{"id":663,"date":"2022-02-02T17:04:35","date_gmt":"2022-02-02T17:04:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=663"},"modified":"2025-12-09T16:53:48","modified_gmt":"2025-12-09T16:53:48","slug":"1-2-2-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-2-2-graphs-of-functions\/","title":{"raw":"1.2: Graphs of Functions","rendered":"1.2: Graphs of Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Identify different types of graphs\r\n<ul>\r\n \t<li>Points<\/li>\r\n \t<li>Lines<\/li>\r\n \t<li>Curves<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Identify features of graphs\r\n<ul>\r\n \t<li>[latex]x[\/latex]-intercepts and [latex]y[\/latex]-intercept<\/li>\r\n \t<li>Maximum and minimum function values<\/li>\r\n \t<li>Local maximums and minimums<\/li>\r\n \t<li>Symmetry<\/li>\r\n \t<li>Asymptotes<\/li>\r\n \t<li>Domain and range<\/li>\r\n \t<li>Intersection points<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Types of Graphs<\/h2>\r\nThe graph of a function can take on many shapes. It may be a line (Figure 1) or a curve (Figure 2). However, it could also be a line segment (Figure 3) or just points on the coordinate plane (Figure 4).\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Linear<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Quadratic<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1722\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1722\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22185423\/desmos-graph-49-300x300.png\" alt=\"Graph of a line\" width=\"270\" height=\"270\" \/> Figure 1. Function as a line.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_659\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-659\" style=\"text-align: start;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02163517\/1-1-1-Parabola-300x300.png\" alt=\"Graph of a curve.\" width=\"270\" height=\"270\" \/> FIgure 2. Function as a curve.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">line segment<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">points<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1723\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1723\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22185908\/desmos-graph-50-300x300.png\" alt=\"Line segment\" width=\"270\" height=\"270\" \/> Figure 3. Function as a line segment.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1724\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1724\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22190117\/desmos-graph-51-300x300.png\" alt=\"Points on a graph\" width=\"270\" height=\"270\" \/> Figure 4. Function as points.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe graph of a line segment has endpoints. When the endpoints are included in the graph they are solid, like the ones in figure 3. When the endpoints are open circles, they are excluded from the graph. Both the domain and the range of the function in figure 3 are [latex][1, 6][\/latex], but they are different sets.\u00a0The domain consists of values of [latex]x[\/latex], and the range consists of values of [latex]f(x)[\/latex].\r\n\r\nWhen a function is graphed as a set of points, it can be written as a set of ordered pairs. The function in figure 4 can be written, [latex]f(x)=\\{ (-2, -2), (1, 1), (2, 2), (3, 3), (6, 6)\\}[\/latex]. Here the domain is the set of [latex]x[\/latex]-values [latex]\\lbrace -2, 1, 2, 3, 6\\rbrace[\/latex] while the range is the set of function values\u00a0[latex]\\lbrace -2, 1, 2, 3, 6\\rbrace[\/latex]. Even though the sets have the same elements, they represent different things i.e. the domain ([latex]x[\/latex]-values) and the range ([latex]f(x)[\/latex]-values).\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nState the domain of the graphed function:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-2020 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-300x300.png\" alt=\"Partial graph of a parabola starting at a closed point (-2,9) and ending at an open point (2,0).\" width=\"300\" height=\"300\" \/><\/p>\r\n\r\n<h4>Solution<\/h4>\r\nThe graph starts at (and includes) the point (\u20132, 8) and ends at (but does not include) the point (2, 0).\r\n\r\nSInce the domain is the set of [latex]x[\/latex]-values, the domain is the set of all [latex]x[\/latex]-values between \u20132 (included) and 2 (not included).\r\n\r\nDomain = [\u20132, 2)\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nState the domain and range of the graphed function:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-750 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-259x300.png\" alt=\"An inverted parabola starting at the closed circle (-3,-6) with vertex (0,3) and ending at the open circle (2,-1).\" width=\"259\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm017\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm017\"]\r\n\r\nDomain = [latex][-3, 2)[\/latex]\r\n\r\nRange = [latex][-6, 3][\/latex]\r\n\r\n[\/hidden-answer]\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<h3>Continuous and Non-continuous Graphs<\/h3>\r\n<em><strong>Continuous graphs<\/strong><\/em> are\u00a0graphs where there is a value of [latex]y[\/latex] for every value of [latex]x[\/latex], and each point is immediately next to the point on either side of it so that the line of the graph is uninterrupted (e.g., figure 5). In other words, if the line is continuous, the graph is continuous. On the other hand, a graph is <em><strong>non-continuous<\/strong><\/em> if there is any break in the graph (e.g., figure 6). There is a break in the graph at [latex]x=0[\/latex]. In other words, if we can draw the graph without taking the pen off the paper, the graph is continuous. Otherwise, it is non-continuous.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">No breaks<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">A break at x=0<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_774\" align=\"aligncenter\" width=\"249\"]<img class=\"wp-image-774\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17030814\/continuous-graph-187x300.png\" alt=\"Graph that is continuous\" width=\"249\" height=\"400\" \/> Figure 5. A continuous graph.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_773\" align=\"aligncenter\" width=\"318\"]<img class=\"wp-image-773\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17030809\/not-continuous-238x300.png\" alt=\"graph that is not continuous\" width=\"318\" height=\"400\" \/> Figure 6. A non-continuous graph.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine if the graph is continuous or not continuous.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter size-medium wp-image-2021\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/05210749\/desmos-graph-2022-05-05T150735.883-300x300.png\" alt=\"graph with a jump\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm792\"]Not continuous.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Features of Graphs<\/h2>\r\n<h3>[latex]x[\/latex]- and [latex]y[\/latex]-Intercepts<\/h3>\r\nThere are some features of graphs that attract our attention. First, we are interested in the intersection point(s) between the graph and the [latex]x[\/latex]-axis. These intersection points are called [latex]x[\/latex]-intercepts. We are also interested in the intersection point between the graph and the [latex]y[\/latex]-axis; the [latex]y[\/latex]-intercept. Figure 7 shows the graph of a function with two [latex]x[\/latex]-intercepts at (1, 0) and (3, 0) and one [latex]y[\/latex]-intercept at (0, 3). Any time the graph represents a function, there can only be at most one [latex]y[\/latex]-intercept. Do you see why? [reveal-answer q=\"hjm039\"]Tell me why.[\/reveal-answer]\r\n[hidden-answer a=\"hjm039\"]For a graph to represent a function, a vertical line must pass through at most one point on the graph. So, if there are more than one [latex]y[\/latex]-intercepts, the line [latex]x=0[\/latex] would pass through all the [latex]y[\/latex]-intercepts and the graph would not be a function.[\/hidden-answer]\r\n\r\n<img class=\"aligncenter wp-image-672 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-300x300.png\" alt=\"A function passing through the y-axis at (0,3), and the x-axis at (1,0) and (3,0).\" width=\"300\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Figure 7. Function with [latex]x[\/latex]- and [latex]y[\/latex]-intercepts.<\/p>\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nIdentify the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts on the graph.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-743 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-300x300.png\" alt=\"parabola passing through (0,-2), (negative one half, 0), and (3.5,0).\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm586\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm586\"]\r\n\r\n[latex]x[\/latex]-intercepts: (0.25, 0) and (3.75, 0)\r\n\r\n[latex]y[\/latex]-intercepts: (0, -1)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Maximum and Minimum Values and Turning Points<\/h3>\r\nThe <em><strong>maximum<\/strong><\/em> and <em><strong>minimum values<\/strong><\/em> of a function are also interesting features of a graph. Sometimes the maximum or minimum values of a function occur at <em><strong>turning points<\/strong><\/em>. For the graph in Figure 8, the turning point of the purple graph on the left is at point [latex](\u20132, 2)[\/latex]. Therefore, the maximum value of this function is 2. The turning point of the green graph on the right is [latex](5, \u20133)[\/latex]. It shows the minimum value of the function, \u20133.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">One turning point<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Two turning points<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_679\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-679 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-300x300.png\" alt=\"Two parabolas, each showing one turning point.\" width=\"300\" height=\"300\" \/> Figure 8. Turning points.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_741\" align=\"aligncenter\" width=\"319\"]<img class=\"wp-image-741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16202644\/Function-with-2-turning-points-300x282.png\" alt=\"Function with 2 turning points\" width=\"319\" height=\"300\" \/> Figure 9. Function with two turning points.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"orphans: 1;\">The graph in figure 9 has two turning points, but neither of these points are maximum or minimum values of the function. Indeed, the range of the function is [latex](-\\infty, +\\infty)[\/latex] so the graphs have no maximum or\u00a0minimum. The turning points on the graph in figure 9 are referred to as a\u00a0<\/span><em style=\"font-size: 16px; orphans: 1;\"><strong>local maximum<\/strong><\/em><span style=\"font-size: 16px; orphans: 1;\">\u00a0and a<\/span><em style=\"font-size: 16px; orphans: 1;\"><strong>\u00a0local minimum<\/strong><\/em><span style=\"font-size: 16px; orphans: 1;\">. They show maximum and minimum values within a certain set of [latex]x[\/latex]-values. For example, the graph of the function in figure 9, has a local minimum value of zero in the domain [latex][-4, 4][\/latex] and a local maximum in the domain [latex][-9, -5][\/latex].<\/span>\r\n\r\nWhen there are local maxima and minima on a graph, we call the overall maximum and minimum of the function the <em><strong>global maximum<\/strong><\/em> and <em><strong>global minimum<\/strong><\/em>.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nIdentify the global maximum and minimum values of the function on the graph.\r\n\r\n<img class=\"aligncenter wp-image-744 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-224x300.png\" alt=\"A polynomial with three tuning points that goes up to infinity on the ends, and has two low points at (-1.5,-2.5) and (1.5,-2.5)\" width=\"224\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm098\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm098\"]\r\n\r\nThere is no global maximum since the graph heads towards [latex]+\\infty[\/latex]\r\n\r\nGlobal minimum = -2.25\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Symmetry<\/h3>\r\n<p style=\"text-align: left;\">A graph of a function is symmetrical is there is a <em><strong>line of symmetry<\/strong><\/em> such that the image of one side is the reflection of the image of the other side regarding the line of symmetry. The graph in Figure 7 is symmetrical about the vertical line [latex]x = 2[\/latex]. The graphs in Figure 8 are symmetrical about vertical the line [latex]x = \u20132[\/latex] for the purple graph on the left and the vertical line [latex]x = 5[\/latex] for the graph on the right. The graph in figure 9 has no line of symmetry.<\/p>\r\nNot all lines of symmetry are vertical. Figure 10 shows the graph of a function that does not have a vertical line of symmetry. However, it does have a line of symmetry; the line [latex]y=x[\/latex]. For example, the points (1, 6) and (6, 1) both lie on the graph and are mirror images of one another across the line\u00a0the line [latex]y=x[\/latex].\r\n\r\n[caption id=\"attachment_770\" align=\"aligncenter\" width=\"298\"]<img class=\"wp-image-770 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-298x300.png\" alt=\"Graph with line of symmetry y=x described above.\" width=\"298\" height=\"300\" \/> Figure 10. Graph with line of symmetry [latex]y=x[\/latex].[\/caption]\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nDetermine the line of symmetry.\r\n\r\n<img class=\"aligncenter size-medium wp-image-744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16213447\/3-turning-points-224x300.png\" alt=\"Graph with 3 turning points\" width=\"224\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm167\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm167\"]\r\n\r\nThe [latex]y[\/latex]-axis; [latex]x=0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Asymptotes<\/h3>\r\nA <em><strong>vertical asymptote<\/strong><\/em> is a vertical line that the graph continually approaches as close as it can get but never actually crosses. At a vertical asymptote the graph gets closer to either positive or negative infinity. A <em><strong>horizontal asymptote<\/strong><\/em>\u00a0is a horizontal line that the graph continually approaches as close as it can get as [latex]x[\/latex] gets closer and closer to either positive or negative infinity.\r\n\r\nFor example,the graph in figure 11 has a vertical asymptote at [latex]x=-2[\/latex]. [latex]x[\/latex] never reaches \u20132, but as it gets closer and closer to \u20132, [latex]f(x)[\/latex] gets closer to [latex]-\\infty[\/latex].\u00a0 The graph in figure 12 has a horizontal asymptote at [latex]y=-4[\/latex]. [latex]f(x)[\/latex] never reaches \u20134, but as [latex]x[\/latex] gets closer to [latex]-\\infty[\/latex], [latex]f(x)[\/latex] gets closer and closer to \u20134.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 47.9079%; text-align: center;\">\r\n<div class=\"mceTemp\">A logarithmic function<\/div><\/th>\r\n<th style=\"width: 52.0921%; text-align: center;\">\r\n<div class=\"mceTemp\">An exponential function<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.9079%;\">\r\n\r\n[caption id=\"attachment_767\" align=\"aligncenter\" width=\"261\"]<img class=\"wp-image-767 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-261x300.png\" alt=\"A continuous function with vertical asymptote x= -2.\" width=\"261\" height=\"300\" \/> Figure 11. Graph of a function with a vertical asymptote.[\/caption]<\/td>\r\n<td style=\"width: 52.0921%;\">\r\n\r\n[caption id=\"attachment_768\" align=\"aligncenter\" width=\"377\"]<img class=\"wp-image-768\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17022622\/y2%5Ex-4-300x239.png\" alt=\"A continuous function with horizontal asymptote y=-4.\" width=\"377\" height=\"300\" \/> Figure 12. Graph of a function with a horizontal asymptote.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe graph in figure 13 has both horizontal and vertical asymptotes. The vertical lines [latex]x=-1[\/latex] and [latex]x=4[\/latex] are vertical asymptotes. Moving from left to right, the graph never makes it to [latex]x=-1[\/latex] but gets incredibly close as [latex]f(x)[\/latex] veers towards infinity. On the other side of\u00a0[latex]x=-1[\/latex] coming from right to left, the graph never makes it to the asymptote [latex]x=-1[\/latex] as [latex]f(x)[\/latex] veers towards negative infinity. The other vertical asymptote is at [latex]x=4[\/latex]. Vertical asymptotes are never crossed.\r\n\r\n[caption id=\"attachment_746\" align=\"aligncenter\" width=\"391\"]<img class=\"wp-image-746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16220512\/asymptotes1-300x233.png\" alt=\"Graph of a rational function with horizontal asymptote y = 3, and vertical asymptotes of x = -1 and x = 4.\" width=\"391\" height=\"304\" \/> Figure 13. Graph with asymptotes.[\/caption]\r\n\r\nIn figure 13, there is also a horizontal asymptote at [latex]y=3[\/latex]. As [latex]x[\/latex] gets closer to either positive or negative infinity, the graph gets closer and closer to the line\u00a0[latex]y=3[\/latex].\r\n\r\nHorizontal asymptotes can be crossed as the graph of the function in figure 14 shows. The vertical asymptotes are the lines [latex]x=-4[\/latex] and [latex]x=2[\/latex], and they are never crossed. The horizontal asymptote is the line [latex]y=3[\/latex]. As [latex]x[\/latex] gets closer and closer to positive or negative infinity, the graph gets closer and closer to\u00a0[latex]y=3[\/latex]. Notice that the graph crosses the horizontal asymptote at the point [latex](0, 3)[\/latex].\r\n\r\n[caption id=\"attachment_769\" align=\"aligncenter\" width=\"441\"]<img class=\"wp-image-769\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17023501\/yxoverx-1x43-300x206.png\" alt=\"Graph of a rational function described above, showing horizontal asymptotes may be crossed within some finite distance of the center of the graph.\" width=\"441\" height=\"303\" \/> Figure 14. Horizontal asymptotes can be crossed.[\/caption]\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDetermine the horizontal and vertical asymptotes of the graphed function:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-771\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17025805\/y6overx-32-300x203.png\" alt=\"The graph f a rational function approaching the lines y = 2, and x=3.\" width=\"440\" height=\"298\" \/><\/p>\r\n[reveal-answer q=\"hjm877\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm877\"]\r\n\r\nHorizontal asymptote: [latex]y=2[\/latex]\r\n\r\nVertical asymptote: [latex]x=3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Multiple Graphs<\/h3>\r\nIf there are two or more graphs on the coordinate plane, we may be interested in their intersection point(s). Figure 15 shows that the intersection point of the two lines is (1, 3), while figure 16 shows two intersection points at (\u20133, 6) and (2, \u20131) of the two functions.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">One intersection point<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Two intersection points<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_687\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-687 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-300x300.png\" alt=\"An example of two lines that intersect at one point, the point (1,3)\" width=\"300\" height=\"300\" \/> Figure 15. Intersection of two functions at one point.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_752\" align=\"aligncenter\" width=\"335\"]<img class=\"wp-image-752\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17013904\/2-intersection-points-300x268.png\" alt=\"An example of an inverted parabola and a line intersecting in two points.\" width=\"335\" height=\"300\" \/> Figure 16. Intersection of two functions at two points.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox tryit\">\r\n<h3>Tr-3,-5)y It 7<\/h3>\r\nState the intersection points of the two functions on the graph.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-759\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17020145\/intersecting-parabolas-197x300.png\" alt=\"A parabola and an inverted parabola intersecting twice, at (-3,-5) and (3,-5).\" width=\"288\" height=\"439\" \/><\/p>\r\n[reveal-answer q=\"hjm460\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm460\"][latex](-3, -5)[\/latex] and [latex](3, -5)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Identify different types of graphs\n<ul>\n<li>Points<\/li>\n<li>Lines<\/li>\n<li>Curves<\/li>\n<\/ul>\n<\/li>\n<li>Identify features of graphs\n<ul>\n<li>[latex]x[\/latex]-intercepts and [latex]y[\/latex]-intercept<\/li>\n<li>Maximum and minimum function values<\/li>\n<li>Local maximums and minimums<\/li>\n<li>Symmetry<\/li>\n<li>Asymptotes<\/li>\n<li>Domain and range<\/li>\n<li>Intersection points<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Types of Graphs<\/h2>\n<p>The graph of a function can take on many shapes. It may be a line (Figure 1) or a curve (Figure 2). However, it could also be a line segment (Figure 3) or just points on the coordinate plane (Figure 4).<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Linear<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Quadratic<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1722\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1722\" class=\"wp-image-1722\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22185423\/desmos-graph-49-300x300.png\" alt=\"Graph of a line\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1722\" class=\"wp-caption-text\">Figure 1. Function as a line.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_659\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-659\" class=\"wp-image-659\" style=\"text-align: start;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02163517\/1-1-1-Parabola-300x300.png\" alt=\"Graph of a curve.\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-659\" class=\"wp-caption-text\">FIgure 2. Function as a curve.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">line segment<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">points<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1723\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1723\" class=\"wp-image-1723\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22185908\/desmos-graph-50-300x300.png\" alt=\"Line segment\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1723\" class=\"wp-caption-text\">Figure 3. Function as a line segment.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1724\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1724\" class=\"wp-image-1724\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22190117\/desmos-graph-51-300x300.png\" alt=\"Points on a graph\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1724\" class=\"wp-caption-text\">Figure 4. Function as points.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graph of a line segment has endpoints. When the endpoints are included in the graph they are solid, like the ones in figure 3. When the endpoints are open circles, they are excluded from the graph. Both the domain and the range of the function in figure 3 are [latex][1, 6][\/latex], but they are different sets.\u00a0The domain consists of values of [latex]x[\/latex], and the range consists of values of [latex]f(x)[\/latex].<\/p>\n<p>When a function is graphed as a set of points, it can be written as a set of ordered pairs. The function in figure 4 can be written, [latex]f(x)=\\{ (-2, -2), (1, 1), (2, 2), (3, 3), (6, 6)\\}[\/latex]. Here the domain is the set of [latex]x[\/latex]-values [latex]\\lbrace -2, 1, 2, 3, 6\\rbrace[\/latex] while the range is the set of function values\u00a0[latex]\\lbrace -2, 1, 2, 3, 6\\rbrace[\/latex]. Even though the sets have the same elements, they represent different things i.e. the domain ([latex]x[\/latex]-values) and the range ([latex]f(x)[\/latex]-values).<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>State the domain of the graphed function:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2020 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-300x300.png\" alt=\"Partial graph of a parabola starting at a closed point (-2,9) and ending at an open point (2,0).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/desmos-graph-2022-05-05T145514.926.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The graph starts at (and includes) the point (\u20132, 8) and ends at (but does not include) the point (2, 0).<\/p>\n<p>SInce the domain is the set of [latex]x[\/latex]-values, the domain is the set of all [latex]x[\/latex]-values between \u20132 (included) and 2 (not included).<\/p>\n<p>Domain = [\u20132, 2)<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>State the domain and range of the graphed function:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-750 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-259x300.png\" alt=\"An inverted parabola starting at the closed circle (-3,-6) with vertex (0,3) and ending at the open circle (2,-1).\" width=\"259\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-259x300.png 259w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-768x891.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-883x1024.png 883w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-65x75.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-225x261.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1-350x406.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Graph-segment1.png 1224w\" sizes=\"auto, (max-width: 259px) 100vw, 259px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm017\">Show Answer<\/span><\/p>\n<div id=\"qhjm017\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex][-3, 2)[\/latex]<\/p>\n<p>Range = [latex][-6, 3][\/latex]<\/p>\n<\/div>\n<\/div>\n<p style=\"text-align: center;\">\n<\/div>\n<h3>Continuous and Non-continuous Graphs<\/h3>\n<p><em><strong>Continuous graphs<\/strong><\/em> are\u00a0graphs where there is a value of [latex]y[\/latex] for every value of [latex]x[\/latex], and each point is immediately next to the point on either side of it so that the line of the graph is uninterrupted (e.g., figure 5). In other words, if the line is continuous, the graph is continuous. On the other hand, a graph is <em><strong>non-continuous<\/strong><\/em> if there is any break in the graph (e.g., figure 6). There is a break in the graph at [latex]x=0[\/latex]. In other words, if we can draw the graph without taking the pen off the paper, the graph is continuous. Otherwise, it is non-continuous.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">No breaks<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">A break at x=0<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_774\" style=\"width: 259px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-774\" class=\"wp-image-774\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17030814\/continuous-graph-187x300.png\" alt=\"Graph that is continuous\" width=\"249\" height=\"400\" \/><\/p>\n<p id=\"caption-attachment-774\" class=\"wp-caption-text\">Figure 5. A continuous graph.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_773\" style=\"width: 328px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-773\" class=\"wp-image-773\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17030809\/not-continuous-238x300.png\" alt=\"graph that is not continuous\" width=\"318\" height=\"400\" \/><\/p>\n<p id=\"caption-attachment-773\" class=\"wp-caption-text\">Figure 6. A non-continuous graph.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine if the graph is continuous or not continuous.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2021\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/05210749\/desmos-graph-2022-05-05T150735.883-300x300.png\" alt=\"graph with a jump\" width=\"300\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm792\">Show Answer<\/span><\/p>\n<div id=\"qhjm792\" class=\"hidden-answer\" style=\"display: none\">Not continuous.<\/div>\n<\/div>\n<\/div>\n<h2>Features of Graphs<\/h2>\n<h3>[latex]x[\/latex]&#8211; and [latex]y[\/latex]-Intercepts<\/h3>\n<p>There are some features of graphs that attract our attention. First, we are interested in the intersection point(s) between the graph and the [latex]x[\/latex]-axis. These intersection points are called [latex]x[\/latex]-intercepts. We are also interested in the intersection point between the graph and the [latex]y[\/latex]-axis; the [latex]y[\/latex]-intercept. Figure 7 shows the graph of a function with two [latex]x[\/latex]-intercepts at (1, 0) and (3, 0) and one [latex]y[\/latex]-intercept at (0, 3). Any time the graph represents a function, there can only be at most one [latex]y[\/latex]-intercept. Do you see why? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm039\">Tell me why.<\/span><\/p>\n<div id=\"qhjm039\" class=\"hidden-answer\" style=\"display: none\">For a graph to represent a function, a vertical line must pass through at most one point on the graph. So, if there are more than one [latex]y[\/latex]-intercepts, the line [latex]x=0[\/latex] would pass through all the [latex]y[\/latex]-intercepts and the graph would not be a function.<\/div>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-672 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-300x300.png\" alt=\"A function passing through the y-axis at (0,3), and the x-axis at (1,0) and (3,0).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-Intercepts-350x350.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 7. Function with [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Identify the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts on the graph.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-743 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-300x300.png\" alt=\"parabola passing through (0,-2), (negative one half, 0), and (3.5,0).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-768x767.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-1024x1022.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola-350x349.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/parabola.png 1128w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm586\">Show Answer<\/span><\/p>\n<div id=\"qhjm586\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x[\/latex]-intercepts: (0.25, 0) and (3.75, 0)<\/p>\n<p>[latex]y[\/latex]-intercepts: (0, -1)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Maximum and Minimum Values and Turning Points<\/h3>\n<p>The <em><strong>maximum<\/strong><\/em> and <em><strong>minimum values<\/strong><\/em> of a function are also interesting features of a graph. Sometimes the maximum or minimum values of a function occur at <em><strong>turning points<\/strong><\/em>. For the graph in Figure 8, the turning point of the purple graph on the left is at point [latex](\u20132, 2)[\/latex]. Therefore, the maximum value of this function is 2. The turning point of the green graph on the right is [latex](5, \u20133)[\/latex]. It shows the minimum value of the function, \u20133.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">One turning point<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Two turning points<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_679\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-679\" class=\"wp-image-679 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-300x300.png\" alt=\"Two parabolas, each showing one turning point.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-MaxMin.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-679\" class=\"wp-caption-text\">Figure 8. Turning points.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_741\" style=\"width: 329px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-741\" class=\"wp-image-741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16202644\/Function-with-2-turning-points-300x282.png\" alt=\"Function with 2 turning points\" width=\"319\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-741\" class=\"wp-caption-text\">Figure 9. Function with two turning points.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"orphans: 1;\">The graph in figure 9 has two turning points, but neither of these points are maximum or minimum values of the function. Indeed, the range of the function is [latex](-\\infty, +\\infty)[\/latex] so the graphs have no maximum or\u00a0minimum. The turning points on the graph in figure 9 are referred to as a\u00a0<\/span><em style=\"font-size: 16px; orphans: 1;\"><strong>local maximum<\/strong><\/em><span style=\"font-size: 16px; orphans: 1;\">\u00a0and a<\/span><em style=\"font-size: 16px; orphans: 1;\"><strong>\u00a0local minimum<\/strong><\/em><span style=\"font-size: 16px; orphans: 1;\">. They show maximum and minimum values within a certain set of [latex]x[\/latex]-values. For example, the graph of the function in figure 9, has a local minimum value of zero in the domain [latex][-4, 4][\/latex] and a local maximum in the domain [latex][-9, -5][\/latex].<\/span><\/p>\n<p>When there are local maxima and minima on a graph, we call the overall maximum and minimum of the function the <em><strong>global maximum<\/strong><\/em> and <em><strong>global minimum<\/strong><\/em>.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Identify the global maximum and minimum values of the function on the graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-744 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-224x300.png\" alt=\"A polynomial with three tuning points that goes up to infinity on the ends, and has two low points at (-1.5,-2.5) and (1.5,-2.5)\" width=\"224\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-224x300.png 224w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-768x1028.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-765x1024.png 765w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-65x87.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-225x301.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points-350x469.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3-turning-points.png 962w\" sizes=\"auto, (max-width: 224px) 100vw, 224px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm098\">Show Answer<\/span><\/p>\n<div id=\"qhjm098\" class=\"hidden-answer\" style=\"display: none\">\n<p>There is no global maximum since the graph heads towards [latex]+\\infty[\/latex]<\/p>\n<p>Global minimum = -2.25<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Symmetry<\/h3>\n<p style=\"text-align: left;\">A graph of a function is symmetrical is there is a <em><strong>line of symmetry<\/strong><\/em> such that the image of one side is the reflection of the image of the other side regarding the line of symmetry. The graph in Figure 7 is symmetrical about the vertical line [latex]x = 2[\/latex]. The graphs in Figure 8 are symmetrical about vertical the line [latex]x = \u20132[\/latex] for the purple graph on the left and the vertical line [latex]x = 5[\/latex] for the graph on the right. The graph in figure 9 has no line of symmetry.<\/p>\n<p>Not all lines of symmetry are vertical. Figure 10 shows the graph of a function that does not have a vertical line of symmetry. However, it does have a line of symmetry; the line [latex]y=x[\/latex]. For example, the points (1, 6) and (6, 1) both lie on the graph and are mirror images of one another across the line\u00a0the line [latex]y=x[\/latex].<\/p>\n<div id=\"attachment_770\" style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-770\" class=\"wp-image-770 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-298x300.png\" alt=\"Graph with line of symmetry y=x described above.\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-768x773.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-1017x1024.png 1017w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-225x227.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry-350x352.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/yx-symmetry.png 1124w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p id=\"caption-attachment-770\" class=\"wp-caption-text\">Figure 10. Graph with line of symmetry [latex]y=x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Determine the line of symmetry.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16213447\/3-turning-points-224x300.png\" alt=\"Graph with 3 turning points\" width=\"224\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm167\">Show Answer<\/span><\/p>\n<div id=\"qhjm167\" class=\"hidden-answer\" style=\"display: none\">\n<p>The [latex]y[\/latex]-axis; [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Asymptotes<\/h3>\n<p>A <em><strong>vertical asymptote<\/strong><\/em> is a vertical line that the graph continually approaches as close as it can get but never actually crosses. At a vertical asymptote the graph gets closer to either positive or negative infinity. A <em><strong>horizontal asymptote<\/strong><\/em>\u00a0is a horizontal line that the graph continually approaches as close as it can get as [latex]x[\/latex] gets closer and closer to either positive or negative infinity.<\/p>\n<p>For example,the graph in figure 11 has a vertical asymptote at [latex]x=-2[\/latex]. [latex]x[\/latex] never reaches \u20132, but as it gets closer and closer to \u20132, [latex]f(x)[\/latex] gets closer to [latex]-\\infty[\/latex].\u00a0 The graph in figure 12 has a horizontal asymptote at [latex]y=-4[\/latex]. [latex]f(x)[\/latex] never reaches \u20134, but as [latex]x[\/latex] gets closer to [latex]-\\infty[\/latex], [latex]f(x)[\/latex] gets closer and closer to \u20134.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 47.9079%; text-align: center;\">\n<div class=\"mceTemp\">A logarithmic function<\/div>\n<\/th>\n<th style=\"width: 52.0921%; text-align: center;\">\n<div class=\"mceTemp\">An exponential function<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 47.9079%;\">\n<div id=\"attachment_767\" style=\"width: 271px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-767\" class=\"wp-image-767 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-261x300.png\" alt=\"A continuous function with vertical asymptote x= -2.\" width=\"261\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-261x300.png 261w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-768x882.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-892x1024.png 892w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-65x75.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-225x258.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2-350x402.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y3lnx2.png 1186w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/p>\n<p id=\"caption-attachment-767\" class=\"wp-caption-text\">Figure 11. Graph of a function with a vertical asymptote.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 52.0921%;\">\n<div id=\"attachment_768\" style=\"width: 387px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-768\" class=\"wp-image-768\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17022622\/y2%5Ex-4-300x239.png\" alt=\"A continuous function with horizontal asymptote y=-4.\" width=\"377\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-768\" class=\"wp-caption-text\">Figure 12. Graph of a function with a horizontal asymptote.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graph in figure 13 has both horizontal and vertical asymptotes. The vertical lines [latex]x=-1[\/latex] and [latex]x=4[\/latex] are vertical asymptotes. Moving from left to right, the graph never makes it to [latex]x=-1[\/latex] but gets incredibly close as [latex]f(x)[\/latex] veers towards infinity. On the other side of\u00a0[latex]x=-1[\/latex] coming from right to left, the graph never makes it to the asymptote [latex]x=-1[\/latex] as [latex]f(x)[\/latex] veers towards negative infinity. The other vertical asymptote is at [latex]x=4[\/latex]. Vertical asymptotes are never crossed.<\/p>\n<div id=\"attachment_746\" style=\"width: 401px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-746\" class=\"wp-image-746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16220512\/asymptotes1-300x233.png\" alt=\"Graph of a rational function with horizontal asymptote y = 3, and vertical asymptotes of x = -1 and x = 4.\" width=\"391\" height=\"304\" \/><\/p>\n<p id=\"caption-attachment-746\" class=\"wp-caption-text\">Figure 13. Graph with asymptotes.<\/p>\n<\/div>\n<p>In figure 13, there is also a horizontal asymptote at [latex]y=3[\/latex]. As [latex]x[\/latex] gets closer to either positive or negative infinity, the graph gets closer and closer to the line\u00a0[latex]y=3[\/latex].<\/p>\n<p>Horizontal asymptotes can be crossed as the graph of the function in figure 14 shows. The vertical asymptotes are the lines [latex]x=-4[\/latex] and [latex]x=2[\/latex], and they are never crossed. The horizontal asymptote is the line [latex]y=3[\/latex]. As [latex]x[\/latex] gets closer and closer to positive or negative infinity, the graph gets closer and closer to\u00a0[latex]y=3[\/latex]. Notice that the graph crosses the horizontal asymptote at the point [latex](0, 3)[\/latex].<\/p>\n<div id=\"attachment_769\" style=\"width: 451px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-769\" class=\"wp-image-769\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17023501\/yxoverx-1x43-300x206.png\" alt=\"Graph of a rational function described above, showing horizontal asymptotes may be crossed within some finite distance of the center of the graph.\" width=\"441\" height=\"303\" \/><\/p>\n<p id=\"caption-attachment-769\" class=\"wp-caption-text\">Figure 14. Horizontal asymptotes can be crossed.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Determine the horizontal and vertical asymptotes of the graphed function:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-771\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17025805\/y6overx-32-300x203.png\" alt=\"The graph f a rational function approaching the lines y = 2, and x=3.\" width=\"440\" height=\"298\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm877\">Show Answer<\/span><\/p>\n<div id=\"qhjm877\" class=\"hidden-answer\" style=\"display: none\">\n<p>Horizontal asymptote: [latex]y=2[\/latex]<\/p>\n<p>Vertical asymptote: [latex]x=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Multiple Graphs<\/h3>\n<p>If there are two or more graphs on the coordinate plane, we may be interested in their intersection point(s). Figure 15 shows that the intersection point of the two lines is (1, 3), while figure 16 shows two intersection points at (\u20133, 6) and (2, \u20131) of the two functions.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">One intersection point<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Two intersection points<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_687\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-687\" class=\"wp-image-687 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-300x300.png\" alt=\"An example of two lines that intersect at one point, the point (1,3)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/1-2-2-TwoLines.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-687\" class=\"wp-caption-text\">Figure 15. Intersection of two functions at one point.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_752\" style=\"width: 345px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-752\" class=\"wp-image-752\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17013904\/2-intersection-points-300x268.png\" alt=\"An example of an inverted parabola and a line intersecting in two points.\" width=\"335\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-752\" class=\"wp-caption-text\">Figure 16. Intersection of two functions at two points.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox tryit\">\n<h3>Tr-3,-5)y It 7<\/h3>\n<p>State the intersection points of the two functions on the graph.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-759\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/17020145\/intersecting-parabolas-197x300.png\" alt=\"A parabola and an inverted parabola intersecting twice, at (-3,-5) and (3,-5).\" width=\"288\" height=\"439\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm460\">Show Answer<\/span><\/p>\n<div id=\"qhjm460\" class=\"hidden-answer\" style=\"display: none\">[latex](-3, -5)[\/latex] and [latex](3, -5)[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/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-663\">\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 of Functions. <strong>Authored by<\/strong>: Leo Chang and 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><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna &amp; Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>. <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":422608,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Graphs of Functions\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna & Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\",\"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-663","chapter","type-chapter","status-publish","hentry"],"part":613,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/663","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":51,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/663\/revisions"}],"predecessor-version":[{"id":4721,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/663\/revisions\/4721"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/613"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/663\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=663"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=663"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=663"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=663"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}