\n<li><strong>X-intercepts<\/strong>: Points where the graph crosses the [latex]x[\/latex]-axis (zeros or roots of the quadratic function).\u00a0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/><figcaption class=\"wp-caption-text\">Graph of a parabola with key features labeled<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>key features of a parabola<\/h3>\n<ul>\n<li><strong>Vertex:<\/strong> The highest or lowest point, indicating a maximum or minimum value.<\/li>\n<li><strong>Axis of Symmetry:<\/strong> A vertical line that divides the parabola into two symmetric parts.<\/li>\n<li><strong>Intercepts:<\/strong>\n<ul>\n<li><strong>Y-intercept:<\/strong> The point where the parabola crosses the [latex]y[\/latex]-axis.<\/li>\n<li><strong>X-intercepts:<\/strong> The points where the parabola crosses the [latex]x[\/latex]-axis, representing the roots of the quadratic equation.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>The places where a function&#8217;s graph crosses the horizontal axis are the places where the function value equals zero. These values are called meaning names such as <em>horizontal intercepts<\/em>, [latex]x[\/latex]<em>-intercepts<\/em>, or the <em>zeros<\/em> of the graph. They can also be referred to as the <em>roots<\/em> of a function.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/><figcaption class=\"wp-caption-text\">Graph of a parabola<\/figcaption><\/figure>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q366804\">Show Solution<\/button><\/p>\n<div id=\"q366804\" class=\"hidden-answer\" style=\"display: none\">\n<p>The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm287786\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287786&theme=lumen&iframe_resize_id=ohm287786&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"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":652,"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1699"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1699\/revisions"}],"predecessor-version":[{"id":4768,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1699\/revisions\/4768"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1699\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1699"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1699"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1699"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1699"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}