{"id":467,"date":"2025-07-10T16:40:57","date_gmt":"2025-07-10T16:40:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=467"},"modified":"2026-01-06T19:16:37","modified_gmt":"2026-01-06T19:16:37","slug":"introduction-to-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-functions-background-youll-need-1\/","title":{"raw":"Introduction to Functions: Background You'll Need 1","rendered":"Introduction to Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify points on a graph<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">The Components of the Coordinate Plane<\/h2>\r\nThe <strong>Cartesian coordinate system<\/strong>, refined by French mathematician Ren\u00e9 Descartes in 1637, serves as a framework for visualizing algebraic relationships. The coordinate plane can be used to plot points and graph lines.\r\n\r\nThis system is comprised of a coordinate plane, which is essentially a grid formed by a horizontal axis, known as the [latex]x[\/latex]-axis, and a vertical axis, referred to as the [latex]y[\/latex]-axis. These axes intersect perpendicularly at a point called the origin, where both [latex]x[\/latex] and [latex]y[\/latex] coordinates are zero.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Cartesian coordinate system<\/h3>\r\nIn the Cartesian coordinate system, the horizontal axis in the coordinate plane is called the <b>[latex]x[\/latex]-axis<\/b>. The vertical axis is called the <b>[latex]y[\/latex]-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at [latex]0[\/latex] on the [latex]x[\/latex]-axis and [latex]0[\/latex] on the [latex]y[\/latex]-axis.\r\n\r\n<\/div>\r\n<\/section><center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064227\/image002.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"417\" height=\"378\" \/><\/center>\r\n<h3>Understanding Quadrants<\/h3>\r\nThe coordinate plane is partitioned into four distinct regions by the [latex]x[\/latex] and [latex]y[\/latex] axes. The quadrants can be seen in the coordinate plane above. Each quadrant has different characteristics for the [latex]x[\/latex] and [latex]y[\/latex] coordinates.\r\n<ul>\r\n \t<li>Quadrant [latex]I[\/latex]: Both [latex]x[\/latex] and [latex]y[\/latex] coordinates are positive.<\/li>\r\n \t<li>Quadrant [latex]II[\/latex]: [latex]x[\/latex] is negative, while [latex]y[\/latex] is positive.<\/li>\r\n \t<li>Quadrant [latex]III[\/latex]: Both [latex]x[\/latex] and [latex]y[\/latex] coordinates are negative.<\/li>\r\n \t<li>Quadrant [latex]IV[\/latex]: [latex]x[\/latex] is positive, but [latex]y[\/latex] is negative.<\/li>\r\n<\/ul>\r\nTo ascertain the quadrant in which a point resides, one must examine the signs of its [latex]x[\/latex] and [latex]y[\/latex] coordinates.\r\n<h3>Locating Points on the Coordinate Plane<\/h3>\r\nIn the Cartesian coordinate system, locations are defined by <strong>ordered pairs.\u00a0<\/strong> An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]-axis (the first value of the ordered pair) and along the [latex]y[\/latex]-axis (the second value of the ordered pair).\r\n\r\nIn an ordered pair, such as ([latex]x ,y [\/latex]), the first value is the [latex]x[\/latex]-coordinate and the second value is the [latex]y[\/latex]-coordinate. The [latex]x[\/latex]-coordinate specifies the horizontal distance from the origin, while the [latex]y[\/latex]-coordinate indicates the vertical distance.\r\n\r\n<section class=\"textbox example\">Consider the point below.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064228\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/><\/center>To identify the location of this point, start at the origin ([latex]0, 0[\/latex]) and move right along the [latex]x[\/latex]-axis until you are under the point.\r\n[latex]\\\\[\/latex]\r\nLook at the label on the [latex]x[\/latex]-axis. The [latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]-coordinate, the first number in the ordered pair.\r\n[latex]\\\\[\/latex]\r\nFrom [latex]4[\/latex] on the [latex]x[\/latex]-axis move up to the point and notice the number with which it aligns on the [latex]y[\/latex]-axis. The [latex]3[\/latex] indicates that, after leaving the [latex]x[\/latex]-axis, you traveled [latex]3[\/latex] units up in the vertical direction, the direction of the [latex]y[\/latex]-axis. This number is the [latex]y[\/latex]-coordinate, the second number in the ordered pair.\r\n[latex]\\\\[\/latex]\r\nWith an [latex]x[\/latex]-coordinate of [latex]4[\/latex] and a [latex]y[\/latex]-coordinate of [latex]3[\/latex], you have the ordered pair ([latex]4, 3[\/latex]).This point lies in Quadrant [latex]I[\/latex].<\/section><section class=\"textbox example\">Describe the point shown as an ordered pair.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064229\/image004-1.jpg\" alt=\"A point that is 2 spaces above the x-axis and 5 spaces to the right of the y-axis.\" width=\"417\" height=\"378\" \/><\/center>[reveal-answer q=\"954779\"]Show Answer[\/reveal-answer][hidden-answer a=\"954779\"]Begin at the origin and move along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]-coordinate and is written first in the ordered pair.\r\n<p style=\"text-align: center;\">[latex](5,y)[\/latex]<\/p>\r\nMove from [latex]5[\/latex] up to the ordered pair and read the number on the [latex]y[\/latex]-axis. This is the [latex]y[\/latex]-coordinate and is written second in the ordered pair.\r\n<p style=\"text-align: center;\">[latex](5, 2)[\/latex]<\/p>\r\nThe point shown as an ordered pair is [latex](5, 2)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317853[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317854[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify points on a graph<\/li>\n<\/ul>\n<\/section>\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">The Components of the Coordinate Plane<\/h2>\n<p>The <strong>Cartesian coordinate system<\/strong>, refined by French mathematician Ren\u00e9 Descartes in 1637, serves as a framework for visualizing algebraic relationships. The coordinate plane can be used to plot points and graph lines.<\/p>\n<p>This system is comprised of a coordinate plane, which is essentially a grid formed by a horizontal axis, known as the [latex]x[\/latex]-axis, and a vertical axis, referred to as the [latex]y[\/latex]-axis. These axes intersect perpendicularly at a point called the origin, where both [latex]x[\/latex] and [latex]y[\/latex] coordinates are zero.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Cartesian coordinate system<\/h3>\n<p>In the Cartesian coordinate system, the horizontal axis in the coordinate plane is called the <b>[latex]x[\/latex]-axis<\/b>. The vertical axis is called the <b>[latex]y[\/latex]-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at [latex]0[\/latex] on the [latex]x[\/latex]-axis and [latex]0[\/latex] on the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/section>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064227\/image002.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"417\" height=\"378\" \/><\/div>\n<h3>Understanding Quadrants<\/h3>\n<p>The coordinate plane is partitioned into four distinct regions by the [latex]x[\/latex] and [latex]y[\/latex] axes. The quadrants can be seen in the coordinate plane above. Each quadrant has different characteristics for the [latex]x[\/latex] and [latex]y[\/latex] coordinates.<\/p>\n<ul>\n<li>Quadrant [latex]I[\/latex]: Both [latex]x[\/latex] and [latex]y[\/latex] coordinates are positive.<\/li>\n<li>Quadrant [latex]II[\/latex]: [latex]x[\/latex] is negative, while [latex]y[\/latex] is positive.<\/li>\n<li>Quadrant [latex]III[\/latex]: Both [latex]x[\/latex] and [latex]y[\/latex] coordinates are negative.<\/li>\n<li>Quadrant [latex]IV[\/latex]: [latex]x[\/latex] is positive, but [latex]y[\/latex] is negative.<\/li>\n<\/ul>\n<p>To ascertain the quadrant in which a point resides, one must examine the signs of its [latex]x[\/latex] and [latex]y[\/latex] coordinates.<\/p>\n<h3>Locating Points on the Coordinate Plane<\/h3>\n<p>In the Cartesian coordinate system, locations are defined by <strong>ordered pairs.\u00a0<\/strong> An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]-axis (the first value of the ordered pair) and along the [latex]y[\/latex]-axis (the second value of the ordered pair).<\/p>\n<p>In an ordered pair, such as ([latex]x ,y[\/latex]), the first value is the [latex]x[\/latex]-coordinate and the second value is the [latex]y[\/latex]-coordinate. The [latex]x[\/latex]-coordinate specifies the horizontal distance from the origin, while the [latex]y[\/latex]-coordinate indicates the vertical distance.<\/p>\n<section class=\"textbox example\">Consider the point below.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064228\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/><\/div>\n<p>To identify the location of this point, start at the origin ([latex]0, 0[\/latex]) and move right along the [latex]x[\/latex]-axis until you are under the point.<br \/>\n[latex]\\\\[\/latex]<br \/>\nLook at the label on the [latex]x[\/latex]-axis. The [latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]-coordinate, the first number in the ordered pair.<br \/>\n[latex]\\\\[\/latex]<br \/>\nFrom [latex]4[\/latex] on the [latex]x[\/latex]-axis move up to the point and notice the number with which it aligns on the [latex]y[\/latex]-axis. The [latex]3[\/latex] indicates that, after leaving the [latex]x[\/latex]-axis, you traveled [latex]3[\/latex] units up in the vertical direction, the direction of the [latex]y[\/latex]-axis. This number is the [latex]y[\/latex]-coordinate, the second number in the ordered pair.<br \/>\n[latex]\\\\[\/latex]<br \/>\nWith an [latex]x[\/latex]-coordinate of [latex]4[\/latex] and a [latex]y[\/latex]-coordinate of [latex]3[\/latex], you have the ordered pair ([latex]4, 3[\/latex]).This point lies in Quadrant [latex]I[\/latex].<\/section>\n<section class=\"textbox example\">Describe the point shown as an ordered pair.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064229\/image004-1.jpg\" alt=\"A point that is 2 spaces above the x-axis and 5 spaces to the right of the y-axis.\" width=\"417\" height=\"378\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q954779\">Show Answer<\/button><\/p>\n<div id=\"q954779\" class=\"hidden-answer\" style=\"display: none\">Begin at the origin and move along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]-coordinate and is written first in the ordered pair.<\/p>\n<p style=\"text-align: center;\">[latex](5,y)[\/latex]<\/p>\n<p>Move from [latex]5[\/latex] up to the ordered pair and read the number on the [latex]y[\/latex]-axis. This is the [latex]y[\/latex]-coordinate and is written second in the ordered pair.<\/p>\n<p style=\"text-align: center;\">[latex](5, 2)[\/latex]<\/p>\n<p>The point shown as an ordered pair is [latex](5, 2)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317853\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317853&theme=lumen&iframe_resize_id=ohm317853&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317854\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317854&theme=lumen&iframe_resize_id=ohm317854&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":36,"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\/467"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/467\/revisions"}],"predecessor-version":[{"id":5210,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/467\/revisions\/5210"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/467\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=467"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=467"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=467"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}