{"id":1205,"date":"2021-10-18T22:35:28","date_gmt":"2021-10-18T22:35:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1205"},"modified":"2022-10-17T17:17:22","modified_gmt":"2022-10-17T17:17:22","slug":"5-3-3-graphs-and-tables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-3-3-graphs-and-tables\/","title":{"raw":"5.3.2: Graphs and Tables","rendered":"5.3.2: Graphs and Tables"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph coordinate pairs from a table of values<\/li>\r\n \t<li>Create a table of values from a graph<\/li>\r\n \t<li>Write a relation as a set of ordered pairs<\/li>\r\n \t<li>Identify the domain and range of a relation<\/li>\r\n \t<li>Identify local maxima and minima<\/li>\r\n \t<li>Identify lines of symmetry<\/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 style=\"margin-top: 0.5em;\">A <b>relation<\/b> is a set of ordered pairs.<\/li>\r\n \t<li>The\u00a0<strong>domain<\/strong> of a relation is the set of all [latex]x-[\/latex]values.<\/li>\r\n \t<li>The\u00a0<strong>range<\/strong> of a relationship is the set of all\u00a0[latex]y-[\/latex]values.<\/li>\r\n \t<li>A <strong>local maximum<\/strong> is the\u00a0[latex]y-[\/latex]value of a point where the graph turns to go back up after moving downwards.<\/li>\r\n \t<li>A <strong>local minimum<\/strong>\u00a0is the [latex]y-[\/latex]value of a point where the graph turns to go back down after moving upwards.<\/li>\r\n \t<li>A\u00a0<strong>maximum<\/strong> is the highest\u00a0[latex]y-[\/latex]value on the graph.<\/li>\r\n \t<li>A\u00a0<strong>minimum<\/strong> is the lowest\u00a0[latex]y-[\/latex]value on the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphs and Tables<\/h2>\r\nGraphs and tables have a symbiotic relationship. Ordered pairs can be written in the rows and columns of a table or graphed as points using the rectangular coordinate system. The same data can be displayed as a table or a graph.\r\n\r\nFigure 1 shows a table of data points that when converted to [latex](x,y)[\/latex] coordinates can be plotted on a rectangular coordinate system to become a graph.\r\n<div class=\"bcc-box bcc-highlight\">\r\n\r\n<img class=\"alignright wp-image-1220\" style=\"font-size: 16px; orphans: 1; widows: 2;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19000919\/Points-on-a-graph-300x223.png\" alt=\"Points on a graph\" width=\"600\" height=\"446\" \/>\r\n<h2><\/h2>\r\n<table style=\"border-collapse: collapse; width: 5.513489139822785%; height: 218px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 2.202643171806166%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th style=\"width: 2.202643171806166%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]-4[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFigure 1\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nA set of ordered pairs is called a <strong><em>relation<\/em><\/strong>. From figure 1, the relation is [latex]\\{(-4,7),(-2,6),(0,5),(2,4),(4,3),(6,2),(8,0)\\}[\/latex]. The set of all values of the <em><strong>independent variable<\/strong><\/em> [latex]x[\/latex] is the\u00a0<em><strong>domain<\/strong><\/em> of the relation, while the set of all the values of the <em><strong>dependent variable<\/strong><\/em> [latex]y[\/latex] is the <em><strong>range<\/strong><\/em> of the relation. For the data in figure 1, domain = [latex]\\{-4,-2,0,2,4,6,8\\}[\/latex] and range = [latex]\\{0,2,3,4,5,6,7\\}[\/latex].\r\n<div class=\"shaded textbox\">\r\n<h3>Relation<\/h3>\r\nA <em><strong>relation<\/strong><\/em> is a set of ordered pairs [latex](x,y)[\/latex].\r\n\r\nThe\u00a0<em><strong>domain<\/strong><\/em> of a relation is the set of all\u00a0[latex]x[\/latex]-values.\r\n\r\nThe <em><strong>range<\/strong><\/em> of a relation is the set of all\u00a0[latex]y[\/latex]-values.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWe can also convert from coordinates on a graph to data points in a table.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the graph to complete the table. Then state the domain and range of the relation.\r\n\r\n<img class=\"aligncenter wp-image-1226\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19002251\/Example-points-to-table-300x244.png\" alt=\"Points on a graph\" width=\"600\" height=\"487\" \/>\r\n<table style=\"border-collapse: collapse; width: 13.970756696497173%; height: 163px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]7[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\n<table style=\"border-collapse: collapse; width: 4.69483568075117%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]7[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]5[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\ndomain = [latex]\\{-4,-1,0,3,5,7\\}[\/latex] and range = [latex]\\{-5,-3,0,2,4\\}[\/latex]\r\n\r\nWe only listed\u00a0[latex]4[\/latex] once because it is not necessary to list it every time it appears in the range.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\nList the domain and range for the following table of values, then graph the data points on a coordinate system.\r\n<table style=\"width: 2.5%;\">\r\n<tbody>\r\n<tr>\r\n<th class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\"><i>x<\/i><\/th>\r\n<th class=\"border\" style=\"width: 2.5%; text-align: center;\"><i>y<\/i><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22123[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22122[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22121[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"594198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594198\"]\r\n\r\nThe domain describes all the [latex]x[\/latex]-values (independent variable values), and we can use set notation with braces { } to make the list.\r\n\r\n[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]\r\n\r\nThe range describes all the [latex]y[\/latex]-values (dependent variable values).\r\n\r\n[latex]\\text{Range}:\\{4\\}[\/latex]\r\n\r\nWe only listed\u00a0[latex]4[\/latex] once because it is not necessary to list it every time it appears in the range.\r\n\r\n<img class=\"aligncenter wp-image-1242\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19164722\/Example-relation-1024x720.png\" alt=\"points with y = 4 in a horizontal line \" width=\"548\" height=\"385\" \/><img class=\"\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe graphs we have seen so far in this section have been of relations with a finite number of points. Some relations have an infinite number of points, with the points lying so close together on a graph they form a curve or a line. Since the relation has an infinite number of points, it is impossible to write every ordered pair in a table. However, a few points can be tabulated to show a pattern.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the graph to state the domain and range of the relation. Then complete the table of values.\r\n\r\n<img class=\"aligncenter wp-image-1246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19171530\/CIrcle-graph.png\" alt=\"Graph of circle with radius 3\" width=\"500\" height=\"497\" \/>\r\n<table style=\"border-collapse: collapse; width: 50%; height: 58px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\"><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-3<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\"><\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">-3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nThe domain is the set of all possible [latex]x[\/latex]-values. In this graph the [latex]x[\/latex]-values\u00a0start at [latex]x=-3[\/latex] and end at [latex]x=3[\/latex]. Both end points are included in the domain. This set of values can be written using interval notation or using set-builder notation.\r\n\r\n[latex]\\text{Domain}:\\left[-3,3\\right][\/latex] or [latex]\\{x|-3\u2264x\u22643,\\;x\\in\\mathbb{R}\\}[\/latex].\r\n\r\nThe range is the set of all possible [latex]y[\/latex]-values. In this graph the [latex]y[\/latex]-values\u00a0start at [latex]y=-3[\/latex] and end at [latex]y=3[\/latex]. Both end points are included in the range. This set of values can be written using interval notation or using set notation.\r\n\r\n[latex]\\text{Range}:\\left[-3,3\\right][\/latex] or [latex]\\{y|-3\u2264y\u22643,\\;y\\in\\mathbb{R}\\}[\/latex].\r\n\r\nTo complete the table we look for the points on the graph with the given\u00a0[latex]x[\/latex] or\u00a0[latex]y[\/latex] values.\r\n<table style=\"border-collapse: collapse; width: 4.694835680751174%; height: 113px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-3<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">-3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nNotice that this graph has an infinite number of <em><strong>lines of symmetry<\/strong><\/em>. A line of symmetry is any line that cuts the graph in half, with each half being a mirror image of the other. In a circle, every diameter is a line of symmetry.\r\n\r\n<\/div>\r\n<div class=\"shaded textbox\">\r\n<h3>LINE OF SYMMETRY<\/h3>\r\n<p style=\"text-align: center;\">A <strong>line of symmetry<\/strong> is any line that cuts the graph in half, with each half being a mirror image of the other.<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the graph to state the domain and range of the relation. Then complete the table of values by identifying the points on the graph. In addition, describe any lines of symmetry.\r\n\r\n<img class=\"alignleft wp-image-1268 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19183822\/Quartic-2-376x1024.png\" alt=\"\" width=\"376\" height=\"1024\" \/>\r\n<table style=\"border-collapse: collapse; width: 4.694835680751174%;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-2<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-1<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.347417840375588%; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Solution<\/h2>\r\nThe domain is the set of all possible [latex]x[\/latex]-values. In this graph the [latex]x[\/latex]-values\u00a0start at [latex]x=-\\infty[\/latex] and end at [latex]x=\\infty[\/latex]. Since [latex]\\infty[\/latex] can never be reached, the end points are not included in the domain. This set of values can be written using interval notation or using set notation.\r\n\r\n[latex]\\text{Domain}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{x|-\\infty\u2264x\u2264\\infty,\\;x\\in\\mathbb{R}\\}[\/latex].\r\n\r\nThe range is the set of all possible [latex]y[\/latex]-values. In this graph the [latex]y[\/latex]-values\u00a0start at [latex]y=-4[\/latex] and end at [latex]y=\\infty[\/latex]. The lowest value of [latex]y=-4[\/latex]\u00a0is included in the range, but [latex]y=\\infty[\/latex] is not. This set of values can be written using interval notation or using set notation.\r\n\r\n[latex]\\text{Range}:\\left[-4,\\infty\\right)[\/latex] or [latex]\\{y|\\;y\u2265-4,\\;y\\in\\mathbb{R}\\}[\/latex].\r\n\r\nTo complete the table we look for the points on the graph with the given\u00a0[latex]x[\/latex]values.\r\n<table style=\"border-collapse: collapse; width: 4.694835680751174%; height: 113px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-2<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-1<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 2.347417840375588%; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis graph has the [latex]y-[\/latex]axis as a line of symmetry.\r\n\r\n&nbsp;\r\n\r\nNotice that this graph has three turning points: a <em><strong>local maximum<\/strong><\/em>\u00a0of [latex]0[\/latex]\u00a0at\u00a0[latex]x=0[\/latex] and two <em><strong>local minima<\/strong><\/em> of\u00a0[latex]-4[\/latex] at\u00a0[latex]x[\/latex]-values close to\u00a0[latex]-1.4[\/latex] and\u00a0[latex]1.4[\/latex]. The exact\u00a0[latex]x[\/latex] values are impossible to tell from the graph. The local maximum of 0 is not the overall maximum of the graph, which is\u00a0[latex]\\infty[\/latex]. Rather is it the maximum value in an area around\u00a0[latex]x=0[\/latex]. On the other hand, the local minima value of\u00a0[latex]-4[\/latex] is also the overall minimum value of the graph.\r\n\r\n<\/div>\r\n<div class=\"shaded textbox\">\r\n<h3>local maxima and minima<\/h3>\r\n<ul>\r\n \t<li>A <strong>local maximum<\/strong> is the\u00a0[latex]y-[\/latex]value of a point where the graph turns to go back up after moving downwards.<\/li>\r\n \t<li>A <strong>local minimum<\/strong>\u00a0is the [latex]y-[\/latex]value of a point where the graph turns to go back down after moving upwards.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\nUse the graph to state the domain and range of the relation. Then state any local maxima or minima.\r\n\r\n<img class=\"aligncenter wp-image-1270\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19190333\/cubic-300x300.png\" alt=\"Graph of a cubic relation\" width=\"500\" height=\"500\" \/>\r\n\r\n[reveal-answer q=\"H000065\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H000065\"]\r\n\r\n[latex]\\text{Domain}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{x|\\;x\\in\\mathbb{R}\\}[\/latex].\r\n\r\n[latex]\\text{Range}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{y|\\;y\\in\\mathbb{R}\\}[\/latex].\r\n\r\nLocal maxima of approximately [latex]6.2[\/latex] at [latex]x\\approx-3.2[\/latex].\r\n\r\nLocal minima of approximately [latex]-6.2[\/latex] at [latex]x\\approx3.2[\/latex].\r\n\r\nThe symbol [latex]\\approx[\/latex] is read \"approximately equal to\".\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph coordinate pairs from a table of values<\/li>\n<li>Create a table of values from a graph<\/li>\n<li>Write a relation as a set of ordered pairs<\/li>\n<li>Identify the domain and range of a relation<\/li>\n<li>Identify local maxima and minima<\/li>\n<li>Identify lines of symmetry<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>KEY words<\/h3>\n<ul>\n<li style=\"margin-top: 0.5em;\">A <b>relation<\/b> is a set of ordered pairs.<\/li>\n<li>The\u00a0<strong>domain<\/strong> of a relation is the set of all [latex]x-[\/latex]values.<\/li>\n<li>The\u00a0<strong>range<\/strong> of a relationship is the set of all\u00a0[latex]y-[\/latex]values.<\/li>\n<li>A <strong>local maximum<\/strong> is the\u00a0[latex]y-[\/latex]value of a point where the graph turns to go back up after moving downwards.<\/li>\n<li>A <strong>local minimum<\/strong>\u00a0is the [latex]y-[\/latex]value of a point where the graph turns to go back down after moving upwards.<\/li>\n<li>A\u00a0<strong>maximum<\/strong> is the highest\u00a0[latex]y-[\/latex]value on the graph.<\/li>\n<li>A\u00a0<strong>minimum<\/strong> is the lowest\u00a0[latex]y-[\/latex]value on the graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphs and Tables<\/h2>\n<p>Graphs and tables have a symbiotic relationship. Ordered pairs can be written in the rows and columns of a table or graphed as points using the rectangular coordinate system. The same data can be displayed as a table or a graph.<\/p>\n<p>Figure 1 shows a table of data points that when converted to [latex](x,y)[\/latex] coordinates can be plotted on a rectangular coordinate system to become a graph.<\/p>\n<div class=\"bcc-box bcc-highlight\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-1220\" style=\"font-size: 16px; orphans: 1; widows: 2;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19000919\/Points-on-a-graph-300x223.png\" alt=\"Points on a graph\" width=\"600\" height=\"446\" \/><\/p>\n<h2><\/h2>\n<table style=\"border-collapse: collapse; width: 5.513489139822785%; height: 218px;\">\n<tbody>\n<tr>\n<th style=\"width: 2.202643171806166%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th style=\"width: 2.202643171806166%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]-4[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 2.202643171806166%; text-align: right;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Figure 1<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>A set of ordered pairs is called a <strong><em>relation<\/em><\/strong>. From figure 1, the relation is [latex]\\{(-4,7),(-2,6),(0,5),(2,4),(4,3),(6,2),(8,0)\\}[\/latex]. The set of all values of the <em><strong>independent variable<\/strong><\/em> [latex]x[\/latex] is the\u00a0<em><strong>domain<\/strong><\/em> of the relation, while the set of all the values of the <em><strong>dependent variable<\/strong><\/em> [latex]y[\/latex] is the <em><strong>range<\/strong><\/em> of the relation. For the data in figure 1, domain = [latex]\\{-4,-2,0,2,4,6,8\\}[\/latex] and range = [latex]\\{0,2,3,4,5,6,7\\}[\/latex].<\/p>\n<div class=\"shaded textbox\">\n<h3>Relation<\/h3>\n<p>A <em><strong>relation<\/strong><\/em> is a set of ordered pairs [latex](x,y)[\/latex].<\/p>\n<p>The\u00a0<em><strong>domain<\/strong><\/em> of a relation is the set of all\u00a0[latex]x[\/latex]-values.<\/p>\n<p>The <em><strong>range<\/strong><\/em> of a relation is the set of all\u00a0[latex]y[\/latex]-values.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>We can also convert from coordinates on a graph to data points in a table.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the graph to complete the table. Then state the domain and range of the relation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1226\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19002251\/Example-points-to-table-300x244.png\" alt=\"Points on a graph\" width=\"600\" height=\"487\" \/><\/p>\n<table style=\"border-collapse: collapse; width: 13.970756696497173%; height: 163px;\">\n<tbody>\n<tr>\n<th class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]7[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<table style=\"border-collapse: collapse; width: 4.69483568075117%;\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]7[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]-4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.3474178403755843%; text-align: right;\">[latex]5[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: right;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>domain = [latex]\\{-4,-1,0,3,5,7\\}[\/latex] and range = [latex]\\{-5,-3,0,2,4\\}[\/latex]<\/p>\n<p>We only listed\u00a0[latex]4[\/latex] once because it is not necessary to list it every time it appears in the range.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>List the domain and range for the following table of values, then graph the data points on a coordinate system.<\/p>\n<table style=\"width: 2.5%;\">\n<tbody>\n<tr>\n<th class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\"><i>x<\/i><\/th>\n<th class=\"border\" style=\"width: 2.5%; text-align: center;\"><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]\u22121[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\" scope=\"row\">[latex]3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.5%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594198\">Show Solution<\/span><\/p>\n<div id=\"q594198\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain describes all the [latex]x[\/latex]-values (independent variable values), and we can use set notation with braces { } to make the list.<\/p>\n<p>[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]<\/p>\n<p>The range describes all the [latex]y[\/latex]-values (dependent variable values).<\/p>\n<p>[latex]\\text{Range}:\\{4\\}[\/latex]<\/p>\n<p>We only listed\u00a0[latex]4[\/latex] once because it is not necessary to list it every time it appears in the range.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1242\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19164722\/Example-relation-1024x720.png\" alt=\"points with y = 4 in a horizontal line\" width=\"548\" height=\"385\" \/><img decoding=\"async\" class=\"\" src=\"src\" alt=\"image\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The graphs we have seen so far in this section have been of relations with a finite number of points. Some relations have an infinite number of points, with the points lying so close together on a graph they form a curve or a line. Since the relation has an infinite number of points, it is impossible to write every ordered pair in a table. However, a few points can be tabulated to show a pattern.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the graph to state the domain and range of the relation. Then complete the table of values.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19171530\/CIrcle-graph.png\" alt=\"Graph of circle with radius 3\" width=\"500\" height=\"497\" \/><\/p>\n<table style=\"border-collapse: collapse; width: 50%; height: 58px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\"><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">3<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-3<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\"><\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">-3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>The domain is the set of all possible [latex]x[\/latex]-values. In this graph the [latex]x[\/latex]-values\u00a0start at [latex]x=-3[\/latex] and end at [latex]x=3[\/latex]. Both end points are included in the domain. This set of values can be written using interval notation or using set-builder notation.<\/p>\n<p>[latex]\\text{Domain}:\\left[-3,3\\right][\/latex] or [latex]\\{x|-3\u2264x\u22643,\\;x\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>The range is the set of all possible [latex]y[\/latex]-values. In this graph the [latex]y[\/latex]-values\u00a0start at [latex]y=-3[\/latex] and end at [latex]y=3[\/latex]. Both end points are included in the range. This set of values can be written using interval notation or using set notation.<\/p>\n<p>[latex]\\text{Range}:\\left[-3,3\\right][\/latex] or [latex]\\{y|-3\u2264y\u22643,\\;y\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>To complete the table we look for the points on the graph with the given\u00a0[latex]x[\/latex] or\u00a0[latex]y[\/latex] values.<\/p>\n<table style=\"border-collapse: collapse; width: 4.694835680751174%; height: 113px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">3<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-3<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">-3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Notice that this graph has an infinite number of <em><strong>lines of symmetry<\/strong><\/em>. A line of symmetry is any line that cuts the graph in half, with each half being a mirror image of the other. In a circle, every diameter is a line of symmetry.<\/p>\n<\/div>\n<div class=\"shaded textbox\">\n<h3>LINE OF SYMMETRY<\/h3>\n<p style=\"text-align: center;\">A <strong>line of symmetry<\/strong> is any line that cuts the graph in half, with each half being a mirror image of the other.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the graph to state the domain and range of the relation. Then complete the table of values by identifying the points on the graph. In addition, describe any lines of symmetry.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1268 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19183822\/Quartic-2-376x1024.png\" alt=\"\" width=\"376\" height=\"1024\" \/><\/p>\n<table style=\"border-collapse: collapse; width: 4.694835680751174%;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-2<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-1<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.347417840375588%; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Solution<\/h2>\n<p>The domain is the set of all possible [latex]x[\/latex]-values. In this graph the [latex]x[\/latex]-values\u00a0start at [latex]x=-\\infty[\/latex] and end at [latex]x=\\infty[\/latex]. Since [latex]\\infty[\/latex] can never be reached, the end points are not included in the domain. This set of values can be written using interval notation or using set notation.<\/p>\n<p>[latex]\\text{Domain}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{x|-\\infty\u2264x\u2264\\infty,\\;x\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>The range is the set of all possible [latex]y[\/latex]-values. In this graph the [latex]y[\/latex]-values\u00a0start at [latex]y=-4[\/latex] and end at [latex]y=\\infty[\/latex]. The lowest value of [latex]y=-4[\/latex]\u00a0is included in the range, but [latex]y=\\infty[\/latex] is not. This set of values can be written using interval notation or using set notation.<\/p>\n<p>[latex]\\text{Range}:\\left[-4,\\infty\\right)[\/latex] or [latex]\\{y|\\;y\u2265-4,\\;y\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>To complete the table we look for the points on the graph with the given\u00a0[latex]x[\/latex]values.<\/p>\n<table style=\"border-collapse: collapse; width: 4.694835680751174%; height: 113px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th class=\"border\" style=\"width: 2.347417840375588%; height: 10px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 2.3474178403755865%; height: 10px; text-align: center;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-2<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">-1<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 2.347417840375588%; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 2.347417840375588%; height: 12px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"width: 2.3474178403755865%; height: 12px; text-align: center;\">\u00a0[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This graph has the [latex]y-[\/latex]axis as a line of symmetry.<\/p>\n<p>&nbsp;<\/p>\n<p>Notice that this graph has three turning points: a <em><strong>local maximum<\/strong><\/em>\u00a0of [latex]0[\/latex]\u00a0at\u00a0[latex]x=0[\/latex] and two <em><strong>local minima<\/strong><\/em> of\u00a0[latex]-4[\/latex] at\u00a0[latex]x[\/latex]-values close to\u00a0[latex]-1.4[\/latex] and\u00a0[latex]1.4[\/latex]. The exact\u00a0[latex]x[\/latex] values are impossible to tell from the graph. The local maximum of 0 is not the overall maximum of the graph, which is\u00a0[latex]\\infty[\/latex]. Rather is it the maximum value in an area around\u00a0[latex]x=0[\/latex]. On the other hand, the local minima value of\u00a0[latex]-4[\/latex] is also the overall minimum value of the graph.<\/p>\n<\/div>\n<div class=\"shaded textbox\">\n<h3>local maxima and minima<\/h3>\n<ul>\n<li>A <strong>local maximum<\/strong> is the\u00a0[latex]y-[\/latex]value of a point where the graph turns to go back up after moving downwards.<\/li>\n<li>A <strong>local minimum<\/strong>\u00a0is the [latex]y-[\/latex]value of a point where the graph turns to go back down after moving upwards.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>Use the graph to state the domain and range of the relation. Then state any local maxima or minima.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1270\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/19190333\/cubic-300x300.png\" alt=\"Graph of a cubic relation\" width=\"500\" height=\"500\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH000065\">Show Answer<\/span><\/p>\n<div id=\"qH000065\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\text{Domain}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{x|\\;x\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>[latex]\\text{Range}:\\left[-\\infty,\\infty\\right][\/latex] or [latex]\\{y|\\;y\\in\\mathbb{R}\\}[\/latex].<\/p>\n<p>Local maxima of approximately [latex]6.2[\/latex] at [latex]x\\approx-3.2[\/latex].<\/p>\n<p>Local minima of approximately [latex]-6.2[\/latex] at [latex]x\\approx3.2[\/latex].<\/p>\n<p>The symbol [latex]\\approx[\/latex] is read &#8220;approximately equal to&#8221;.<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1205\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Graphs and Tables. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":370291,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Graphs and Tables\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1205","chapter","type-chapter","status-publish","hentry"],"part":657,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1205","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":61,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1205\/revisions"}],"predecessor-version":[{"id":2700,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1205\/revisions\/2700"}],"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\/1205\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1205"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1205"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1205"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}