{"id":1194,"date":"2021-10-18T21:27:03","date_gmt":"2021-10-18T21:27:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1194"},"modified":"2022-10-05T16:37:21","modified_gmt":"2022-10-05T16:37:21","slug":"5-3-the-cartesian-plane","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-3-the-cartesian-plane\/","title":{"raw":"5.3.1: The Rectangular Coordinate System","rendered":"5.3.1: The Rectangular Coordinate System"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify quadrants on the Cartesian plane<\/li>\r\n \t<li>Identify axes on the Cartesian plane<\/li>\r\n \t<li>Identify points on the Cartesian plane as ordered pairs<\/li>\r\n \t<li>Plot ordered pairs as points on a rectangular coordinate system<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>KEY words<\/h3>\r\n<ul>\r\n \t<li><strong>Coordinates<\/strong>: an ordered pair written in the form [latex](x,y)[\/latex]<\/li>\r\n \t<li style=\"margin-top: 0.5em;\"><strong>Origin<\/strong>: the point (0, 0) where the axes cross<\/li>\r\n \t<li><strong>Perpendicular<\/strong>: sitting at right angles<\/li>\r\n \t<li><strong>Quadrant<\/strong>: A quarter of the coordinate plane separated by the axes<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Rectangular Coordinate System<\/h2>\r\nThe <strong>rectangular\u00a0<\/strong><b>coordinate system<\/b>\u00a0was developed in 1637 and refined by the French mathematician Ren\u00e9 Descartes. The rectangular coordinate system is often referred to as the Cartesian plane (in honor of Descartes). This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.\r\n\r\nYou have likely used a coordinate system before. Many maps use a grid system to identify locations. The map in figure 1 uses a horizontal and vertical grid to convey information about an object\u2019s location. The numbers [latex]1,2,3[\/latex], and [latex]4[\/latex] across the bottom of the map and the letters A, B, C, and D along the left side identify the columns and rows of the grid, respectively. Every location on the map can be identified by a number and a letter that identifies the cell in the grid.\r\n\r\nThe general location of any item on this map can be found by using the letter and number of its grid cell.\u00a0For example, the Student Center is in section 2B. It is located in the grid section above the number [latex]2[\/latex] and next to the letter B. The Stadium is in section 4D, and the library is in 2C.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"658\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224708\/CNX_BMath_Figure_11_01_001.png\" alt=\"The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library. At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium.\" width=\"658\" height=\"403\" \/> Figure 1. A campus map.[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the map in figure 1.\r\n<ol id=\"eip-id1164754238718\" class=\"circled\">\r\n \t<li>Find the grid section of the Residence Halls.<\/li>\r\n \t<li>What is located in grid section 3B?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol id=\"eip-id1164753925474\" class=\"circled\">\r\n \t<li>Read the number below the Residence Halls, [latex]4[\/latex], and the letter to the side, A. So the Residence Halls are in grid section 4A.<\/li>\r\n \t<li>Find [latex]3[\/latex] across the bottom of the map and B along the side. Look below the [latex]3[\/latex] and next to the B. The engineering building is in grid section 3B.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n\r\nThe <em><strong>rectangular coordinate system<\/strong><\/em> consists of two basic elements: the <em><strong>rectangular coordinate plane<\/strong><\/em> and ordered pairs plotted as\u00a0<strong><em>points<\/em><\/strong> on the plane.\u00a0 Figure 2 shows the rectangular coordinate <strong><em>plane<\/em><\/strong>. It consists of a <em><strong>horizontal<\/strong> <b>axis<\/b><\/em> and a <em><strong>vertical axis.<\/strong><\/em>\u00a0Each axis is a number line and the number lines intersect at right angles forming a two-dimensional plane. The axes are <em><strong>perpendicular<\/strong><\/em> to each other and intersect where zero lies on both axes.\r\n\r\nThe horizontal axis in the coordinate plane is called the <em><strong>[latex]x-axis[\/latex]<\/strong><\/em>. The vertical axis is called the <strong><em>[latex]y-axis[\/latex]<\/em><\/strong>. 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-axis[\/latex] and [latex]0[\/latex] on the [latex]y-axis[\/latex].\r\n\r\nThe intersecting [latex]x[\/latex]<i>-<\/i> and [latex]y[\/latex]<i>-<\/i>axes of the coordinate plane divide it\u00a0into four sections. These four sections are called <em><b>quadrants<\/b><\/em>. Quadrants are named using the Roman numerals I, II, III, and IV beginning with the top right quadrant and moving counter clockwise.\r\n<div class=\"textbox shaded\">\r\n<h3>The rectangular coordinate PLANE<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img 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\" \/> Figure 2. The rectangular coordinate plane.[\/caption]\r\n\r\n<\/div>\r\nLocations on the coordinate plane are described as <em><b>ordered pairs<\/b><\/em>. An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]<i>-<\/i>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 called the <em><strong>[latex]x[\/latex]<\/strong><b><em>-<\/em>coordinate<\/b><\/em> and the second value is the <em><strong>[latex]y[\/latex]<\/strong><b>-coordinate<\/b><\/em>. Note that the [latex]x[\/latex]<i>-<\/i>coordinate is listed before the [latex]y[\/latex]<i>-<\/i>coordinate. Since the origin has an [latex]x[\/latex]<i>-<\/i>coordinate of [latex]0[\/latex] and a [latex]y[\/latex]<i>-<\/i>coordinate of [latex]0[\/latex], its ordered pair is written [latex](0, 0)[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Ordered Pair<\/h3>\r\nAn ordered pair, [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point in a rectangular coordinate system.\r\n\r\n[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]\r\n\r\n<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 \" \/>\r\n\r\n<\/div>\r\n<h3 id=\"Plotting Points in the Coordinate Plane\" class=\"no-indent\" style=\"text-align: left;\">Plotting Points<\/h3>\r\nPoints can be plotted on the rectangular coordinate plane by first locating the\u00a0[latex]x[\/latex] value then locating the\u00a0[latex]y[\/latex] value.\r\n\r\nFor example, to plot the point [latex]\\left(2,5\\right)[\/latex], first locate [latex]2[\/latex] on the [latex]x[\/latex]-axis then move vertically to the level of [latex]5[\/latex] on the [latex]y[\/latex]-axis.\u00a0We plot the point directly above\u00a0[latex]2[\/latex] on the \u00a0[latex]x[\/latex]-axis and at the level of [latex]5[\/latex] on the [latex]y[\/latex]-axis, as shown in figure 3.\r\n\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 \" width=\"301\" height=\"308\" \/> Figure 3. Plotting the point (2, 5).[\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nPlot [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(3,1\\right)[\/latex] in the same rectangular coordinate system.\r\n[reveal-answer q=\"501893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"501893\"]\r\n\r\nSolution\r\nThe coordinate values are the same for both points, but the [latex]x[\/latex] and [latex]y[\/latex] values are reversed. Let\u2019s begin with point [latex]\\left(1,3\\right)[\/latex]. The [latex]x\\text{-coordinate}[\/latex] is [latex]1[\/latex] so find [latex]1[\/latex] on the [latex]x\\text{-axis}[\/latex] and then move vertically to the level where we find [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex]. Plot the point [latex]\\left(1,3\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224723\/CNX_BMath_Figure_11_01_006_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 3 on the y axis. A vertical dotted line passes through 1 on the x axis. The dotted lines intersect at a point labeled \" \/>\r\nTo plot the point [latex]\\left(3,1\\right)[\/latex], we start by locating [latex]3[\/latex] on the [latex]x\\text{-axis}[\/latex] then we move vertically to the level of [latex]1[\/latex] on the [latex]y\\text{-axis}[\/latex] and plot the point [latex]\\left(3,1\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224725\/CNX_BMath_Figure_11_01_007_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 1 on the y-axis. A vertical dotted line passes through 3 on the x axis. The dotted line intersects at a point labeled \" \/>\r\nNotice that the order of the coordinates does matter, so, [latex]\\left(1,3\\right)[\/latex] is not the same point as [latex]\\left(3,1\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146882[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen one (or both) of the coordinates of an ordered pair is negative, we move in the negative direction along one or both axes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nPlot the point [latex](\u22124,\u22122)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064233\/image006.jpg\" alt=\"Graph with blue arrow pointing from origin to four units to the left. A red arrow points down 2 spaces to the point negative 4, negative 2.\" width=\"417\" height=\"378\" \/>\r\n\r\nThe [latex]<i>x-<\/i>[\/latex]coordinate is [latex]\u22124[\/latex] because it comes first in the ordered pair. Start at the origin and move [latex]4[\/latex] units in a negative direction (left) along the <i>x-<\/i>axis.\r\n\r\nThe [latex]<i>y-<\/i>[\/latex]coordinate is [latex]\u22122[\/latex] because it comes second in the ordered pair. Now move [latex]2[\/latex] units in a negative direction (down). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]\u22122[\/latex] on that axis.\r\n\r\nNow draw a point at that location and label it.\r\n\r\n<\/div>\r\nNOTE: A point on the coordinate plane represents a single location. The point has no dimension (no length nor width); although it is physically impossible to draw a dot without dimension.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nHow do the signs affect the location of the points?\r\n\r\nPlot each point:\r\n\r\n1. [latex]\\left(-5,2\\right)[\/latex]\r\n2. [latex]\\left(-5,-2\\right)[\/latex]\r\n3. [latex]\\left(5,2\\right)[\/latex]\r\n4. [latex]\\left(5,-2\\right)[\/latex]\r\n\r\nAs we locate the [latex]x\\text{-coordinate}[\/latex] and the [latex]y\\text{-coordinate}[\/latex], we must be careful with the signs.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224736\/CNX_BMath_Figure_11_01_028_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point \" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146885[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Watch the video below for more examples of how to plot ordered pairs.<\/p>\r\nhttps:\/\/youtu.be\/p_MESleS3mw\r\n<h3 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Identify quadrants and use them to plot points<\/h3>\r\nWhen we described the rectangular coordinate plane, we mentioned the four quadrants, I, II, III, and IV.\u00a0 These quadrants can be useful for locating points because ordered pairs within any particular quadrant share certain characteristics. Consider the points in each quadrant in figure 4.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064235\/image008-1.jpg\" alt=\"A graph with many plotted points in different quadrants. Quadrant 1 has the point (1,3); the point (2,2); and the point (4,1). Quadrant 2 has the point negative 1, one; the point negative 2, 5; and the point negative 4, one. Quadrant 3 has the point negative 2, negative 3; the point negative 3, negative 3; and the point negative 1, negative 5. Quadrant 4 has the point 2, negative 1; the point 1, negative 3; and the point 4, negative 4.\" width=\"417\" height=\"378\" \/> Figure 4. Points within quadrants.[\/caption]\r\n\r\nWithin each quadrant, the signs of the [latex]<i>x-[\/latex]<\/i>coordinates and [latex]<i>y-[\/latex]<\/i>coordinates of each ordered pair are the same. They follow a pattern, shown in figure 5.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064237\/image009-1.jpg\" alt=\"Graph with quadrants. Quadrant 1 is positive, positive. Quadrant 2 is negative, positive. Quadrant 3 is negative, negative. Quadrant 4 is positive, negative.\" width=\"417\" height=\"378\" \/> Figure 5. Patterns of ordered pars within quadrants.[\/caption]\r\n\r\nThe example below details how to determine the quadrant location of a point just by thinking about the signs of its coordinates. Thinking about the quadrant location before plotting a point can help us prevent mistakes. It is also useful knowledge for checking that we have plotted a point correctly.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIn which quadrant is the point [latex](\u22127,10)[\/latex] located?\r\n\r\n[reveal-answer q=\"222353\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222353\"]\r\n\r\nLook at the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,+)[\/latex].\r\n<p style=\"text-align: center;\">[latex](\u22127,10)[\/latex]<\/p>\r\nUsing the table or grid above, locate the pattern [latex](\u2212,+)[\/latex].\r\n\r\nPoints with the pattern [latex](\u2212,+)[\/latex] are in Quadrant II.\r\n<h4>Answer<\/h4>\r\nThe point [latex](\u22127,10)[\/latex] is in Quadrant II.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIn which quadrant is the point [latex](\u221210,\u22125)[\/latex] located?\r\n<h4>Solution<\/h4>\r\nLook at the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,\u2212)[\/latex].\r\n<p style=\"text-align: center;\">[latex](\u221210,\u22125)[\/latex]<\/p>\r\nPoints with the pattern [latex](\u2212,\u2212)[\/latex] are in Quadrant III.\r\n\r\nUsing the table or grid above, locate the pattern [latex](\u2212,\u2212)[\/latex].\r\n<h4>Answer<\/h4>\r\nThe point [latex](\u221210,\u22125)[\/latex] is in Quadrant III.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nPlot each point in the rectangular coordinate plane and identify the quadrant in which the point is located:\r\n\r\n1. [latex]\\left(-1,3\\right)[\/latex]\r\n2. [latex]\\left(-3,-4\\right)[\/latex]\r\n3. [latex]\\left(2,-3\\right)[\/latex]\r\n4. [latex]\\left(3,{\\dfrac{5}{2}}\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nThe first number of the coordinate pair is the [latex]x\\text{-coordinate}[\/latex], and the second number is the [latex]y\\text{-coordinate}[\/latex].\r\n1. Since [latex]x=-1,y=3[\/latex], the point [latex]\\left(-1,3\\right)[\/latex] is in Quadrant II.\r\n2. Since [latex]x=-3,y=-4[\/latex], the point [latex]\\left(-3,-4\\right)[\/latex] is in Quadrant III.\r\n3. Since [latex]x=2,y=-1[\/latex], the point [latex]\\left(2,-1\\right)[\/latex] is in Quadrant lV.\r\n4. Since [latex]x=3,y={\\dfrac{5}{2}}[\/latex], the point [latex]\\left(3,{\\dfrac{5}{2}}\\right)[\/latex] is in Quadrant I. It may be helpful to write [latex]{\\dfrac{5}{2}}[\/latex] as the mixed number, [latex]2{\\dfrac{1}{2}}[\/latex], or decimal, [latex]2.5[\/latex]. Then we know that the point is halfway between [latex]2[\/latex] and [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224731\/CNX_BMath_Figure_11_01_010.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point \" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146883[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"video1\">Watch the video below to see more examples of how to identify the quadrant that a point is located in.<\/p>\r\nhttps:\/\/youtu.be\/iTsJsPgcE4E\r\n<h3>\u00a0Points on the Axes<\/h3>\r\nWhen a point is plotted on one of the axes, one of the coordinates must be zero.\u00a0Figure 6 shows the graph of the ordered pair [latex](0,4)[\/latex] located on the [latex]y[\/latex]-axis.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064238\/image010-1.jpg\" alt=\"Graph of the point (0,4). The point is on the y-axis.\" width=\"417\" height=\"378\" \/> Figure 6. Plotting a point on the [latex]y[\/latex]-axis.[\/caption]A point located on one of the axes is not considered to be in a quadrant. It is simply on one of the axes. Whenever the [latex]x[\/latex]<i>-<\/i>coordinate is [latex]0[\/latex], the point is located on the [latex]y[\/latex]<i>-<\/i>axis. Similarly, any point that has a [latex]y[\/latex]<i>-<\/i>coordinate of [latex]0[\/latex] is located on the [latex]x[\/latex]<i>-<\/i>axis.\r\n<div class=\"textbox shaded\">\r\n<h3>Points on the Axes<\/h3>\r\nPoints with a [latex]y\\text{-coordinate}[\/latex] equal to [latex]0[\/latex] are on the [latex]x\\text{-axis}[\/latex], and have coordinates [latex]\\left(a,0\\right)[\/latex].\r\n\r\nPoints with an [latex]x\\text{-coordinate}[\/latex] equal to [latex]0[\/latex] are on the [latex]y\\text{-axis}[\/latex], and have coordinates [latex]\\left(0,b\\right)[\/latex].\r\n\r\n<\/div>\r\nWhen the point falls on both axes, both coordinates are zero. Therefore, its ordered pair is [latex]\\left(0,0\\right)[\/latex] . This point is called the <strong><em>origin<\/em><\/strong>.\r\n<div class=\"textbox shaded\">\r\n<h3>The Origin<\/h3>\r\nThe point [latex]\\left(0,0\\right)[\/latex] is called the <strong>origin<\/strong>. It is the point where the <em>[latex]x[\/latex]<\/em>-axis and <em>[latex]y[\/latex]<\/em>-axis intersect.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLE<\/h3>\r\nPlot each point on a coordinate plane:\r\n\r\n1. [latex]\\left(0,5\\right)[\/latex]\r\n2. [latex]\\left(4,0\\right)[\/latex]\r\n3. [latex]\\left(-3,0\\right)[\/latex]\r\n4. [latex]\\left(0,0\\right)[\/latex]\r\n5. [latex]\\left(0,-1\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\n<ol id=\"eip-id1164754129469\" class=\"circled\">\r\n \t<li>Since [latex]x=0[\/latex], the point whose coordinates are [latex]\\left(0,5\\right)[\/latex] is on the [latex]y\\text{-axis}[\/latex].<\/li>\r\n \t<li>Since [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(4,0\\right)[\/latex] is on the [latex]x\\text{-axis}[\/latex].<\/li>\r\n \t<li>Since [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(-3,0\\right)[\/latex] is on the [latex]x\\text{-axis}[\/latex].<\/li>\r\n \t<li>Since [latex]x=0[\/latex] and [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(0,0\\right)[\/latex] is the origin.<\/li>\r\n \t<li>Since [latex]x=0[\/latex], the point whose coordinates are [latex]\\left(0,-1\\right)[\/latex] is on the [latex]y\\text{-axis}[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224744\/CNX_BMath_Figure_11_01_031_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point \" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question]146886[\/ohm_question]\r\n\r\n<\/div>\r\nIn algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the [latex]x[\/latex]-coordinate of a point on a graph, read the number on the [latex]x[\/latex]-axis directly above or below the point. To identify the [latex]y[\/latex]-coordinate of a point, read the number on the [latex]y[\/latex]-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order [latex]\\left(x,y\\right)[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDescribe the point shown as an ordered pair.\r\n\r\n<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\" \/>\r\n<h4>Solution<\/h4>\r\nBegin at the origin and move along the [latex]x[\/latex]<i>-<\/i>axis. This is the [latex]x[\/latex]<i>-<\/i>coordinate and is written first in the ordered pair.\r\n<p style=\"text-align: center;\">[latex]\\left(5, y\\right)[\/latex]<\/p>\r\nMove from 5 up to the ordered pair and read the number on the [latex]y[\/latex]<i>-<\/i>axis. This is the [latex]y[\/latex]<i>-<\/i>coordinate and is written second in the ordered pair.\r\n<p style=\"text-align: center;\">[latex](5,2)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe point shown as an ordered pair is [latex](5,2)[\/latex].\r\n\r\n<\/div>\r\n<h3>Describing a point shown as an ordered pair<\/h3>\r\nhttps:\/\/youtu.be\/c9WVU34MY5Q\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nName the ordered pair of each point shown:\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224748\/CNX_BMath_Figure_11_01_017.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point \" \/>\r\n<h4>Solution<\/h4>\r\nPoint A is above [latex]-3[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]-3[\/latex]. The point is to the left of [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]3[\/latex]. The coordinates of the point are [latex]\\left(-3,3\\right)[\/latex].\r\n\r\nPoint B is below [latex]-1[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]-1[\/latex]. The point is to the left of [latex]-3[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]-3[\/latex]. The coordinates of the point are [latex]\\left(-1,-3\\right)[\/latex].\r\n\r\nPoint C is above [latex]2[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]2[\/latex]. The point is to the right of [latex]4[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]4[\/latex]. The coordinates of the point are [latex]\\left(2,4\\right)[\/latex].\r\n\r\nPoint D is below [latex]4[\/latex] on the [latex]x-\\text{axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]4[\/latex]. The point is to the right of [latex]-4[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]-4[\/latex]. The coordinates of the point are [latex]\\left(4,-4\\right)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146915[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146919[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWatch the following video for another example of how to determine the ordered pair for points on the coordinate plane.\r\n\r\nhttps:\/\/youtu.be\/c9WVU34MY5Q","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify quadrants on the Cartesian plane<\/li>\n<li>Identify axes on the Cartesian plane<\/li>\n<li>Identify points on the Cartesian plane as ordered pairs<\/li>\n<li>Plot ordered pairs as points on a rectangular coordinate system<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>KEY words<\/h3>\n<ul>\n<li><strong>Coordinates<\/strong>: an ordered pair written in the form [latex](x,y)[\/latex]<\/li>\n<li style=\"margin-top: 0.5em;\"><strong>Origin<\/strong>: the point (0, 0) where the axes cross<\/li>\n<li><strong>Perpendicular<\/strong>: sitting at right angles<\/li>\n<li><strong>Quadrant<\/strong>: A quarter of the coordinate plane separated by the axes<\/li>\n<\/ul>\n<\/div>\n<h2>The Rectangular Coordinate System<\/h2>\n<p>The <strong>rectangular\u00a0<\/strong><b>coordinate system<\/b>\u00a0was developed in 1637 and refined by the French mathematician Ren\u00e9 Descartes. The rectangular coordinate system is often referred to as the Cartesian plane (in honor of Descartes). This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.<\/p>\n<p>You have likely used a coordinate system before. Many maps use a grid system to identify locations. The map in figure 1 uses a horizontal and vertical grid to convey information about an object\u2019s location. The numbers [latex]1,2,3[\/latex], and [latex]4[\/latex] across the bottom of the map and the letters A, B, C, and D along the left side identify the columns and rows of the grid, respectively. Every location on the map can be identified by a number and a letter that identifies the cell in the grid.<\/p>\n<p>The general location of any item on this map can be found by using the letter and number of its grid cell.\u00a0For example, the Student Center is in section 2B. It is located in the grid section above the number [latex]2[\/latex] and next to the letter B. The Stadium is in section 4D, and the library is in 2C.<\/p>\n<div style=\"width: 668px\" 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\/24224708\/CNX_BMath_Figure_11_01_001.png\" alt=\"The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library. At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium.\" width=\"658\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. A campus map.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the map in figure 1.<\/p>\n<ol id=\"eip-id1164754238718\" class=\"circled\">\n<li>Find the grid section of the Residence Halls.<\/li>\n<li>What is located in grid section 3B?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol id=\"eip-id1164753925474\" class=\"circled\">\n<li>Read the number below the Residence Halls, [latex]4[\/latex], and the letter to the side, A. So the Residence Halls are in grid section 4A.<\/li>\n<li>Find [latex]3[\/latex] across the bottom of the map and B along the side. Look below the [latex]3[\/latex] and next to the B. The engineering building is in grid section 3B.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The <em><strong>rectangular coordinate system<\/strong><\/em> consists of two basic elements: the <em><strong>rectangular coordinate plane<\/strong><\/em> and ordered pairs plotted as\u00a0<strong><em>points<\/em><\/strong> on the plane.\u00a0 Figure 2 shows the rectangular coordinate <strong><em>plane<\/em><\/strong>. It consists of a <em><strong>horizontal<\/strong> <b>axis<\/b><\/em> and a <em><strong>vertical axis.<\/strong><\/em>\u00a0Each axis is a number line and the number lines intersect at right angles forming a two-dimensional plane. The axes are <em><strong>perpendicular<\/strong><\/em> to each other and intersect where zero lies on both axes.<\/p>\n<p>The horizontal axis in the coordinate plane is called the <em><strong>[latex]x-axis[\/latex]<\/strong><\/em>. The vertical axis is called the <strong><em>[latex]y-axis[\/latex]<\/em><\/strong>. 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-axis[\/latex] and [latex]0[\/latex] on the [latex]y-axis[\/latex].<\/p>\n<p>The intersecting [latex]x[\/latex]<i>&#8211;<\/i> and [latex]y[\/latex]<i>&#8211;<\/i>axes of the coordinate plane divide it\u00a0into four sections. These four sections are called <em><b>quadrants<\/b><\/em>. Quadrants are named using the Roman numerals I, II, III, and IV beginning with the top right quadrant and moving counter clockwise.<\/p>\n<div class=\"textbox shaded\">\n<h3>The rectangular coordinate PLANE<\/h3>\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The rectangular coordinate plane.<\/p>\n<\/div>\n<\/div>\n<p>Locations on the coordinate plane are described as <em><b>ordered pairs<\/b><\/em>. An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]<i>&#8211;<\/i>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 called the <em><strong>[latex]x[\/latex]<\/strong><b><em>&#8211;<\/em>coordinate<\/b><\/em> and the second value is the <em><strong>[latex]y[\/latex]<\/strong><b>-coordinate<\/b><\/em>. Note that the [latex]x[\/latex]<i>&#8211;<\/i>coordinate is listed before the [latex]y[\/latex]<i>&#8211;<\/i>coordinate. Since the origin has an [latex]x[\/latex]<i>&#8211;<\/i>coordinate of [latex]0[\/latex] and a [latex]y[\/latex]<i>&#8211;<\/i>coordinate of [latex]0[\/latex], its ordered pair is written [latex](0, 0)[\/latex].<\/p>\n<div class=\"textbox shaded\">\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>[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><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\" \/><\/p>\n<\/div>\n<h3 id=\"Plotting Points in the Coordinate Plane\" class=\"no-indent\" style=\"text-align: left;\">Plotting Points<\/h3>\n<p>Points can be plotted on the rectangular coordinate plane by first locating the\u00a0[latex]x[\/latex] value then locating the\u00a0[latex]y[\/latex] value.<\/p>\n<p>For example, to plot the point [latex]\\left(2,5\\right)[\/latex], first locate [latex]2[\/latex] on the [latex]x[\/latex]-axis then move vertically to the level of [latex]5[\/latex] on the [latex]y[\/latex]-axis.\u00a0We plot the point directly above\u00a0[latex]2[\/latex] on the \u00a0[latex]x[\/latex]-axis and at the level of [latex]5[\/latex] on the [latex]y[\/latex]-axis, as shown in figure 3.<\/p>\n<div style=\"width: 311px\" 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\" width=\"301\" height=\"308\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Plotting the point (2, 5).<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Plot [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(3,1\\right)[\/latex] in the same rectangular coordinate system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q501893\">Show Solution<\/span><\/p>\n<div id=\"q501893\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThe coordinate values are the same for both points, but the [latex]x[\/latex] and [latex]y[\/latex] values are reversed. Let\u2019s begin with point [latex]\\left(1,3\\right)[\/latex]. The [latex]x\\text{-coordinate}[\/latex] is [latex]1[\/latex] so find [latex]1[\/latex] on the [latex]x\\text{-axis}[\/latex] and then move vertically to the level where we find [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex]. Plot the point [latex]\\left(1,3\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224723\/CNX_BMath_Figure_11_01_006_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 3 on the y axis. A vertical dotted line passes through 1 on the x axis. The dotted lines intersect at a point labeled\" \/><br \/>\nTo plot the point [latex]\\left(3,1\\right)[\/latex], we start by locating [latex]3[\/latex] on the [latex]x\\text{-axis}[\/latex] then we move vertically to the level of [latex]1[\/latex] on the [latex]y\\text{-axis}[\/latex] and plot the point [latex]\\left(3,1\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224725\/CNX_BMath_Figure_11_01_007_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. A horizontal dotted line passes through 1 on the y-axis. A vertical dotted line passes through 3 on the x axis. The dotted line intersects at a point labeled\" \/><br \/>\nNotice that the order of the coordinates does matter, so, [latex]\\left(1,3\\right)[\/latex] is not the same point as [latex]\\left(3,1\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146882\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146882&theme=oea&iframe_resize_id=ohm146882&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When one (or both) of the coordinates of an ordered pair is negative, we move in the negative direction along one or both axes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Plot the point [latex](\u22124,\u22122)[\/latex].<\/p>\n<p><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\/04064233\/image006.jpg\" alt=\"Graph with blue arrow pointing from origin to four units to the left. A red arrow points down 2 spaces to the point negative 4, negative 2.\" width=\"417\" height=\"378\" \/><\/p>\n<p>The [latex]<i>x-<\/i>[\/latex]coordinate is [latex]\u22124[\/latex] because it comes first in the ordered pair. Start at the origin and move [latex]4[\/latex] units in a negative direction (left) along the <i>x-<\/i>axis.<\/p>\n<p>The [latex]<i>y-<\/i>[\/latex]coordinate is [latex]\u22122[\/latex] because it comes second in the ordered pair. Now move [latex]2[\/latex] units in a negative direction (down). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]\u22122[\/latex] on that axis.<\/p>\n<p>Now draw a point at that location and label it.<\/p>\n<\/div>\n<p>NOTE: A point on the coordinate plane represents a single location. The point has no dimension (no length nor width); although it is physically impossible to draw a dot without dimension.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>How do the signs affect the location of the points?<\/p>\n<p>Plot each point:<\/p>\n<p>1. [latex]\\left(-5,2\\right)[\/latex]<br \/>\n2. [latex]\\left(-5,-2\\right)[\/latex]<br \/>\n3. [latex]\\left(5,2\\right)[\/latex]<br \/>\n4. [latex]\\left(5,-2\\right)[\/latex]<\/p>\n<p>As we locate the [latex]x\\text{-coordinate}[\/latex] and the [latex]y\\text{-coordinate}[\/latex], we must be careful with the signs.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224736\/CNX_BMath_Figure_11_01_028_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146885\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146885&theme=oea&iframe_resize_id=ohm146885&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Watch the video below for more examples of how to plot ordered pairs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Plot Points Given as Ordered Pairs on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/p_MESleS3mw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Identify quadrants and use them to plot points<\/h3>\n<p>When we described the rectangular coordinate plane, we mentioned the four quadrants, I, II, III, and IV.\u00a0 These quadrants can be useful for locating points because ordered pairs within any particular quadrant share certain characteristics. Consider the points in each quadrant in figure 4.<\/p>\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064235\/image008-1.jpg\" alt=\"A graph with many plotted points in different quadrants. Quadrant 1 has the point (1,3); the point (2,2); and the point (4,1). Quadrant 2 has the point negative 1, one; the point negative 2, 5; and the point negative 4, one. Quadrant 3 has the point negative 2, negative 3; the point negative 3, negative 3; and the point negative 1, negative 5. Quadrant 4 has the point 2, negative 1; the point 1, negative 3; and the point 4, negative 4.\" width=\"417\" height=\"378\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. Points within quadrants.<\/p>\n<\/div>\n<p>Within each quadrant, the signs of the [latex]<i>x-[\/latex]<\/i>coordinates and [latex]<i>y-[\/latex]<\/i>coordinates of each ordered pair are the same. They follow a pattern, shown in figure 5.<\/p>\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064237\/image009-1.jpg\" alt=\"Graph with quadrants. Quadrant 1 is positive, positive. Quadrant 2 is negative, positive. Quadrant 3 is negative, negative. Quadrant 4 is positive, negative.\" width=\"417\" height=\"378\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Patterns of ordered pars within quadrants.<\/p>\n<\/div>\n<p>The example below details how to determine the quadrant location of a point just by thinking about the signs of its coordinates. Thinking about the quadrant location before plotting a point can help us prevent mistakes. It is also useful knowledge for checking that we have plotted a point correctly.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>In which quadrant is the point [latex](\u22127,10)[\/latex] located?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222353\">Show Solution<\/span><\/p>\n<div id=\"q222353\" class=\"hidden-answer\" style=\"display: none\">\n<p>Look at the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,+)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex](\u22127,10)[\/latex]<\/p>\n<p>Using the table or grid above, locate the pattern [latex](\u2212,+)[\/latex].<\/p>\n<p>Points with the pattern [latex](\u2212,+)[\/latex] are in Quadrant II.<\/p>\n<h4>Answer<\/h4>\n<p>The point [latex](\u22127,10)[\/latex] is in Quadrant II.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>In which quadrant is the point [latex](\u221210,\u22125)[\/latex] located?<\/p>\n<h4>Solution<\/h4>\n<p>Look at the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,\u2212)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex](\u221210,\u22125)[\/latex]<\/p>\n<p>Points with the pattern [latex](\u2212,\u2212)[\/latex] are in Quadrant III.<\/p>\n<p>Using the table or grid above, locate the pattern [latex](\u2212,\u2212)[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The point [latex](\u221210,\u22125)[\/latex] is in Quadrant III.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Plot each point in the rectangular coordinate plane and identify the quadrant in which the point is located:<\/p>\n<p>1. [latex]\\left(-1,3\\right)[\/latex]<br \/>\n2. [latex]\\left(-3,-4\\right)[\/latex]<br \/>\n3. [latex]\\left(2,-3\\right)[\/latex]<br \/>\n4. [latex]\\left(3,{\\dfrac{5}{2}}\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>The first number of the coordinate pair is the [latex]x\\text{-coordinate}[\/latex], and the second number is the [latex]y\\text{-coordinate}[\/latex].<br \/>\n1. Since [latex]x=-1,y=3[\/latex], the point [latex]\\left(-1,3\\right)[\/latex] is in Quadrant II.<br \/>\n2. Since [latex]x=-3,y=-4[\/latex], the point [latex]\\left(-3,-4\\right)[\/latex] is in Quadrant III.<br \/>\n3. Since [latex]x=2,y=-1[\/latex], the point [latex]\\left(2,-1\\right)[\/latex] is in Quadrant lV.<br \/>\n4. Since [latex]x=3,y={\\dfrac{5}{2}}[\/latex], the point [latex]\\left(3,{\\dfrac{5}{2}}\\right)[\/latex] is in Quadrant I. It may be helpful to write [latex]{\\dfrac{5}{2}}[\/latex] as the mixed number, [latex]2{\\dfrac{1}{2}}[\/latex], or decimal, [latex]2.5[\/latex]. Then we know that the point is halfway between [latex]2[\/latex] and [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex].<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224731\/CNX_BMath_Figure_11_01_010.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146883\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146883&theme=oea&iframe_resize_id=ohm146883&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"video1\">Watch the video below to see more examples of how to identify the quadrant that a point is located in.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Identify the Quadrant of a Point on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iTsJsPgcE4E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>\u00a0Points on the Axes<\/h3>\n<p>When a point is plotted on one of the axes, one of the coordinates must be zero.\u00a0Figure 6 shows the graph of the ordered pair [latex](0,4)[\/latex] located on the [latex]y[\/latex]-axis.<\/p>\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064238\/image010-1.jpg\" alt=\"Graph of the point (0,4). The point is on the y-axis.\" width=\"417\" height=\"378\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. Plotting a point on the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<p>A point located on one of the axes is not considered to be in a quadrant. It is simply on one of the axes. Whenever the [latex]x[\/latex]<i>&#8211;<\/i>coordinate is [latex]0[\/latex], the point is located on the [latex]y[\/latex]<i>&#8211;<\/i>axis. Similarly, any point that has a [latex]y[\/latex]<i>&#8211;<\/i>coordinate of [latex]0[\/latex] is located on the [latex]x[\/latex]<i>&#8211;<\/i>axis.<\/p>\n<div class=\"textbox shaded\">\n<h3>Points on the Axes<\/h3>\n<p>Points with a [latex]y\\text{-coordinate}[\/latex] equal to [latex]0[\/latex] are on the [latex]x\\text{-axis}[\/latex], and have coordinates [latex]\\left(a,0\\right)[\/latex].<\/p>\n<p>Points with an [latex]x\\text{-coordinate}[\/latex] equal to [latex]0[\/latex] are on the [latex]y\\text{-axis}[\/latex], and have coordinates [latex]\\left(0,b\\right)[\/latex].<\/p>\n<\/div>\n<p>When the point falls on both axes, both coordinates are zero. Therefore, its ordered pair is [latex]\\left(0,0\\right)[\/latex] . This point is called the <strong><em>origin<\/em><\/strong>.<\/p>\n<div class=\"textbox shaded\">\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 <em>[latex]x[\/latex]<\/em>-axis and <em>[latex]y[\/latex]<\/em>-axis intersect.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>Plot each point on a coordinate plane:<\/p>\n<p>1. [latex]\\left(0,5\\right)[\/latex]<br \/>\n2. [latex]\\left(4,0\\right)[\/latex]<br \/>\n3. [latex]\\left(-3,0\\right)[\/latex]<br \/>\n4. [latex]\\left(0,0\\right)[\/latex]<br \/>\n5. [latex]\\left(0,-1\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<ol id=\"eip-id1164754129469\" class=\"circled\">\n<li>Since [latex]x=0[\/latex], the point whose coordinates are [latex]\\left(0,5\\right)[\/latex] is on the [latex]y\\text{-axis}[\/latex].<\/li>\n<li>Since [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(4,0\\right)[\/latex] is on the [latex]x\\text{-axis}[\/latex].<\/li>\n<li>Since [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(-3,0\\right)[\/latex] is on the [latex]x\\text{-axis}[\/latex].<\/li>\n<li>Since [latex]x=0[\/latex] and [latex]y=0[\/latex], the point whose coordinates are [latex]\\left(0,0\\right)[\/latex] is the origin.<\/li>\n<li>Since [latex]x=0[\/latex], the point whose coordinates are [latex]\\left(0,-1\\right)[\/latex] is on the [latex]y\\text{-axis}[\/latex].<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224744\/CNX_BMath_Figure_11_01_031_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point\" \/><\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146886\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146886&theme=oea&iframe_resize_id=ohm146886&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the [latex]x[\/latex]-coordinate of a point on a graph, read the number on the [latex]x[\/latex]-axis directly above or below the point. To identify the [latex]y[\/latex]-coordinate of a point, read the number on the [latex]y[\/latex]-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order [latex]\\left(x,y\\right)[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Describe the point shown as an ordered pair.<\/p>\n<p><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\" \/><\/p>\n<h4>Solution<\/h4>\n<p>Begin at the origin and move along the [latex]x[\/latex]<i>&#8211;<\/i>axis. This is the [latex]x[\/latex]<i>&#8211;<\/i>coordinate and is written first in the ordered pair.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(5, y\\right)[\/latex]<\/p>\n<p>Move from 5 up to the ordered pair and read the number on the [latex]y[\/latex]<i>&#8211;<\/i>axis. This is the [latex]y[\/latex]<i>&#8211;<\/i>coordinate and is written second in the ordered pair.<\/p>\n<p style=\"text-align: center;\">[latex](5,2)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The point shown as an ordered pair is [latex](5,2)[\/latex].<\/p>\n<\/div>\n<h3>Describing a point shown as an ordered pair<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Determine the Ordered Pairs for Points Plotted on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/c9WVU34MY5Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Name the ordered pair of each point shown:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224748\/CNX_BMath_Figure_11_01_017.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point\" \/><\/p>\n<h4>Solution<\/h4>\n<p>Point A is above [latex]-3[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]-3[\/latex]. The point is to the left of [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]3[\/latex]. The coordinates of the point are [latex]\\left(-3,3\\right)[\/latex].<\/p>\n<p>Point B is below [latex]-1[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]-1[\/latex]. The point is to the left of [latex]-3[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]-3[\/latex]. The coordinates of the point are [latex]\\left(-1,-3\\right)[\/latex].<\/p>\n<p>Point C is above [latex]2[\/latex] on the [latex]x\\text{-axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]2[\/latex]. The point is to the right of [latex]4[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]4[\/latex]. The coordinates of the point are [latex]\\left(2,4\\right)[\/latex].<\/p>\n<p>Point D is below [latex]4[\/latex] on the [latex]x-\\text{axis}[\/latex], so the [latex]x\\text{-coordinate}[\/latex] of the point is [latex]4[\/latex]. The point is to the right of [latex]-4[\/latex] on the [latex]y\\text{-axis}[\/latex], so the [latex]y\\text{-coordinate}[\/latex] of the point is [latex]-4[\/latex]. The coordinates of the point are [latex]\\left(4,-4\\right)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146915\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146915&theme=oea&iframe_resize_id=ohm146915&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146919\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146919&theme=oea&iframe_resize_id=ohm146919&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Watch the following video for another example of how to determine the ordered pair for points on the coordinate plane.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Determine the Ordered Pairs for Points Plotted on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/c9WVU34MY5Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":370291,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1194","chapter","type-chapter","status-publish","hentry"],"part":657,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions"}],"predecessor-version":[{"id":2684,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions\/2684"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/657"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}