{"id":491,"date":"2025-07-10T17:44:32","date_gmt":"2025-07-10T17:44:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=491"},"modified":"2026-01-07T22:37:35","modified_gmt":"2026-01-07T22:37:35","slug":"working-with-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/working-with-functions-background-youll-need-2\/","title":{"raw":"Working with Functions: Background You'll Need 2","rendered":"Working with Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Graph points on a coordinate plane<\/span><\/li>\r\n<\/ul>\r\n<\/section>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis.\r\n\r\nPerpendicular 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.\r\n\r\n[caption id=\"attachment_3322\" align=\"aligncenter\" width=\"418\"]<img class=\"wp-image-3322 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.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=\"418\" height=\"370\" \/> coordinate plane with labels for each quadrant[\/caption]\r\n\r\nThe 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]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.\r\n\r\n[caption id=\"attachment_3324\" align=\"aligncenter\" width=\"357\"]<img class=\"wp-image-3324 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. \" width=\"357\" height=\"372\" \/> x,y, coordinate plane ranging from -5 to 5[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Cartesian coordinate system<\/h3>\r\n<p id=\"fs-id1400039\">A two-dimensional plane where the<\/p>\r\n\r\n<ul id=\"fs-id573737\">\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<p id=\"fs-id3085633\">A point in the plane is defined as an ordered pair,\u00a0[latex](x,y)[\/latex],\u00a0such that\u00a0[latex]x[\/latex] is determined by its horizontal distance from the origin and\u00a0[latex]y[\/latex] is determined by its vertical distance from the origin.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317890[\/ohm_question]<\/section>\r\n<h2>Plotting Points<\/h2>\r\nEach 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].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">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.\r\n\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\" \/> <b>An illustration of how to plot the point (3,-1).<\/b>[\/caption]\r\n\r\n<\/section><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=\"923766\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"923766\"]\r\n<ul>\r\n \t<li>To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is [latex]\u20132[\/latex], so move two units to the left. The <em>y<\/em>-coordinate is [latex]4[\/latex], so then move four units up in the positive <em>y <\/em>direction.<\/li>\r\n \t<li>To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is [latex]3[\/latex], so move three units to the right. The <em>y<\/em>-coordinate is also [latex]3[\/latex], so move three units up in the positive <em>y <\/em>direction.<\/li>\r\n \t<li>To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is [latex]0[\/latex]. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is [latex]\u20133[\/latex], so move three units down in the negative <em>y<\/em> direction.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"272\"]<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=\"272\" height=\"247\" \/> Graph showing how to plot (-2, 4), (3,3), and (0,-3)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317891[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Graph points on a coordinate plane<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis.<\/p>\n<p>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<figure id=\"attachment_3322\" aria-describedby=\"caption-attachment-3322\" style=\"width: 418px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3322 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.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=\"418\" height=\"370\" \/><figcaption id=\"caption-attachment-3322\" class=\"wp-caption-text\">coordinate plane with labels for each quadrant<\/figcaption><\/figure>\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]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.<\/p>\n<figure id=\"attachment_3324\" aria-describedby=\"caption-attachment-3324\" style=\"width: 357px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3324 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"357\" height=\"372\" \/><figcaption id=\"caption-attachment-3324\" class=\"wp-caption-text\">x,y, coordinate plane ranging from -5 to 5<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3>Cartesian coordinate system<\/h3>\n<p id=\"fs-id1400039\">A two-dimensional plane where the<\/p>\n<ul id=\"fs-id573737\">\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 id=\"fs-id3085633\">A point in the plane is defined as an ordered pair,\u00a0[latex](x,y)[\/latex],\u00a0such that\u00a0[latex]x[\/latex] is determined by its horizontal distance from the origin and\u00a0[latex]y[\/latex] is determined by its vertical distance from the origin.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317890\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317890&theme=lumen&iframe_resize_id=ohm317890&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Plotting Points<\/h2>\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].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">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<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\"><b>An illustration of how to plot the point (3,-1).<\/b><\/figcaption><\/figure>\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=\"q923766\">Show Answer<\/button><\/p>\n<div id=\"q923766\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is [latex]\u20132[\/latex], so move two units to the left. The <em>y<\/em>-coordinate is [latex]4[\/latex], so then move four units up in the positive <em>y <\/em>direction.<\/li>\n<li>To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is [latex]3[\/latex], so move three units to the right. The <em>y<\/em>-coordinate is also [latex]3[\/latex], so move three units up in the positive <em>y <\/em>direction.<\/li>\n<li>To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is [latex]0[\/latex]. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is [latex]\u20133[\/latex], so move three units down in the negative <em>y<\/em> direction.<\/li>\n<\/ul>\n<figure style=\"width: 272px\" 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\/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=\"272\" height=\"247\" \/><figcaption class=\"wp-caption-text\">Graph showing how to plot (-2, 4), (3,3), and (0,-3)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317891\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317891&theme=lumen&iframe_resize_id=ohm317891&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":498,"module-header":"- Select Header -","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/491"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/491\/revisions"}],"predecessor-version":[{"id":5228,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/491\/revisions\/5228"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/498"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/491\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=491"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=491"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=491"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}