{"id":3712,"date":"2023-05-26T16:36:18","date_gmt":"2023-05-26T16:36:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3712"},"modified":"2025-08-29T04:11:43","modified_gmt":"2025-08-29T04:11:43","slug":"complex-graphical-analysis-and-the-limits-of-modeling-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/complex-graphical-analysis-and-the-limits-of-modeling-background-youll-need-1\/","title":{"raw":"Advanced Data Interpretation: Background You\u2019ll Need 1","rendered":"Advanced Data Interpretation: Background You\u2019ll Need 1"},"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;Determine the exact location of a point on a graph&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6657,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Determine the exact location of a point on a graph<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The <strong>rectangular coordinate system<\/strong> is also called the [latex]x\\text{-}y[\/latex] plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named Ren\u00e9 Descartes.)<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the rectangular coordinate system<\/h3>\r\n\r\nThe rectangular coordinate system is a two-dimensional plane defined by a pair of perpendicular axes, usually labeled [latex]x[\/latex] (horizontal) and [latex]y[\/latex] (vertical), used to plot points, lines, and curves by assigning them coordinates based on their distance from these axes.\r\n\r\n<p>&nbsp;<\/p>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224713\/CNX_BMath_Figure_11_01_002.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. An arrow points to the horizontal axis with the label \" \/><\/center><\/section>\r\n<p>In the rectangular coordinate system, every point is represented by an <strong>ordered pair<\/strong>. The first number in the ordered pair is the [latex]x[\/latex]-coordinate of the point, and the second number is the [latex]y[\/latex]-coordinate of the point.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Ordered Pair<\/h3>\r\n<p>An ordered pair, [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point in a rectangular coordinate system.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>[latex]\\begin{array}{c}\\text{The first number is the }x\\text{-coordinate}.\\hfill \\\\ \\text{The second number is the }y\\text{-coordinate}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224718\/CNX_BMath_Figure_11_01_027_img.png\" alt=\"The ordered pair x y is labeled with the first coordinate x labeled as \" \/><\/center><\/section>\r\n<p>So how do the coordinates of a point help you locate a point on the [latex]x\\text{-}y[\/latex] plane?<\/p>\r\n<p>Let\u2019s try locating the point [latex]\\left(2,5\\right)[\/latex] . In this ordered pair, the [latex]x[\/latex] -coordinate is [latex]2[\/latex] and the [latex]y[\/latex] -coordinate is [latex]5[\/latex] .<\/p>\r\n<p>We start by locating the [latex]x[\/latex] value, [latex]2[\/latex], on the [latex]x\\text{-axis.}[\/latex] Then we lightly sketch a vertical line through [latex]x=2[\/latex], as shown in the image below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"308\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224719\/CNX_BMath_Figure_11_01_004.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.\" width=\"308\" height=\"321\" \/> Figure 1. Locate 2 on the x-axis and sketch a vertical line through it[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Now we locate the [latex]y[\/latex] value, [latex]5[\/latex], on the [latex]y[\/latex] -axis and sketch a horizontal line through [latex]y=5[\/latex] . The point where these two lines meet is the point with coordinates [latex]\\left(2,5\\right)[\/latex]. We plot the point there, as shown in the image below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"301\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224721\/CNX_BMath_Figure_11_01_005.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. An arrow starts at the end of the first arrow at 2 on the x-axis and goes vertically 5 units to a point labeled (2, 5)\" width=\"301\" height=\"308\" \/> Figure 2. Find 5 on the y-axis and sketch a horizontal line. The coordinates (2,5) are where the lines intersect[\/caption]\r\n<\/center>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6974[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine the exact location of a point on a graph&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6657,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Determine the exact location of a point on a graph<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The <strong>rectangular coordinate system<\/strong> is also called the [latex]x\\text{-}y[\/latex] plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named Ren\u00e9 Descartes.)<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>the rectangular coordinate system<\/h3>\n<p>The rectangular coordinate system is a two-dimensional plane defined by a pair of perpendicular axes, usually labeled [latex]x[\/latex] (horizontal) and [latex]y[\/latex] (vertical), used to plot points, lines, and curves by assigning them coordinates based on their distance from these axes.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224713\/CNX_BMath_Figure_11_01_002.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. An arrow points to the horizontal axis with the label\" \/><\/div>\n<\/section>\n<p>In the rectangular coordinate system, every point is represented by an <strong>ordered pair<\/strong>. The first number in the ordered pair is the [latex]x[\/latex]-coordinate of the point, and the second number is the [latex]y[\/latex]-coordinate of the point.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Ordered Pair<\/h3>\n<p>An ordered pair, [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point in a rectangular coordinate system.<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\\begin{array}{c}\\text{The first number is the }x\\text{-coordinate}.\\hfill \\\\ \\text{The second number is the }y\\text{-coordinate}.\\hfill \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224718\/CNX_BMath_Figure_11_01_027_img.png\" alt=\"The ordered pair x y is labeled with the first coordinate x labeled as\" \/><\/div>\n<\/section>\n<p>So how do the coordinates of a point help you locate a point on the [latex]x\\text{-}y[\/latex] plane?<\/p>\n<p>Let\u2019s try locating the point [latex]\\left(2,5\\right)[\/latex] . In this ordered pair, the [latex]x[\/latex] -coordinate is [latex]2[\/latex] and the [latex]y[\/latex] -coordinate is [latex]5[\/latex] .<\/p>\n<p>We start by locating the [latex]x[\/latex] value, [latex]2[\/latex], on the [latex]x\\text{-axis.}[\/latex] Then we lightly sketch a vertical line through [latex]x=2[\/latex], as shown in the image below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224719\/CNX_BMath_Figure_11_01_004.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.\" width=\"308\" height=\"321\" \/><figcaption class=\"wp-caption-text\">Figure 1. Locate 2 on the x-axis and sketch a vertical line through it<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now we locate the [latex]y[\/latex] value, [latex]5[\/latex], on the [latex]y[\/latex] -axis and sketch a horizontal line through [latex]y=5[\/latex] . The point where these two lines meet is the point with coordinates [latex]\\left(2,5\\right)[\/latex]. We plot the point there, as shown in the image below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 301px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224721\/CNX_BMath_Figure_11_01_005.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. An arrow starts at the end of the first arrow at 2 on the x-axis and goes vertically 5 units to a point labeled (2, 5)\" width=\"301\" height=\"308\" \/><figcaption class=\"wp-caption-text\">Figure 2. Find 5 on the y-axis and sketch a horizontal line. The coordinates (2,5) are where the lines intersect<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6974\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6974&theme=lumen&iframe_resize_id=ohm6974&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"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":88,"module-header":"background_you_need","content_attributions":[{"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\/3712"}],"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":17,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3712\/revisions"}],"predecessor-version":[{"id":15906,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3712\/revisions\/15906"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/88"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3712\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3712"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3712"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3712"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3712"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}