{"id":16502,"date":"2019-10-03T17:08:31","date_gmt":"2019-10-03T17:08:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/plot-ordered-pairs\/"},"modified":"2020-10-22T09:15:00","modified_gmt":"2020-10-22T09:15:00","slug":"plot-ordered-pairs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/plot-ordered-pairs\/","title":{"raw":"9.1.a - Plotting Points on the Coordinate Plane","rendered":"9.1.a &#8211; Plotting Points on the Coordinate Plane"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify quadrants on the coordinate plane<\/li>\r\n \t<li>Plot points on a rectangular coordinate system<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left\">Introduction to the Coordinate Plane<\/h2>\r\nYou have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object or location? Many maps, such as the Campus Map shown below, use a grid system to identify locations. This map uses a horizontal and vertical grid to convey information about an object\u2019s location. Do you see the numbers [latex]1,2,3[\/latex], and [latex]4[\/latex] across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.\r\n\r\nThe general location of any item on this map can be found by using the letter and number of its grid square.\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. In which grid section is the Stadium? The Stadium is in section 4D.\r\n\r\n<img class=\"aligncenter\" 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.\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the map above.\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 4C?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"668289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668289\"]\r\n\r\nSolution\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]4[\/latex] across the bottom of the map and C along the side. Look below the [latex]4[\/latex] and next to the C. Tiger Field is in grid section 4C.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Rectangular Coordinate System<\/h2>\r\nThe coordinate plane has similar elements to the grid shown above.\u00a0 Look at the Rectangular Coordinate System show in the box below. It consists of a horizontal <b>axis<\/b> and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)\r\n\r\nThe horizontal axis in the coordinate plane is called the [latex]x-axis[\/latex]. The vertical axis is called the [latex]y-axis[\/latex]. 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 <i>x-<\/i> and <i>y-<\/i>axes of the coordinate plane divide it\u00a0into four sections. These four sections are called <b>quadrants<\/b>. 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 system<\/h3>\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\/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\" \/>\r\n\r\n<\/div>\r\nLocations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the <i>x-<\/i>axis (the first value of the ordered pair) and along the [latex]y-axis[\/latex] (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 <b>x-coordinate<\/b> and the second value is the <b>y-coordinate<\/b>. Note that the <i>x-<\/i>coordinate is listed before the <i>y-<\/i>coordinate. Since the origin has an <i>x-<\/i>coordinate of [latex]0[\/latex] and a <i>y-<\/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<h2 id=\"Plotting Points in the Coordinate Plane\" class=\"no-indent\" style=\"text-align: left\">Plotting Points in the Coordinate Plane<\/h2>\r\nNow that you know the components of a rectangular system, let's learn how to plot ordered pairs, that is locate a point on the coordinate system given an ordered pair. Just remember, the process start at the origin\u2014the beginning!\r\n\r\nLet\u2019s try locating the point [latex]\\left(2,5\\right)[\/latex]. In this ordered pair, the [latex]x[\/latex]-coordinate is [latex]2[\/latex] and the [latex]y[\/latex]-coordinate is [latex]5[\/latex] .\r\n\r\nWe start by locating the [latex]x[\/latex] value, [latex]2[\/latex], on the [latex]x\\text{-axis.}[\/latex] Then we lightly sketch a vertical line through [latex]x=2[\/latex], as shown in the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224719\/CNX_BMath_Figure_11_01_004.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.\" \/>\r\nNow we locate the [latex]y[\/latex] value, [latex]5[\/latex], on the [latex]y[\/latex] -axis and sketch a horizontal line through [latex]y=5[\/latex] . The point where these two lines meet is the point with coordinates [latex]\\left(2,5\\right)[\/latex]. We plot the point there, as shown in the image below.\r\n\r\n<img class=\"aligncenter\" 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 \" \/>\r\n\r\n<span style=\"color: #ff0000\">Reminder:<\/span>\u00a0 Any ordered pair, such as (2,5), represents one single point on the graph.\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 sketch a vertical line through [latex]x=1[\/latex]. The [latex]y\\text{-coordinate}[\/latex] is [latex]3[\/latex] so we find [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex] and sketch a horizontal line through [latex]y=3[\/latex]. Where the two lines meet, we 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] and sketch a vertical line through [latex]x=3[\/latex]. Then we find [latex]1[\/latex] on the [latex]y\\text{-axis}[\/latex] and sketch a horizontal line through [latex]y=1[\/latex]. Where the two lines meet, we 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&nbsp;\r\n\r\nNow, let's take another look at how to plot the ordered pair (1,3).\u00a0 The method show below is slightly different from the method shown above, but yields the same result.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nPlot the point [latex](1, 3)[\/latex].\r\n\r\n[reveal-answer q=\"28562\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"28562\"]\r\n\r\nThe <i>x-<\/i>coordinate is [latex]1[\/latex] because it comes first in the ordered pair. Start at the origin and move a distance of [latex]1[\/latex] unit in a positive direction (to the right) from the origin along the <i>x-<\/i>axis.\r\n\r\nThe <i>y-<\/i>coordinate is [latex]3[\/latex] because it comes second in the ordered pair. From here move directly [latex]3[\/latex] units in a positive direction (up). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]3[\/latex] on that axis.\r\n<h4>Answer<\/h4>\r\nDraw a point at this location and label the point [latex](1, 3)[\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064232\/image005.jpg\" alt=\"Graph. A blue arrow goes from the origin (the point 0,0) to one unit to the right. A red arrow goes from the blue arrow to 3 units above the x-axis. That point is (1,3).\" width=\"417\" height=\"378\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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\nIn the previous examples, both the <i>x-<\/i> and <i>y-<\/i>coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.\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 <i>x-<\/i>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 <i>y-<\/i>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\nThe steps for plotting a point are summarized below.\r\n<div class=\"textbox shaded\">\r\n<h3>Steps for Plotting an Ordered Pair (<i>x<\/i>, <i>y<\/i>) in the Coordinate Plane<\/h3>\r\n<ul>\r\n \t<li>Determine the <i>x-<\/i>coordinate. Beginning at the origin, move horizontally, the direction of the <i>x<\/i>-axis, the distance given by the <i>x-<\/i>coordinate. If the <i>x-<\/i>coordinate is positive, move to the right; if the <i>x-<\/i>coordinate is negative, move to the left.<\/li>\r\n \t<li>Determine the <i>y-<\/i>coordinate. Beginning at the <i>x-<\/i>coordinate, move vertically, the direction of the <i>y<\/i>-axis, the distance given by the <i>y-<\/i>coordinate. If the <i>y-<\/i>coordinate is positive, move up; if the <i>y-<\/i>coordinate is negative, move down.<\/li>\r\n \t<li>Draw a point at the ending location. Label the point with the ordered pair.<\/li>\r\n \t<li>An ordered pair is represented by a <strong>single<\/strong> point on the graph.<\/li>\r\n<\/ul>\r\n<\/div>\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[reveal-answer q=\"83169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"83169\"]\r\n\r\nSolution\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[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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<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<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left\">Identify quadrants and use them to plot points<\/h2>\r\nRecall that when we described the rectangular coordinate system above, we mentioned the four quadrants, 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. Look at each quadrant in the graph below. What do you notice about the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates of the points within each quadrant?\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\/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\" \/>\r\n\r\nWithin each quadrant, the signs of the <i>x-<\/i>coordinates and <i>y-<\/i>coordinates of each ordered pair are the same. They also follow a pattern, which is outlined in the table below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Quadrant<\/th>\r\n<th>General Form of Point in this Quadrant<\/th>\r\n<th>Example<\/th>\r\n<th>Description<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>I<\/td>\r\n<td>[latex](+,+)[\/latex]<\/td>\r\n<td>[latex](5,4)[\/latex]<\/td>\r\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a positive direction (right) and along the <i>y-<\/i>axis in a positive direction (up).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>II<\/td>\r\n<td>[latex](\u2212,+)[\/latex]<\/td>\r\n<td>[latex](\u22125,4)[\/latex]<\/td>\r\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a negative direction (left) and along the <i>y-<\/i>axis in a positive direction (up).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>III<\/td>\r\n<td>[latex](\u2212,\u2212)[\/latex]<\/td>\r\n<td>[latex](\u22125,\u22124)[\/latex]<\/td>\r\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a negative direction (left) and along the <i>y-<\/i>axis in a negative direction (down).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>IV<\/td>\r\n<td>[latex](+,\u2212)[\/latex]<\/td>\r\n<td>[latex](5,\u22124)[\/latex]<\/td>\r\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a positive direction (right) and along the <i>y-<\/i>axis in a negative direction (down).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOnce you know about the quadrants in the coordinate plane, you can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here\u2019s another way to think about it.\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\/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\" \/>\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 you prevent a mistake. It is also useful knowledge for checking that you 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\r\n[reveal-answer q=\"999799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"999799\"]\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,\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.[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nPlot each point in the rectangular coordinate system 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[reveal-answer q=\"933080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933080\"]\r\n\r\nSolution\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[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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\">You can 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\nWhat happens if an ordered pair has an <em>x<\/em>- or <i>y-<\/i>coordinate of zero? The example below shows the graph of the ordered pair [latex](0,4)[\/latex]. Notice how the point [latex]\\left(0,4\\right)[\/latex] is located on the [latex]y[\/latex]-axis.\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\/04064238\/image010-1.jpg\" alt=\"Graph of the point (0,4). The point is on the y-axis.\" width=\"417\" height=\"378\" \/>\r\n\r\nA 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 <i>x-<\/i>coordinate is [latex]0[\/latex], the point is located on the <i>y-<\/i>axis. Similarly, any point that has a <i>y-<\/i>coordinate of [latex]0[\/latex] will be located on the <i>x-<\/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\nWhat is the ordered pair of the point where the axes cross, or interesect? At that point both coordinates are zero, so its ordered pair is [latex]\\left(0,0\\right)[\/latex] . The point has a special name. It is called the <em>origin<\/em>.\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>x<\/em>-axis and <em>y<\/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 grid:\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[reveal-answer q=\"803167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803167\"]\r\n\r\nSolution\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 \" \/>[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\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\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify quadrants on the coordinate plane<\/li>\n<li>Plot points on a rectangular coordinate system<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left\">Introduction to the Coordinate Plane<\/h2>\n<p>You have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object or location? Many maps, such as the Campus Map shown below, use a grid system to identify locations. This map uses a horizontal and vertical grid to convey information about an object\u2019s location. Do you see the numbers [latex]1,2,3[\/latex], and [latex]4[\/latex] across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.<\/p>\n<p>The general location of any item on this map can be found by using the letter and number of its grid square.\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. In which grid section is the Stadium? The Stadium is in section 4D.<\/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\/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.\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the map above.<\/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 4C?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668289\">Show Solution<\/span><\/p>\n<div id=\"q668289\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\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]4[\/latex] across the bottom of the map and C along the side. Look below the [latex]4[\/latex] and next to the C. Tiger Field is in grid section 4C.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Rectangular Coordinate System<\/h2>\n<p>The coordinate plane has similar elements to the grid shown above.\u00a0 Look at the Rectangular Coordinate System show in the box below. It consists of a horizontal <b>axis<\/b> and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)<\/p>\n<p>The horizontal axis in the coordinate plane is called the [latex]x-axis[\/latex]. The vertical axis is called the [latex]y-axis[\/latex]. 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 <i>x-<\/i> and <i>y-<\/i>axes of the coordinate plane divide it\u00a0into four sections. These four sections are called <b>quadrants<\/b>. 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 system<\/h3>\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\/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<\/div>\n<p>Locations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the <i>x-<\/i>axis (the first value of the ordered pair) and along the [latex]y-axis[\/latex] (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 <b>x-coordinate<\/b> and the second value is the <b>y-coordinate<\/b>. Note that the <i>x-<\/i>coordinate is listed before the <i>y-<\/i>coordinate. Since the origin has an <i>x-<\/i>coordinate of [latex]0[\/latex] and a <i>y-<\/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<h2 id=\"Plotting Points in the Coordinate Plane\" class=\"no-indent\" style=\"text-align: left\">Plotting Points in the Coordinate Plane<\/h2>\n<p>Now that you know the components of a rectangular system, let&#8217;s learn how to plot ordered pairs, that is locate a point on the coordinate system given an ordered pair. Just remember, the process start at the origin\u2014the beginning!<\/p>\n<p>Let\u2019s try locating the point [latex]\\left(2,5\\right)[\/latex]. In this ordered pair, the [latex]x[\/latex]-coordinate is [latex]2[\/latex] and the [latex]y[\/latex]-coordinate is [latex]5[\/latex] .<\/p>\n<p>We start by locating the [latex]x[\/latex] value, [latex]2[\/latex], on the [latex]x\\text{-axis.}[\/latex] Then we lightly sketch a vertical line through [latex]x=2[\/latex], as shown in the image below.<\/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\/24224719\/CNX_BMath_Figure_11_01_004.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. There is a vertical dotted line passing through 2 on the x-axis.\" \/><br \/>\nNow we locate the [latex]y[\/latex] value, [latex]5[\/latex], on the [latex]y[\/latex] -axis and sketch a horizontal line through [latex]y=5[\/latex] . The point where these two lines meet is the point with coordinates [latex]\\left(2,5\\right)[\/latex]. We plot the point there, as shown in the image below.<\/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\/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\" \/><\/p>\n<p><span style=\"color: #ff0000\">Reminder:<\/span>\u00a0 Any ordered pair, such as (2,5), represents one single point on the graph.<\/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 sketch a vertical line through [latex]x=1[\/latex]. The [latex]y\\text{-coordinate}[\/latex] is [latex]3[\/latex] so we find [latex]3[\/latex] on the [latex]y\\text{-axis}[\/latex] and sketch a horizontal line through [latex]y=3[\/latex]. Where the two lines meet, we 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] and sketch a vertical line through [latex]x=3[\/latex]. Then we find [latex]1[\/latex] on the [latex]y\\text{-axis}[\/latex] and sketch a horizontal line through [latex]y=1[\/latex]. Where the two lines meet, we 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<p>&nbsp;<\/p>\n<p>Now, let&#8217;s take another look at how to plot the ordered pair (1,3).\u00a0 The method show below is slightly different from the method shown above, but yields the same result.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Plot the point [latex](1, 3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q28562\">Show Solution<\/span><\/p>\n<div id=\"q28562\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <i>x-<\/i>coordinate is [latex]1[\/latex] because it comes first in the ordered pair. Start at the origin and move a distance of [latex]1[\/latex] unit in a positive direction (to the right) from the origin along the <i>x-<\/i>axis.<\/p>\n<p>The <i>y-<\/i>coordinate is [latex]3[\/latex] because it comes second in the ordered pair. From here move directly [latex]3[\/latex] units in a positive direction (up). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]3[\/latex] on that axis.<\/p>\n<h4>Answer<\/h4>\n<p>Draw a point at this location and label the point [latex](1, 3)[\/latex].<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\/04064232\/image005.jpg\" alt=\"Graph. A blue arrow goes from the origin (the point 0,0) to one unit to the right. A red arrow goes from the blue arrow to 3 units above the x-axis. That point is (1,3).\" width=\"417\" height=\"378\" \/><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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>In the previous examples, both the <i>x-<\/i> and <i>y-<\/i>coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.<\/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 <i>x-<\/i>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 <i>y-<\/i>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>The steps for plotting a point are summarized below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Steps for Plotting an Ordered Pair (<i>x<\/i>, <i>y<\/i>) in the Coordinate Plane<\/h3>\n<ul>\n<li>Determine the <i>x-<\/i>coordinate. Beginning at the origin, move horizontally, the direction of the <i>x<\/i>-axis, the distance given by the <i>x-<\/i>coordinate. If the <i>x-<\/i>coordinate is positive, move to the right; if the <i>x-<\/i>coordinate is negative, move to the left.<\/li>\n<li>Determine the <i>y-<\/i>coordinate. Beginning at the <i>x-<\/i>coordinate, move vertically, the direction of the <i>y<\/i>-axis, the distance given by the <i>y-<\/i>coordinate. If the <i>y-<\/i>coordinate is positive, move up; if the <i>y-<\/i>coordinate is negative, move down.<\/li>\n<li>Draw a point at the ending location. Label the point with the ordered pair.<\/li>\n<li>An ordered pair is represented by a <strong>single<\/strong> point on the graph.<\/li>\n<\/ul>\n<\/div>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q83169\">Show Solution<\/span><\/p>\n<div id=\"q83169\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nAs 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>\n<\/div>\n<p>&nbsp;<\/p>\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 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<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left\">Identify quadrants and use them to plot points<\/h2>\n<p>Recall that when we described the rectangular coordinate system above, we mentioned the four quadrants, 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. Look at each quadrant in the graph below. What do you notice about the signs of the <i>x-<\/i> and <i>y-<\/i>coordinates of the points within each quadrant?<\/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\/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>Within each quadrant, the signs of the <i>x-<\/i>coordinates and <i>y-<\/i>coordinates of each ordered pair are the same. They also follow a pattern, which is outlined in the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Quadrant<\/th>\n<th>General Form of Point in this Quadrant<\/th>\n<th>Example<\/th>\n<th>Description<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>I<\/td>\n<td>[latex](+,+)[\/latex]<\/td>\n<td>[latex](5,4)[\/latex]<\/td>\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a positive direction (right) and along the <i>y-<\/i>axis in a positive direction (up).<\/td>\n<\/tr>\n<tr>\n<td>II<\/td>\n<td>[latex](\u2212,+)[\/latex]<\/td>\n<td>[latex](\u22125,4)[\/latex]<\/td>\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a negative direction (left) and along the <i>y-<\/i>axis in a positive direction (up).<\/td>\n<\/tr>\n<tr>\n<td>III<\/td>\n<td>[latex](\u2212,\u2212)[\/latex]<\/td>\n<td>[latex](\u22125,\u22124)[\/latex]<\/td>\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a negative direction (left) and along the <i>y-<\/i>axis in a negative direction (down).<\/td>\n<\/tr>\n<tr>\n<td>IV<\/td>\n<td>[latex](+,\u2212)[\/latex]<\/td>\n<td>[latex](5,\u22124)[\/latex]<\/td>\n<td>Starting from the origin, go along the <i>x-<\/i>axis in a positive direction (right) and along the <i>y-<\/i>axis in a negative direction (down).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Once you know about the quadrants in the coordinate plane, you can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here\u2019s another way to think about it.<\/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\/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>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 you prevent a mistake. It is also useful knowledge for checking that you 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q999799\">Show Solution<\/span><\/p>\n<div id=\"q999799\" 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,\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><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Plot each point in the rectangular coordinate system 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933080\">Show Solution<\/span><\/p>\n<div id=\"q933080\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\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].<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>\n<\/div>\n<p>&nbsp;<\/p>\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\">You can 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>What happens if an ordered pair has an <em>x<\/em>&#8211; or <i>y-<\/i>coordinate of zero? The example below shows the graph of the ordered pair [latex](0,4)[\/latex]. Notice how the point [latex]\\left(0,4\\right)[\/latex] is located on the [latex]y[\/latex]-axis.<\/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\/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>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 <i>x-<\/i>coordinate is [latex]0[\/latex], the point is located on the <i>y-<\/i>axis. Similarly, any point that has a <i>y-<\/i>coordinate of [latex]0[\/latex] will be located on the <i>x-<\/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>What is the ordered pair of the point where the axes cross, or interesect? At that point both coordinates are zero, so its ordered pair is [latex]\\left(0,0\\right)[\/latex] . The point has a special name. It is called the <em>origin<\/em>.<\/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>x<\/em>-axis and <em>y<\/em>-axis intersect.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>Plot each point on a coordinate grid:<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803167\">Show Solution<\/span><\/p>\n<div id=\"q803167\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\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\" \/><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\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>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-16502\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Determine the Ordered Pairs for Points Plotted on the Coordinate Plane. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/c9WVU34MY5Q\">https:\/\/youtu.be\/c9WVU34MY5Q<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Plot Points Given as Ordered Pairs on the Coordinate Plane. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/p_MESleS3mw\">https:\/\/youtu.be\/p_MESleS3mw<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determine the Ordered Pairs for Points Plotted on the Coordinate Plane\",\"author\":\"mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/c9WVU34MY5Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Plot Points Given as Ordered Pairs on the Coordinate Plane\",\"author\":\"mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/p_MESleS3mw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","CANDELA_OUTCOMES_GUID":"04dcacf26ac2496eb29551e7708e024a, 551290c8af9c4829b7d3c3719fcce0c0, a618c895a56741438582f65ed2cac7f0","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16502","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16502","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16502\/revisions"}],"predecessor-version":[{"id":20328,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16502\/revisions\/20328"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16502\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16502"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16502"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16502"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=16502"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}