{"id":3719,"date":"2023-05-26T17:14:41","date_gmt":"2023-05-26T17:14:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3719"},"modified":"2024-10-18T20:57:42","modified_gmt":"2024-10-18T20:57:42","slug":"complex-graphical-analysis-and-the-limits-of-modeling-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/complex-graphical-analysis-and-the-limits-of-modeling-background-youll-need-2\/","title":{"raw":"Advanced Data Interpretation: Background You\u2019ll Need 2","rendered":"Advanced Data Interpretation: Background You\u2019ll 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;Determine the region a point is 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 region a point is on a graph<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis form the rectangular coordinate system. These axes divide a plane into four areas, called <strong>quadrants<\/strong>. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See the image below.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the four quadrants of the rectangular coordinate system<\/h3>\r\n\r\nIn the rectangular coordinate system, the four <strong>quadrants <\/strong>are defined by the intersection of the [latex]x[\/latex] (horizontal) and [latex]y[\/latex] (vertical) axes, each representing a distinct combination of positive and negative values:\r\n\r\n<ul>\r\n\t<li>Quadrant I (positive [latex]x [\/latex] and [latex]y[\/latex])<\/li>\r\n\t<li>Quadrant II (negative [latex]x[\/latex], positive [latex]y[\/latex])<\/li>\r\n\t<li>Quadrant III (negative [latex]x[\/latex] and [latex]y[\/latex])<\/li>\r\n\t<li>Quadrant IV (positive [latex]x[\/latex], negative [latex]y[\/latex]).<\/li>\r\n<\/ul>\r\n\r\nThese quadrants help in categorizing the position and movement of points or graphs in the coordinate plane.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224716\/CNX_BMath_Figure_11_01_003.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled I, the top-left portion is labeled II, the bottom-left portion is labeled III, and the bottom-right portion is labeled IV. \" width=\"345\" height=\"354\" \/><\/center><\/section>\r\n<p>When we located where a specific point is on the cartesian <span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify the quadrant a point is in on the cartesian coordinate plane&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6785,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">coordinate plane we can specify which quadrant that point lies in.<\/span><\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6975[\/ohm2_question]<\/section>\r\n<p><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify the quadrant a point is in on the cartesian coordinate plane&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6785,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">A tricky situation is<\/span> when one of the coordinates are zero. Where is the point [latex]\\left(0,4\\right)[\/latex] located? Where is the point [latex]\\left(-2,0\\right)[\/latex] located? The point [latex]\\left(0,4\\right)[\/latex] is on the [latex]y[\/latex]-axis and the point [latex]\\left(-2,0\\right)[\/latex] is on the [latex]x[\/latex]-axis.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Points on the Axes<\/h3>\r\n<p>Points with a [latex]y[\/latex]-coordinate equal to [latex]0[\/latex] are on the [latex]x[\/latex]-axis, and have coordinates [latex]\\left(a,0\\right)[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Points with an [latex]x[\/latex]-coordinate equal to [latex]0[\/latex] are on the [latex]y[\/latex]-axis, and have coordinates [latex]\\left(0,b\\right)[\/latex].<\/p>\r\n<\/section>\r\n<p>What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is [latex]\\left(0,0\\right)[\/latex] . The point has a special name. It is called the <strong>origin<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the origin<\/h3>\r\n<p>The point [latex]\\left(0,0\\right)[\/latex] is called the <strong>origin<\/strong>. It is the point where the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis intersect.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6977[\/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 region a point is 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 region a point is on a graph<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis form the rectangular coordinate system. These axes divide a plane into four areas, called <strong>quadrants<\/strong>. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See the image below.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>the four quadrants of the rectangular coordinate system<\/h3>\n<p>In the rectangular coordinate system, the four <strong>quadrants <\/strong>are defined by the intersection of the [latex]x[\/latex] (horizontal) and [latex]y[\/latex] (vertical) axes, each representing a distinct combination of positive and negative values:<\/p>\n<ul>\n<li>Quadrant I (positive [latex]x[\/latex] and [latex]y[\/latex])<\/li>\n<li>Quadrant II (negative [latex]x[\/latex], positive [latex]y[\/latex])<\/li>\n<li>Quadrant III (negative [latex]x[\/latex] and [latex]y[\/latex])<\/li>\n<li>Quadrant IV (positive [latex]x[\/latex], negative [latex]y[\/latex]).<\/li>\n<\/ul>\n<p>These quadrants help in categorizing the position and movement of points or graphs in the coordinate plane.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224716\/CNX_BMath_Figure_11_01_003.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled I, the top-left portion is labeled II, the bottom-left portion is labeled III, and the bottom-right portion is labeled IV.\" width=\"345\" height=\"354\" \/><\/div>\n<\/section>\n<p>When we located where a specific point is on the cartesian <span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify the quadrant a point is in on the cartesian coordinate plane&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6785,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">coordinate plane we can specify which quadrant that point lies in.<\/span><\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6975\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6975&theme=lumen&iframe_resize_id=ohm6975&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify the quadrant a point is in on the cartesian coordinate plane&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6785,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">A tricky situation is<\/span> when one of the coordinates are zero. Where is the point [latex]\\left(0,4\\right)[\/latex] located? Where is the point [latex]\\left(-2,0\\right)[\/latex] located? The point [latex]\\left(0,4\\right)[\/latex] is on the [latex]y[\/latex]-axis and the point [latex]\\left(-2,0\\right)[\/latex] is on the [latex]x[\/latex]-axis.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Points on the Axes<\/h3>\n<p>Points with a [latex]y[\/latex]-coordinate equal to [latex]0[\/latex] are on the [latex]x[\/latex]-axis, and have coordinates [latex]\\left(a,0\\right)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Points with an [latex]x[\/latex]-coordinate equal to [latex]0[\/latex] are on the [latex]y[\/latex]-axis, and have coordinates [latex]\\left(0,b\\right)[\/latex].<\/p>\n<\/section>\n<p>What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is [latex]\\left(0,0\\right)[\/latex] . The point has a special name. It is called the <strong>origin<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>the origin<\/h3>\n<p>The point [latex]\\left(0,0\\right)[\/latex] is called the <strong>origin<\/strong>. It is the point where the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis intersect.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6977\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6977&theme=lumen&iframe_resize_id=ohm6977&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":3,"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\/3719"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3719\/revisions"}],"predecessor-version":[{"id":12789,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3719\/revisions\/12789"}],"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\/3719\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3719"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3719"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3719"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}