{"id":2850,"date":"2023-05-16T13:49:24","date_gmt":"2023-05-16T13:49:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2850"},"modified":"2025-08-29T04:35:19","modified_gmt":"2025-08-29T04:35:19","slug":"graph-theory-background-youll-need-part-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/graph-theory-background-youll-need-part-2\/","title":{"raw":"Graph Theory: Background You'll Need 2","rendered":"Graph Theory: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Plot points and read graphs on a grid&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:5057,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;10&quot;:2,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Arial&quot;}\">Plot points on a grid<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Plot Points<\/h2>\r\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the [latex]x[\/latex]-axis and the[latex]y[\/latex]-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure 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\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/> Figure 1. The perpendicular x-axis and y-axis in the Cartesian coordinate system creates four quadrants[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex].<\/p>\r\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of [latex]x[\/latex] and [latex]y[\/latex]-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.<\/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\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/> Figure 2. The coordinate plane with the coordinates (3,-1) highlighted[\/caption]\r\n<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Cartesian coordinate system<\/h3>\r\n\r\nA two-dimensional plane where the\r\n\r\n<ul>\r\n\t<li>[latex]x[\/latex]-axis is the horizontal axis<\/li>\r\n\t<li>[latex]y[\/latex]-axis is the vertical axis<\/li>\r\n<\/ul>\r\n\r\nA point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that [latex]x[\/latex] is determined by its horizontal distance from the origin and [latex]y[\/latex] is determined by its vertical distance from the origin.<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane. [reveal-answer q=\"380739\"]Show Solution[\/reveal-answer] [hidden-answer a=\"380739\"] To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The [latex]x[\/latex]-coordinate is [latex]\u20132[\/latex], so move two units to the left. The [latex]y[\/latex]-coordinate is [latex]4[\/latex], so then move four units up in the positive [latex]y[\/latex] direction. To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The [latex]x[\/latex]-coordinate is [latex]3[\/latex], so move three units to the right. The [latex]y[\/latex]-coordinate is also [latex]3[\/latex], so move three units up in the positive [latex]y[\/latex] direction. To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The [latex]x[\/latex]-coordinate is [latex]0[\/latex]. This tells us not to move in either direction along the [latex]x[\/latex]-axis. The [latex]y[\/latex]-coordinate is [latex]\u20133[\/latex], so move three units down in the negative [latex]y[\/latex] direction.<center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<strong>Analysis of the Solution<\/strong> Note that when either coordinate is zero, the point must be on an axis. If the [latex]x[\/latex]-coordinate is zero, the point is on the [latex]y[\/latex]-axis. If the [latex]y[\/latex]-coordinate is zero, the point is on the [latex]x[\/latex]-axis. [\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]6199[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Plot points and read graphs on a grid&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:5057,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;10&quot;:2,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Arial&quot;}\">Plot points on a grid<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Plot Points<\/h2>\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the [latex]x[\/latex]-axis and the[latex]y[\/latex]-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure 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\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\">Figure 1. The perpendicular x-axis and y-axis in the Cartesian coordinate system creates four quadrants<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of [latex]x[\/latex] and [latex]y[\/latex]-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.<\/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\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\">Figure 2. The coordinate plane with the coordinates (3,-1) highlighted<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Cartesian coordinate system<\/h3>\n<p>A two-dimensional plane where the<\/p>\n<ul>\n<li>[latex]x[\/latex]-axis is the horizontal axis<\/li>\n<li>[latex]y[\/latex]-axis is the vertical axis<\/li>\n<\/ul>\n<p>A point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that [latex]x[\/latex] is determined by its horizontal distance from the origin and [latex]y[\/latex] is determined by its vertical distance from the origin.<\/p><\/div>\n<\/section>\n<section class=\"textbox example\">Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q380739\">Show Solution<\/button> <\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\"> To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The [latex]x[\/latex]-coordinate is [latex]\u20132[\/latex], so move two units to the left. The [latex]y[\/latex]-coordinate is [latex]4[\/latex], so then move four units up in the positive [latex]y[\/latex] direction. To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The [latex]x[\/latex]-coordinate is [latex]3[\/latex], so move three units to the right. The [latex]y[\/latex]-coordinate is also [latex]3[\/latex], so move three units up in the positive [latex]y[\/latex] direction. To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The [latex]x[\/latex]-coordinate is [latex]0[\/latex]. This tells us not to move in either direction along the [latex]x[\/latex]-axis. The [latex]y[\/latex]-coordinate is [latex]\u20133[\/latex], so move three units down in the negative [latex]y[\/latex] direction.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/><\/div>\n<p>&nbsp;<\/p>\n<p><strong>Analysis of the Solution<\/strong> Note that when either coordinate is zero, the point must be on an axis. If the [latex]x[\/latex]-coordinate is zero, the point is on the [latex]y[\/latex]-axis. If the [latex]y[\/latex]-coordinate is zero, the point is on the [latex]x[\/latex]-axis. <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm6199\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6199&theme=lumen&iframe_resize_id=ohm6199&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"background_you_need","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc-attribution","description":"Prealgebra","author":"","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757"}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2850"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2850\/revisions"}],"predecessor-version":[{"id":15918,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2850\/revisions\/15918"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2850\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2850"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2850"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2850"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}