{"id":973,"date":"2024-05-01T18:27:54","date_gmt":"2024-05-01T18:27:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=973"},"modified":"2025-08-21T23:08:26","modified_gmt":"2025-08-21T23:08:26","slug":"graphing-and-analyzing-linear-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphing-and-analyzing-linear-functions-learn-it-1\/","title":{"raw":"Graphing and Analyzing Linear Equations: Learn It 1","rendered":"Graphing and Analyzing Linear Equations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Learn to plot points and draw lines on a graph using a set of coordinates.<\/li>\r\n \t<li>Find the x-intercept and y-intercept of graphs.<\/li>\r\n \t<li>Determine the slope based on the steepness and direction of a line.<\/li>\r\n \t<li>Use formulas to calculate the distances and midpoints between points.<\/li>\r\n<\/ul>\r\n<\/section>An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.\r\n\r\nWhile there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.\r\n\r\nThe 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]293759[\/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\">[ohm2_question hide_question_numbers=1]18915[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Learn to plot points and draw lines on a graph using a set of coordinates.<\/li>\n<li>Find the x-intercept and y-intercept of graphs.<\/li>\n<li>Determine the slope based on the steepness and direction of a line.<\/li>\n<li>Use formulas to calculate the distances and midpoints between points.<\/li>\n<\/ul>\n<\/section>\n<p>An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p>While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.<\/p>\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\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.jpg 418w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-300x266.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-65x58.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-225x199.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-350x310.jpg 350w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><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\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948.jpg 357w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948-288x300.jpg 288w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948-65x68.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948-225x234.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140259\/1707ad850f34971da177bf292fbb51e8f2128948-350x365.jpg 350w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><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=\"ohm293759\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=293759&theme=lumen&iframe_resize_id=ohm293759&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=\"ohm18915\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18915&theme=lumen&iframe_resize_id=ohm18915&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"learn_it","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/973"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/973\/revisions"}],"predecessor-version":[{"id":7966,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/973\/revisions\/7966"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/973\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=973"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=973"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=973"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}