{"id":1002,"date":"2023-07-14T15:36:07","date_gmt":"2023-07-14T15:36:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=1002"},"modified":"2023-08-10T22:15:49","modified_gmt":"2023-08-10T22:15:49","slug":"understanding-a-graph-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/understanding-a-graph-of-a-function\/","title":{"raw":"\u25aa   Understanding a Graph of a Function","rendered":"\u25aa   Understanding a Graph of a Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Relate the solutions of an equation to the [latex]x[\/latex]-intercepts of its function.<\/li>\r\n \t<li>Recognize distinct parts of a graph of a function: below\u00a0the [latex]x[\/latex]-axis, on\u00a0the [latex]x[\/latex]-axis, and above\u00a0the [latex]x[\/latex]-axis.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>[latex]x[\/latex]-intercepts of a Function and Solutions of an Equation<\/h2>\r\nHow can we find the [latex]x[\/latex]-intercepts (or real zeros) of a function [latex]y=f(x)[\/latex]? First, we need to set [latex]y=0[\/latex] because all [latex]x[\/latex]-intercepts are on the [latex]x[\/latex]-axis and any points on the [latex]x[\/latex]-axis have zero for its [latex]y[\/latex]-coordinate. Then, we need to solve the equation [latex]f(x)=0[\/latex]. For example, to find the\u00a0[latex]x[\/latex]-intercepts (or real zeros) of a function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex],\u00a0we need to solve the equation [latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex]. From this equation, we can find [latex]x=-3, 0, 5[\/latex] as its solutions and can write them as [latex](-3, 0)[\/latex], [latex](0, 0)[\/latex], and [latex](5, 0)[\/latex] because those solutions are the [latex]x[\/latex] values when its [latex]y[\/latex] value is zero.\u00a0So, we can conclude that the solutions of the equation\u00a0[latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex] is actually the\u00a0[latex]x[\/latex]-intercepts (or real zeros) of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex]. We can confirm this relation from the graph of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex] as well.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"280\"]<img class=\"aligncenter wp-image-997 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2.png\" alt=\"A graph of y=1\/7x(x+3)(x-5)\" width=\"280\" height=\"280\" \/> Figure 2. Graph of [latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex][\/caption]In Figure 2, we can find the solutions of the equation\u00a0[latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex] by locating the [latex]x[\/latex]-intercepts of the graph of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex].\r\n<div class=\"textbox\">\r\n<h3>General Note: Solutions of an Equation and [latex]x[\/latex]-intercepts of its Function<\/h3>\r\nThe solutions of an equation [latex]f(x)=0[\/latex] are the [latex]x[\/latex]-intercepts of the function [latex]y=f(x)[\/latex].\r\n\r\n<\/div>\r\n<h2>Distinct Parts of a Graph of a Function<\/h2>\r\nNow let's consider more points on the graph. As we can see in Figure 2, some parts of the graph are above the [latex]x[\/latex]-axis, some parts of the graph are on\u00a0the [latex]x[\/latex]-axis, and some parts of the graph are below\u00a0the [latex]x[\/latex]-axis. Try the following DESMOS activity to explore the relationship between those distinct parts of the function and the sign of its [latex]y[\/latex] values:\r\n<div class=\"textbox tryit\">\r\n<h3>DESMOS Activity<\/h3>\r\n<a href=\"https:\/\/www.desmos.com\/calculator\/8hvobsa4n9\" target=\"_blank\" rel=\"noopener\">Y Values of a Function on its Graph<\/a>\u00a0or <a href=\"https:\/\/student.desmos.com\/activitybuilder\/student-greeting\/64b1c1e5d012890e719d5bd2\" target=\"_blank\" rel=\"noopener\">Y Values of a Function on its Graph<\/a>\r\n\r\n<\/div>\r\nFrom the DESMOS activity above, we can conclude the followings:\r\n<div class=\"textbox\">\r\n<h3>General Note: Distinct Parts of a Graph of a Function and Their [latex]y[\/latex] Values<\/h3>\r\n(a) When a graph is below the [latex]x[\/latex]-axis, [latex]y[\/latex] values are negative. So, [latex]f(x)&lt;0[\/latex].\r\n\r\n(b) When a graph is on the [latex]x[\/latex]-axis, [latex]y[\/latex] values are zero. So, [latex]f(x)=0[\/latex].\r\n\r\n(c) When a graph is above the [latex]x[\/latex]-axis, [latex]y[\/latex] values are positive. So, [latex]f(x)&gt;0[\/latex].\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Relate the solutions of an equation to the [latex]x[\/latex]-intercepts of its function.<\/li>\n<li>Recognize distinct parts of a graph of a function: below\u00a0the [latex]x[\/latex]-axis, on\u00a0the [latex]x[\/latex]-axis, and above\u00a0the [latex]x[\/latex]-axis.<\/li>\n<\/ul>\n<\/div>\n<h2>[latex]x[\/latex]-intercepts of a Function and Solutions of an Equation<\/h2>\n<p>How can we find the [latex]x[\/latex]-intercepts (or real zeros) of a function [latex]y=f(x)[\/latex]? First, we need to set [latex]y=0[\/latex] because all [latex]x[\/latex]-intercepts are on the [latex]x[\/latex]-axis and any points on the [latex]x[\/latex]-axis have zero for its [latex]y[\/latex]-coordinate. Then, we need to solve the equation [latex]f(x)=0[\/latex]. For example, to find the\u00a0[latex]x[\/latex]-intercepts (or real zeros) of a function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex],\u00a0we need to solve the equation [latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex]. From this equation, we can find [latex]x=-3, 0, 5[\/latex] as its solutions and can write them as [latex](-3, 0)[\/latex], [latex](0, 0)[\/latex], and [latex](5, 0)[\/latex] because those solutions are the [latex]x[\/latex] values when its [latex]y[\/latex] value is zero.\u00a0So, we can conclude that the solutions of the equation\u00a0[latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex] is actually the\u00a0[latex]x[\/latex]-intercepts (or real zeros) of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex]. We can confirm this relation from the graph of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex] as well.<\/p>\n<div style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-997\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2.png\" alt=\"A graph of y=1\/7x(x+3)(x-5)\" width=\"280\" height=\"280\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2.png 395w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2-225x225.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.5-Function-Equation-Inequality_Graph2-350x350.png 350w\" sizes=\"auto, (max-width: 280px) 100vw, 280px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Graph of [latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex]<\/p>\n<\/div>\n<p>In Figure 2, we can find the solutions of the equation\u00a0[latex]\\frac{1}{7}x(x+3)(x-5)=0[\/latex] by locating the [latex]x[\/latex]-intercepts of the graph of the function\u00a0[latex]f(x)=\\frac{1}{7}x(x+3)(x-5)[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>General Note: Solutions of an Equation and [latex]x[\/latex]-intercepts of its Function<\/h3>\n<p>The solutions of an equation [latex]f(x)=0[\/latex] are the [latex]x[\/latex]-intercepts of the function [latex]y=f(x)[\/latex].<\/p>\n<\/div>\n<h2>Distinct Parts of a Graph of a Function<\/h2>\n<p>Now let&#8217;s consider more points on the graph. As we can see in Figure 2, some parts of the graph are above the [latex]x[\/latex]-axis, some parts of the graph are on\u00a0the [latex]x[\/latex]-axis, and some parts of the graph are below\u00a0the [latex]x[\/latex]-axis. Try the following DESMOS activity to explore the relationship between those distinct parts of the function and the sign of its [latex]y[\/latex] values:<\/p>\n<div class=\"textbox tryit\">\n<h3>DESMOS Activity<\/h3>\n<p><a href=\"https:\/\/www.desmos.com\/calculator\/8hvobsa4n9\" target=\"_blank\" rel=\"noopener\">Y Values of a Function on its Graph<\/a>\u00a0or <a href=\"https:\/\/student.desmos.com\/activitybuilder\/student-greeting\/64b1c1e5d012890e719d5bd2\" target=\"_blank\" rel=\"noopener\">Y Values of a Function on its Graph<\/a><\/p>\n<\/div>\n<p>From the DESMOS activity above, we can conclude the followings:<\/p>\n<div class=\"textbox\">\n<h3>General Note: Distinct Parts of a Graph of a Function and Their [latex]y[\/latex] Values<\/h3>\n<p>(a) When a graph is below the [latex]x[\/latex]-axis, [latex]y[\/latex] values are negative. So, [latex]f(x)<0[\/latex].\n\n(b) When a graph is on the [latex]x[\/latex]-axis, [latex]y[\/latex] values are zero. So, [latex]f(x)=0[\/latex].\n\n(c) When a graph is above the [latex]x[\/latex]-axis, [latex]y[\/latex] values are positive. So, [latex]f(x)>0[\/latex].<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1002\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Understanding a Graph of a Function. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":705214,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Understanding a Graph of a Function\",\"author\":\"Michelle Eunhee Chung\",\"organization\":\"Georgia State University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mchung12"],"pb_section_license":""},"chapter-type":[],"contributor":[62],"license":[],"class_list":["post-1002","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1002","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/705214"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1002\/revisions"}],"predecessor-version":[{"id":1036,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1002\/revisions\/1036"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/91"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1002\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=1002"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1002"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1002"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=1002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}