{"id":39,"date":"2023-11-08T13:32:01","date_gmt":"2023-11-08T13:32:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-graphing-using-ordered-pairs\/"},"modified":"2026-02-05T09:07:39","modified_gmt":"2026-02-05T09:07:39","slug":"1-1-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/1-1-graphs\/","title":{"raw":"1.1 Graphs","rendered":"1.1 Graphs"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the graph of equations of the forms:\r\n<ul>\r\n \t<li>[latex]y=x[\/latex]<\/li>\r\n \t<li>[latex]y=x^2[\/latex]<\/li>\r\n \t<li>[latex]y=x^3[\/latex]<\/li>\r\n \t<li>[latex]y=\\sqrt{x}[\/latex]<\/li>\r\n \t<li>[latex]y=\\dfrac{1}{x}[\/latex]<\/li>\r\n \t<li>[latex]y=|x|[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Create a table of values to graph equations of the forms:\r\n<ul>\r\n \t<li>[latex]y=x[\/latex]<\/li>\r\n \t<li>[latex]y=x^2[\/latex]<\/li>\r\n \t<li>[latex]y=x^3[\/latex]<\/li>\r\n \t<li>[latex]y=\\sqrt{x}[\/latex]<\/li>\r\n \t<li>[latex]y=\\dfrac{1}{x}[\/latex]<\/li>\r\n \t<li>[latex]y=|x|[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nA graph is more than just an image; it's a powerful tool for visually representing the precise relationship between two variables. Mastering the ability to interpret and create graphs is a vital skill that you'll carry into your chosen field and beyond, enhancing your understanding and communication in both your professional and civic life.\r\n\r\nMathematics is a crucial tool for making sense of the world around us. Whether you're a scientist, social scientist, engineer, business leader, health care provider, or politician, strong math skills are essential to achieving your goals. This course will provide you with the mathematical foundation needed to excel in a wide range of disciplines.\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Think about it...<\/h3>\r\nClimate change is one of the most serious threats facing our planet today. The graph below contains information about data collected at an observatory on the Big Island, Mauna Loa, Hawaii. The red graph illustrates the amount of CO<sub>2<\/sub> (carbon dioxide) in the air measured in ppm (parts per million). This number tells us how many parts of carbon dioxide there are in one million parts of air. The green graph illustrates the amount of CO<sub>2<\/sub> in the ocean water surrounding the Hawaiian islands. The blue graph illustrates the pH of the ocean water surrounding the Hawaiian islands.\r\n\r\nGraphs illustrate a precise relationship between two quantities. Make sure to identify the two relationships being compared in each of the three graphs. Observe the labelled axes.\r\n\r\n<strong>Keep in mind:<\/strong>\r\n<ul>\r\n \t<li>The [latex]x[\/latex]--axis is used to describe the independent variable or input.<\/li>\r\n \t<li>The [latex]y[\/latex]-axis is used to describe the dependent variable or output.<\/li>\r\n \t<li>A point on the graph shows how an input and an output are related.<\/li>\r\n<\/ul>\r\n<strong>Questions about the graphs:<\/strong>\r\n\r\na) What do you notice about these graphs-describe in your own words?\r\n\r\nb) On all three graphs, the data shows fluctuations on the graphs. What hypothesis can you make about why these graphs are fluctuating?\r\n\r\nc) Does it appear that the trend of one graph is effecting the other graphs? why or why not? If so, what might be some possible reasons for this effect. If not, explain your reasoning.\r\n\r\nd) Why might some of the graphs be increasing while others are decreasing?\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/1-1-graphs\/screenshot-2024-08-03-at-1-07-01%e2%80%afpm\/\" rel=\"attachment wp-att-1720\"><img class=\"alignnone wp-image-1720 size-large\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-1024x768.png\" alt=\"A graph showing changes in atmospheric CO\u2082, ocean CO\u2082, and ocean pH in the North Pacific over time. The red data shows atmospheric CO\u2082 at Mauna Loa, fluctuating, but rising steadily from about 315 ppm in 1958 to over 420 ppm in 2024. The green data shows ocean CO\u2082 (pCO\u2082) at Station ALOHA, increasing from about 315 \u00b5atm in 1988 to over 400 \u00b5atm in 2024. The blue data shows ocean pH at Station ALOHA, which decreases from about 8.2 to just below 8.05 over the same time.\" width=\"1024\" height=\"768\" \/><\/a>\r\n\r\nHere is a link to an interactive graph of the atmospheric CO<sub>2<\/sub> levels in Mauna Loa, Hawaii. It may be helpful to use this to see the precise the data points on this graph.\r\n\r\n<a href=\"https:\/\/gml.noaa.gov\/ccgg\/trends\/graph.html\">https:\/\/gml.noaa.gov\/ccgg\/trends\/graph.html<\/a>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nAbove we used graphs to describe relationships between quantities but there are other ways we can describe these \u00a0mathematical relationships. Mathematical ideas can be communicated in many ways using different representations.\u00a0Some representations include using tables of values, lists of ordered pairs, words, graphs, expressions, and equations to describe mathematical relationships. In this course, it is important to develop skills using all these representations. We will start with ordered pairs and tables of values.\r\n<h2>The coordinate plane and plotting points<\/h2>\r\nYou have likely used a coordinate plane before.\u00a0The coordinate plane consists of a horizontal axis and a vertical axis; number lines that intersect at right angles. (They are perpendicular to each other.)\r\n\r\n<img class=\"aligncenter size-medium wp-image-1828\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"424\" height=\"422\" \/>\r\n\r\nThe horizontal axis in the coordinate plane is called the [latex]x[\/latex]<b>-axis<\/b>. The vertical axis is called the [latex]y[\/latex]<b>-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at\u00a0[latex]0[\/latex] on the [latex]x[\/latex]<i>-<\/i>axis and\u00a0[latex]0[\/latex] on the [latex]y[\/latex]<i>-<\/i>axis.\r\n\r\nLocations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]<i>-<\/i>axis (the first value of the ordered pair) and along the [latex]y[\/latex]-axis (the second value of the ordered pair).\r\n\r\nIn an ordered pair, such as [latex](x,y)[\/latex], the first value is called the [latex]x[\/latex]<b>-coordinate<\/b>\u00a0or the input and the second value is the [latex]y[\/latex]<b>-coordinate <\/b>or the output. Note that the [latex]x[\/latex]<i>-<\/i>coordinate is listed before the [latex]y[\/latex]<i>-<\/i>coordinate. Since the origin has an [latex]x[\/latex]<i>-<\/i>coordinate of\u00a0[latex]0[\/latex] and a [latex]y[\/latex]<i>-<\/i>coordinate of\u00a0[latex]0[\/latex], its ordered pair is written as [latex](0,0)[\/latex].\r\n\r\nConsider the point below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182920\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/>\r\n\r\nTo identify the location of this point, start at the origin [latex](0,0)[\/latex] and move right along the [latex]x[\/latex]<i>-<\/i>axis until you are under the point. Look at the label on the [latex]x[\/latex]<i>-<\/i>axis. The\u00a0[latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]<i>-<\/i>coordinate; the first number in the ordered pair.\r\n\r\nFrom\u00a0[latex]4[\/latex] on the [latex]x[\/latex]<i>-<\/i>axis move up to the point and notice the number with which it aligns on the [latex]y[\/latex]<i>-<\/i>axis. The\u00a0[latex]3[\/latex] indicates that, after leaving the [latex]x[\/latex]-axis, you traveled\u00a0[latex]3[\/latex] units up in the vertical direction, the direction of the [latex]y[\/latex]-axis. This number is the [latex]y[\/latex]<i>-<\/i>coordinate, the second number in the ordered pair. With an [latex]x[\/latex]<i>-<\/i>coordinate of\u00a0[latex]4[\/latex] and a [latex]y[\/latex]<i>-<\/i>coordinate of\u00a0[latex]3[\/latex], you have the ordered pair [latex](4,3)[\/latex].\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nPractice identifying points from a graph.\r\n\r\n[ohm_question]92753[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using a Table of Values to Graph a Linear Equation<\/h2>\r\nA helpful first step in graphing an equation is to make a table of values. This is particularly useful when you do not know the general shape the graph will have. You probably already know that the graph of a linear equation will be a straight line, but let us make a table first to see how it can be helpful.\r\n\r\nMake a table of values for [latex]y=3x+2[\/latex]. It is a good idea to include negative values, positive values, and zero as inputs. Evaluate the given equation for each input value of [latex]x[\/latex].\r\n<table style=\"width: 75%; border: 1px solid black; font-size: 110%;\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 15%; border: 1px solid #999999;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3x+2[\/latex]<\/th>\r\n<th style=\"width: 25%; border: 1px solid #999999;\">[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\r\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{\u22122})+2=-6+2=-4[\/latex]<\/td>\r\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](-2,-4)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\r\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{\u22121})+2=-3+2=-1[\/latex]<\/td>\r\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](-1,-1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{0})+2=0+2=2[\/latex]<\/td>\r\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](0,2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\r\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{1})+2=3+2=5[\/latex]<\/td>\r\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](1,5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{3}[\/latex]<\/td>\r\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{3})+2=9+2=11[\/latex]<\/td>\r\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](3,11)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow use the ordered pairs to help you draw the graph of the equation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232426\/image006.gif\" alt=\"Line graph on coordinate plane. Line has dots on (negative 2, negative 4), (negative 1, negative 1), (0,2), (1,5) and (3,11).\" width=\"322\" height=\"353\" \/>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nComplete the table of values and plot the linear equation.\r\n<p style=\"text-align: center;\">[ohm_question]170086[\/ohm_question]<\/p>\r\n\r\n<\/div>\r\n<h2>Using a Table of Values to Graph a Quadratic Equation<\/h2>\r\nNot all graphs are straight lines. Next, let's explore the shape of the graph of a Quadratic Equation. Let's use a table of values to graph a basic quadratic equation: [latex]y=x^2[\/latex].\r\nStart by creating a table of values, then plot the ordered pairs.\r\n<div align=\"center\">\r\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-2})^2=4[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-2,4)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-1})^2=1[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-1,1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{0})^2=0[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,0)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{1})^2=1[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{2})^2=4[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](2,4)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points [latex](-2,4), (-1,1), (0,0), (1,1), (2,4)[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232437\/image013.gif\" alt=\"Graph with the point negative 2, 4; the point negative 1, 1; the point 0, 0; the point 1,1; the point 2,4.\" width=\"322\" height=\"353\" \/>\r\n\r\nSince the points are <i>not<\/i> on a line, you cannot use a straight edge. Connect the points as best as you can using a <i>smooth curve<\/i> (not a series of straight lines). You may want to find and plot additional points (such as the ones in blue here). Placing arrows on the tips of the lines implies that they continue in that direction forever.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232441\/image014.gif\" alt=\"Coordinate plane with U shaped curve or parabola connecting the blue points (negative 3,9), (3,9) with red points (negative 2,4), (2,4) and bottom of U shape at (0,0).\" width=\"322\" height=\"353\" \/>\r\n\r\nNotice that the shape is similar to the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The lowest point on this graph is called the vertex. The vertical line that goes through the vertex is called the line of reflection. In this case, that line is the [latex]y[\/latex]-axis.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nComplete the table of values and graph [latex]y=x^2-3[\/latex].\r\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2-3[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"72815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"72815\"]\r\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2-3[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-2})^2-3=1[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-2,1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-1})^2-3=-2[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-1,-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{0})^2-3=-3[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,-3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{1})^2-3=-2[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{2})^2-3=1[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](2,1)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nGraph the points [latex](-2,1), (-1,-2), (0,-3), (1,-2), (2,1)[\/latex] Then connect the points with a smooth curve.\r\n<div align=\"center\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/jym4noksrm?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2><\/h2>\r\n<h2>Using a Table of Values to Graph a Square Root Equation<\/h2>\r\nTo graph a square root equation [latex]y=\\sqrt{x}[\/latex] using a table of values, in this example we will only choose [latex]x[\/latex]-values that are perfect squares. Doing this makes it much easier to evaluate and plot the points.\r\n<div align=\"center\">\r\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 72px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{0}}=0[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,0)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{1}}=1[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{4}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{4}}=2[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](4,2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{9}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{9}}=3[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](9,3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{16}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{16}}=4[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](16,4)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that we only used positive or zero values for the inputs to the square root. Recall that [latex]\\sqrt{x}[\/latex] means to find the number whose square is [latex]x[\/latex]. For example, [latex]\\sqrt{49}[\/latex] means \"find the number whose square is\u00a0[latex]49[\/latex].\"\u00a0Since there is no real number that we can square and get a negative, the equation [latex]y=\\sqrt{x}[\/latex] will be defined for [latex]x&gt;0[\/latex].\r\n\r\nHere's the graph of [latex]y=\\sqrt{x}[\/latex].\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232457\/image026.gif\" alt=\"Curved line branching up and right from the point 0,0.\" width=\"258\" height=\"288\" \/><\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nComplete the table of values and graph [latex]y=\\sqrt{(x+4)}[\/latex].\r\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 70%; border: 1px solid #999999; height: 12px;\">[latex]y=\\sqrt{(x+4)}[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-4}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-3}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{5}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{12}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"897380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"897380\"]\r\n\r\nOn this example, notice the [latex]x[\/latex]-values that we are choosing are not perfect squares like in the explanation above. This time we are choosing [latex]x[\/latex]-values that will make the radicand simplify to be a perfect square. An [latex]x[\/latex] value of [latex]-3[\/latex] was picked below. Notice when I substitute [latex]-3[\/latex] in place of [latex]x[\/latex] in the equation we get [latex]-3+4[\/latex] under the square root. When we simplify that we get [latex]1[\/latex] and that is a perfect square. Check out the other [latex]x[\/latex]-values and notice what happens when you simplify before you take the square root.\r\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 82px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 70%; border: 1px solid #999999; height: 12px;\">[latex]y=\\sqrt{(x+4)}[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-4}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{-4}\\color{black}{)+4}}=\\sqrt{0}=0[\/latex]<\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](-4,0)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-3}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{-3}\\color{black}{)+4}}=\\sqrt{1}=1[\/latex]<\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](-3,1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{0}\\color{black}{)+4}}=\\sqrt{4}=2[\/latex]<\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](0,2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{5}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{5}\\color{black}{)+4}}=\\sqrt{9}=3[\/latex]<\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](5,3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{12}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{12}\\color{black}{)+4}}=\\sqrt{16}=4[\/latex]<\/td>\r\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](12,4)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div align=\"center\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/eqom3dfziw?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2><\/h2>\r\n<h2>Graphs of Common Functions<\/h2>\r\nThroughout our course, you'll use graphs to aid in understanding relationships between input ([latex]x[\/latex]) and output ([latex]y[\/latex]) variables. Learn to recognize these graphs and their features quickly by name, formula, and graph.\r\n<table style=\"border-collapse: collapse; width: 100%; font-size: 110%; height: 126px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Name<\/strong><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Equation<\/strong><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Graph<\/strong><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Table<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 25%; height: 28px; text-align: center;\">Constant<\/td>\r\n<td style=\"width: 25%; height: 28px; text-align: center;\">[latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\r\n<td style=\"width: 25%; height: 28px; text-align: center;\">\u00a0\u00a0\u00a0\u00a0 <img class=\"alignnone wp-image-1607 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1-300x281.jpg\" alt=\"A horizontal line crossing y-axis at 2 on a coordinate plane. Additional points: (negative 2,2) and (2,2).\" width=\"300\" height=\"281\" \/><\/td>\r\n<td style=\"width: 25%; height: 28px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 55.642%; height: 84px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">0<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Linear<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=x[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1609 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1-300x278.jpg\" alt=\"Graph of a diagonal line on a coordinate plane that goes through the origin (0,0). Additional points are (2,2) and (negative 2, negative 2).\" width=\"300\" height=\"278\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 54.8638%; height: 84px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">-2<\/td>\r\n<td style=\"width: 50%;\">-2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2<\/td>\r\n<td style=\"width: 50%;\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Absolute Value<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=|x|[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1606 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg\" alt=\"A graph of an upward facing absolute function (v-shape) on a coordinate plane with the vertex at (0,0). Additional points shown at (negative 2,2) and (2,2).\" width=\"300\" height=\"273\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 49.4163%; height: 64px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">0<\/td>\r\n<td style=\"width: 50%; height: 11px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Quadratic<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y={x}^{2}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1610 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1-300x282.jpg\" alt=\"Graph of parabola or U-shape on a coordinate plane representing y = x squared. Curve comes from positive infinity on left, curves through the origin (0, 0) and rises back up to positive infinity on right.  Additional points at (negative 2,4), (2,4), (negative 1,1) and (1,1).\" width=\"300\" height=\"282\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 59.5331%; height: 111px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">-2<\/td>\r\n<td style=\"width: 50%;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">-1<\/td>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2<\/td>\r\n<td style=\"width: 50%;\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Cubic<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y={x}^{3}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1608 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1-300x277.jpg\" alt=\"Curved graph on a coordinate plane representing y equals x cube that starts at negative infinity left and increases to the right, passing smoothly through the origin (0, 0). Additional points at (negative 1, negative 1), (negative 0.5, just below x-axis), (0.5, just above x-axis) and (1, 1).\" width=\"300\" height=\"277\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 59.5331%; height: 122px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">-1<\/td>\r\n<td style=\"width: 50%;\">-1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">-0.5<\/td>\r\n<td style=\"width: 50%;\">-0.125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">0.5<\/td>\r\n<td style=\"width: 50%;\">0.125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Reciprocal<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=\\frac{1}{x}[\/latex]\r\n\r\nNotice that this equation is not defined for all inputs. We'll talk more about this in the next section.<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1611 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1-300x278.jpg\" alt=\"A rectangular hyperbola from equation 1 over x consists of 2 curves in the first and third quadrants. The graph has asymptotes at x and y axes meaning the curves get infinitely close to these lines by never touch them.\" width=\"300\" height=\"278\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 47.8599%; height: 127px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">-0.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">-1<\/td>\r\n<td style=\"width: 50%; height: 11px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">-0.5<\/td>\r\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">0.5<\/td>\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">1<\/td>\r\n<td style=\"width: 50%; height: 11px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"width: 50%; height: 11px;\">2<\/td>\r\n<td style=\"width: 50%; height: 11px;\">0.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">Square root<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=\\sqrt{x}[\/latex]\r\n\r\nNotice that this equation is not defined for all inputs. We'll talk more about this in the next section.<\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img class=\"alignnone wp-image-1612 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1-300x282.jpg\" alt=\"A graph on a coordinate plane starting at the origin (0, 0) and curving slowly upward to the right with additional points shown at (1, 1) and (4, 2).\" width=\"300\" height=\"282\" \/><\/td>\r\n<td style=\"width: 25%; height: 14px; text-align: center;\">\r\n<table style=\"border-collapse: collapse; width: 47.8599%; height: 88px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<td style=\"width: 50%;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">4<\/td>\r\n<td style=\"width: 50%;\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111722&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"650\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the graph of equations of the forms:\n<ul>\n<li>[latex]y=x[\/latex]<\/li>\n<li>[latex]y=x^2[\/latex]<\/li>\n<li>[latex]y=x^3[\/latex]<\/li>\n<li>[latex]y=\\sqrt{x}[\/latex]<\/li>\n<li>[latex]y=\\dfrac{1}{x}[\/latex]<\/li>\n<li>[latex]y=|x|[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Create a table of values to graph equations of the forms:\n<ul>\n<li>[latex]y=x[\/latex]<\/li>\n<li>[latex]y=x^2[\/latex]<\/li>\n<li>[latex]y=x^3[\/latex]<\/li>\n<li>[latex]y=\\sqrt{x}[\/latex]<\/li>\n<li>[latex]y=\\dfrac{1}{x}[\/latex]<\/li>\n<li>[latex]y=|x|[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>A graph is more than just an image; it&#8217;s a powerful tool for visually representing the precise relationship between two variables. Mastering the ability to interpret and create graphs is a vital skill that you&#8217;ll carry into your chosen field and beyond, enhancing your understanding and communication in both your professional and civic life.<\/p>\n<p>Mathematics is a crucial tool for making sense of the world around us. Whether you&#8217;re a scientist, social scientist, engineer, business leader, health care provider, or politician, strong math skills are essential to achieving your goals. This course will provide you with the mathematical foundation needed to excel in a wide range of disciplines.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Think about it&#8230;<\/h3>\n<p>Climate change is one of the most serious threats facing our planet today. The graph below contains information about data collected at an observatory on the Big Island, Mauna Loa, Hawaii. The red graph illustrates the amount of CO<sub>2<\/sub> (carbon dioxide) in the air measured in ppm (parts per million). This number tells us how many parts of carbon dioxide there are in one million parts of air. The green graph illustrates the amount of CO<sub>2<\/sub> in the ocean water surrounding the Hawaiian islands. The blue graph illustrates the pH of the ocean water surrounding the Hawaiian islands.<\/p>\n<p>Graphs illustrate a precise relationship between two quantities. Make sure to identify the two relationships being compared in each of the three graphs. Observe the labelled axes.<\/p>\n<p><strong>Keep in mind:<\/strong><\/p>\n<ul>\n<li>The [latex]x[\/latex]&#8211;axis is used to describe the independent variable or input.<\/li>\n<li>The [latex]y[\/latex]-axis is used to describe the dependent variable or output.<\/li>\n<li>A point on the graph shows how an input and an output are related.<\/li>\n<\/ul>\n<p><strong>Questions about the graphs:<\/strong><\/p>\n<p>a) What do you notice about these graphs-describe in your own words?<\/p>\n<p>b) On all three graphs, the data shows fluctuations on the graphs. What hypothesis can you make about why these graphs are fluctuating?<\/p>\n<p>c) Does it appear that the trend of one graph is effecting the other graphs? why or why not? If so, what might be some possible reasons for this effect. If not, explain your reasoning.<\/p>\n<p>d) Why might some of the graphs be increasing while others are decreasing?<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/1-1-graphs\/screenshot-2024-08-03-at-1-07-01%e2%80%afpm\/\" rel=\"attachment wp-att-1720\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1720 size-large\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-1024x768.png\" alt=\"A graph showing changes in atmospheric CO\u2082, ocean CO\u2082, and ocean pH in the North Pacific over time. The red data shows atmospheric CO\u2082 at Mauna Loa, fluctuating, but rising steadily from about 315 ppm in 1958 to over 420 ppm in 2024. The green data shows ocean CO\u2082 (pCO\u2082) at Station ALOHA, increasing from about 315 \u00b5atm in 1988 to over 400 \u00b5atm in 2024. The blue data shows ocean pH at Station ALOHA, which decreases from about 8.2 to just below 8.05 over the same time.\" width=\"1024\" height=\"768\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-1024x768.png 1024w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-300x225.png 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-768x576.png 768w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-65x49.png 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-225x169.png 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM-350x263.png 350w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/Screenshot-2024-08-03-at-1.07.01\u202fPM.png 1298w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p>\n<p>Here is a link to an interactive graph of the atmospheric CO<sub>2<\/sub> levels in Mauna Loa, Hawaii. It may be helpful to use this to see the precise the data points on this graph.<\/p>\n<p><a href=\"https:\/\/gml.noaa.gov\/ccgg\/trends\/graph.html\">https:\/\/gml.noaa.gov\/ccgg\/trends\/graph.html<\/a><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Above we used graphs to describe relationships between quantities but there are other ways we can describe these \u00a0mathematical relationships. Mathematical ideas can be communicated in many ways using different representations.\u00a0Some representations include using tables of values, lists of ordered pairs, words, graphs, expressions, and equations to describe mathematical relationships. In this course, it is important to develop skills using all these representations. We will start with ordered pairs and tables of values.<\/p>\n<h2>The coordinate plane and plotting points<\/h2>\n<p>You have likely used a coordinate plane before.\u00a0The coordinate plane consists of a horizontal axis and a vertical axis; number lines that intersect at right angles. (They are perpendicular to each other.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1828\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"424\" height=\"422\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1.jpg 1412w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-150x150.jpg 150w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-300x300.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-768x766.jpg 768w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-1024x1021.jpg 1024w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-65x65.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-225x224.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/xy-plane-rev1-350x349.jpg 350w\" sizes=\"auto, (max-width: 424px) 100vw, 424px\" \/><\/p>\n<p>The horizontal axis in the coordinate plane is called the [latex]x[\/latex]<b>-axis<\/b>. The vertical axis is called the [latex]y[\/latex]<b>-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at\u00a0[latex]0[\/latex] on the [latex]x[\/latex]<i>&#8211;<\/i>axis and\u00a0[latex]0[\/latex] on the [latex]y[\/latex]<i>&#8211;<\/i>axis.<\/p>\n<p>Locations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the [latex]x[\/latex]<i>&#8211;<\/i>axis (the first value of the ordered pair) and along the [latex]y[\/latex]-axis (the second value of the ordered pair).<\/p>\n<p>In an ordered pair, such as [latex](x,y)[\/latex], the first value is called the [latex]x[\/latex]<b>-coordinate<\/b>\u00a0or the input and the second value is the [latex]y[\/latex]<b>-coordinate <\/b>or the output. Note that the [latex]x[\/latex]<i>&#8211;<\/i>coordinate is listed before the [latex]y[\/latex]<i>&#8211;<\/i>coordinate. Since the origin has an [latex]x[\/latex]<i>&#8211;<\/i>coordinate of\u00a0[latex]0[\/latex] and a [latex]y[\/latex]<i>&#8211;<\/i>coordinate of\u00a0[latex]0[\/latex], its ordered pair is written as [latex](0,0)[\/latex].<\/p>\n<p>Consider the point below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182920\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/><\/p>\n<p>To identify the location of this point, start at the origin [latex](0,0)[\/latex] and move right along the [latex]x[\/latex]<i>&#8211;<\/i>axis until you are under the point. Look at the label on the [latex]x[\/latex]<i>&#8211;<\/i>axis. The\u00a0[latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the [latex]x[\/latex]-axis. This is the [latex]x[\/latex]<i>&#8211;<\/i>coordinate; the first number in the ordered pair.<\/p>\n<p>From\u00a0[latex]4[\/latex] on the [latex]x[\/latex]<i>&#8211;<\/i>axis move up to the point and notice the number with which it aligns on the [latex]y[\/latex]<i>&#8211;<\/i>axis. The\u00a0[latex]3[\/latex] indicates that, after leaving the [latex]x[\/latex]-axis, you traveled\u00a0[latex]3[\/latex] units up in the vertical direction, the direction of the [latex]y[\/latex]-axis. This number is the [latex]y[\/latex]<i>&#8211;<\/i>coordinate, the second number in the ordered pair. With an [latex]x[\/latex]<i>&#8211;<\/i>coordinate of\u00a0[latex]4[\/latex] and a [latex]y[\/latex]<i>&#8211;<\/i>coordinate of\u00a0[latex]3[\/latex], you have the ordered pair [latex](4,3)[\/latex].<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Practice identifying points from a graph.<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm92753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&theme=oea&iframe_resize_id=ohm92753&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using a Table of Values to Graph a Linear Equation<\/h2>\n<p>A helpful first step in graphing an equation is to make a table of values. This is particularly useful when you do not know the general shape the graph will have. You probably already know that the graph of a linear equation will be a straight line, but let us make a table first to see how it can be helpful.<\/p>\n<p>Make a table of values for [latex]y=3x+2[\/latex]. It is a good idea to include negative values, positive values, and zero as inputs. Evaluate the given equation for each input value of [latex]x[\/latex].<\/p>\n<table style=\"width: 75%; border: 1px solid black; font-size: 110%;\">\n<thead>\n<tr>\n<th style=\"width: 15%; border: 1px solid #999999;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3x+2[\/latex]<\/th>\n<th style=\"width: 25%; border: 1px solid #999999;\">[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{\u22122})+2=-6+2=-4[\/latex]<\/td>\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](-2,-4)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{\u22121})+2=-3+2=-1[\/latex]<\/td>\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](-1,-1)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{0})+2=0+2=2[\/latex]<\/td>\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](0,2)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{1})+2=3+2=5[\/latex]<\/td>\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](1,5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15%; border: 1px solid #999999;\">[latex]\\color{green}{3}[\/latex]<\/td>\n<td style=\"width: 60%; border: 1px solid #999999;\">[latex]y=3(\\color{green}{3})+2=9+2=11[\/latex]<\/td>\n<td style=\"width: 25%; border: 1px solid #999999;\">[latex](3,11)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now use the ordered pairs to help you draw the graph of the equation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232426\/image006.gif\" alt=\"Line graph on coordinate plane. Line has dots on (negative 2, negative 4), (negative 1, negative 1), (0,2), (1,5) and (3,11).\" width=\"322\" height=\"353\" \/><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Complete the table of values and plot the linear equation.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" id=\"ohm170086\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=170086&theme=oea&iframe_resize_id=ohm170086&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using a Table of Values to Graph a Quadratic Equation<\/h2>\n<p>Not all graphs are straight lines. Next, let&#8217;s explore the shape of the graph of a Quadratic Equation. Let&#8217;s use a table of values to graph a basic quadratic equation: [latex]y=x^2[\/latex].<br \/>\nStart by creating a table of values, then plot the ordered pairs.<\/p>\n<div style=\"margin: auto;\">\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-2})^2=4[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-2,4)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-1})^2=1[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-1,1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{0})^2=0[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,0)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{1})^2=1[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{2})^2=4[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](2,4)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points [latex](-2,4), (-1,1), (0,0), (1,1), (2,4)[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232437\/image013.gif\" alt=\"Graph with the point negative 2, 4; the point negative 1, 1; the point 0, 0; the point 1,1; the point 2,4.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Since the points are <i>not<\/i> on a line, you cannot use a straight edge. Connect the points as best as you can using a <i>smooth curve<\/i> (not a series of straight lines). You may want to find and plot additional points (such as the ones in blue here). Placing arrows on the tips of the lines implies that they continue in that direction forever.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232441\/image014.gif\" alt=\"Coordinate plane with U shaped curve or parabola connecting the blue points (negative 3,9), (3,9) with red points (negative 2,4), (2,4) and bottom of U shape at (0,0).\" width=\"322\" height=\"353\" \/><\/p>\n<p>Notice that the shape is similar to the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The lowest point on this graph is called the vertex. The vertical line that goes through the vertex is called the line of reflection. In this case, that line is the [latex]y[\/latex]-axis.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Complete the table of values and graph [latex]y=x^2-3[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2-3[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\"><\/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=\"q72815\">Show Solution<\/span><\/p>\n<div id=\"q72815\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 300px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y=x^2-3[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22122}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-2})^2-3=1[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-2,1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{\u22121}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{-1})^2-3=-2[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](-1,-2)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{0})^2-3=-3[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,-3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{1})^2-3=-2[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,-2)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{2}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=(\\color{green}{2})^2-3=1[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](2,1)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Graph the points [latex](-2,1), (-1,-2), (0,-3), (1,-2), (2,1)[\/latex] Then connect the points with a smooth curve.<\/p>\n<div style=\"margin: auto;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/jym4noksrm?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2><\/h2>\n<h2>Using a Table of Values to Graph a Square Root Equation<\/h2>\n<p>To graph a square root equation [latex]y=\\sqrt{x}[\/latex] using a table of values, in this example we will only choose [latex]x[\/latex]-values that are perfect squares. Doing this makes it much easier to evaluate and plot the points.<\/p>\n<div style=\"margin: auto;\">\n<table style=\"border-collapse: collapse; width: 35%; border: 1px solid black; height: 72px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 40%; border: 1px solid #999999; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{0}}=0[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](0,0)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{1}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{1}}=1[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](1,1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{4}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{4}}=2[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](4,2)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{9}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{9}}=3[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](9,3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 30%; border: 1px solid #999999;\">[latex]\\color{green}{16}[\/latex]<\/td>\n<td style=\"height: 14px; width: 40%; border: 1px solid #999999;\">[latex]y=\\sqrt{\\color{green}{16}}=4[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999;\">[latex](16,4)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that we only used positive or zero values for the inputs to the square root. Recall that [latex]\\sqrt{x}[\/latex] means to find the number whose square is [latex]x[\/latex]. For example, [latex]\\sqrt{49}[\/latex] means &#8220;find the number whose square is\u00a0[latex]49[\/latex].&#8221;\u00a0Since there is no real number that we can square and get a negative, the equation [latex]y=\\sqrt{x}[\/latex] will be defined for [latex]x>0[\/latex].<\/p>\n<p>Here&#8217;s the graph of [latex]y=\\sqrt{x}[\/latex].<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232457\/image026.gif\" alt=\"Curved line branching up and right from the point 0,0.\" width=\"258\" height=\"288\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Complete the table of values and graph [latex]y=\\sqrt{(x+4)}[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 70%; border: 1px solid #999999; height: 12px;\">[latex]y=\\sqrt{(x+4)}[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-4}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-3}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{5}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{12}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\"><\/td>\n<td style=\"width: 244px; border: 1px solid #999999;\"><\/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=\"q897380\">Show Solution<\/span><\/p>\n<div id=\"q897380\" class=\"hidden-answer\" style=\"display: none\">\n<p>On this example, notice the [latex]x[\/latex]-values that we are choosing are not perfect squares like in the explanation above. This time we are choosing [latex]x[\/latex]-values that will make the radicand simplify to be a perfect square. An [latex]x[\/latex] value of [latex]-3[\/latex] was picked below. Notice when I substitute [latex]-3[\/latex] in place of [latex]x[\/latex] in the equation we get [latex]-3+4[\/latex] under the square root. When we simplify that we get [latex]1[\/latex] and that is a perfect square. Check out the other [latex]x[\/latex]-values and notice what happens when you simplify before you take the square root.<\/p>\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 82px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 70%; border: 1px solid #999999; height: 12px;\">[latex]y=\\sqrt{(x+4)}[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px;\">\u00a0[latex](x,y)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-4}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{-4}\\color{black}{)+4}}=\\sqrt{0}=0[\/latex]<\/td>\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](-4,0)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{-3}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{-3}\\color{black}{)+4}}=\\sqrt{1}=1[\/latex]<\/td>\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](-3,1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{0}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{0}\\color{black}{)+4}}=\\sqrt{4}=2[\/latex]<\/td>\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](0,2)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{5}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{5}\\color{black}{)+4}}=\\sqrt{9}=3[\/latex]<\/td>\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](5,3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 254px; border: 1px solid #999999;\">[latex]\\color{green}{12}[\/latex]<\/td>\n<td style=\"height: 14px; width: 234px; border: 1px solid #999999;\">[latex]y=\\sqrt{(\\color{green}{12}\\color{black}{)+4}}=\\sqrt{16}=4[\/latex]<\/td>\n<td style=\"width: 244px; border: 1px solid #999999; height: 14px;\">[latex](12,4)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div style=\"margin: auto;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/eqom3dfziw?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2><\/h2>\n<h2>Graphs of Common Functions<\/h2>\n<p>Throughout our course, you&#8217;ll use graphs to aid in understanding relationships between input ([latex]x[\/latex]) and output ([latex]y[\/latex]) variables. Learn to recognize these graphs and their features quickly by name, formula, and graph.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; font-size: 110%; height: 126px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Name<\/strong><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Equation<\/strong><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Graph<\/strong><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><strong>Table<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 25%; height: 28px; text-align: center;\">Constant<\/td>\n<td style=\"width: 25%; height: 28px; text-align: center;\">[latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<td style=\"width: 25%; height: 28px; text-align: center;\">\u00a0\u00a0\u00a0\u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1607 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1-300x281.jpg\" alt=\"A horizontal line crossing y-axis at 2 on a coordinate plane. Additional points: (negative 2,2) and (2,2).\" width=\"300\" height=\"281\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1-300x281.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1-65x61.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1-225x211.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/constant-1.jpg 315w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 28px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 55.642%; height: 84px;\">\n<tbody>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">0<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Linear<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=x[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1609 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1-300x278.jpg\" alt=\"Graph of a diagonal line on a coordinate plane that goes through the origin (0,0). Additional points are (2,2) and (negative 2, negative 2).\" width=\"300\" height=\"278\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1-300x278.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1-65x60.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1-225x209.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-1.jpg 320w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 54.8638%; height: 84px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">-2<\/td>\n<td style=\"width: 50%;\">-2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">0<\/td>\n<td style=\"width: 50%;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2<\/td>\n<td style=\"width: 50%;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Absolute Value<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=|x|[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1606 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg\" alt=\"A graph of an upward facing absolute function (v-shape) on a coordinate plane with the vertex at (0,0). Additional points shown at (negative 2,2) and (2,2).\" width=\"300\" height=\"273\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-65x59.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-225x205.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1.jpg 324w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 49.4163%; height: 64px;\">\n<tbody>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">0<\/td>\n<td style=\"width: 50%; height: 11px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Quadratic<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y={x}^{2}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1610 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1-300x282.jpg\" alt=\"Graph of parabola or U-shape on a coordinate plane representing y = x squared. Curve comes from positive infinity on left, curves through the origin (0, 0) and rises back up to positive infinity on right.  Additional points at (negative 2,4), (2,4), (negative 1,1) and (1,1).\" width=\"300\" height=\"282\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1-300x282.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1-65x61.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1-225x211.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/quadratic-1.jpg 316w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 59.5331%; height: 111px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">-2<\/td>\n<td style=\"width: 50%;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">-1<\/td>\n<td style=\"width: 50%;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">0<\/td>\n<td style=\"width: 50%;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">1<\/td>\n<td style=\"width: 50%;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2<\/td>\n<td style=\"width: 50%;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Cubic<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y={x}^{3}[\/latex]<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1608 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1-300x277.jpg\" alt=\"Curved graph on a coordinate plane representing y equals x cube that starts at negative infinity left and increases to the right, passing smoothly through the origin (0, 0). Additional points at (negative 1, negative 1), (negative 0.5, just below x-axis), (0.5, just above x-axis) and (1, 1).\" width=\"300\" height=\"277\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1-300x277.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1-65x60.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1-225x208.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/cubic-1.jpg 319w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 59.5331%; height: 122px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">-1<\/td>\n<td style=\"width: 50%;\">-1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">-0.5<\/td>\n<td style=\"width: 50%;\">-0.125<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">0<\/td>\n<td style=\"width: 50%;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">0.5<\/td>\n<td style=\"width: 50%;\">0.125<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">1<\/td>\n<td style=\"width: 50%;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Reciprocal<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=\\frac{1}{x}[\/latex]<\/p>\n<p>Notice that this equation is not defined for all inputs. We&#8217;ll talk more about this in the next section.<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1611 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1-300x278.jpg\" alt=\"A rectangular hyperbola from equation 1 over x consists of 2 curves in the first and third quadrants. The graph has asymptotes at x and y axes meaning the curves get infinitely close to these lines by never touch them.\" width=\"300\" height=\"278\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1-300x278.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1-65x60.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1-225x208.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/reciprocal-1.jpg 320w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 47.8599%; height: 127px;\">\n<tbody>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%; height: 11px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\n<td style=\"width: 50%; height: 11px;\">-0.5<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">-1<\/td>\n<td style=\"width: 50%; height: 11px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">-0.5<\/td>\n<td style=\"width: 50%; height: 11px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">0.5<\/td>\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">1<\/td>\n<td style=\"width: 50%; height: 11px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"width: 50%; height: 11px;\">2<\/td>\n<td style=\"width: 50%; height: 11px;\">0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 25%; height: 14px; text-align: center;\">Square root<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">[latex]y=\\sqrt{x}[\/latex]<\/p>\n<p>Notice that this equation is not defined for all inputs. We&#8217;ll talk more about this in the next section.<\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1612 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1-300x282.jpg\" alt=\"A graph on a coordinate plane starting at the origin (0, 0) and curving slowly upward to the right with additional points shown at (1, 1) and (4, 2).\" width=\"300\" height=\"282\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1-300x282.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1-65x61.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1-225x211.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/square-root-1.jpg 315w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 25%; height: 14px; text-align: center;\">\n<table style=\"border-collapse: collapse; width: 47.8599%; height: 88px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">0<\/td>\n<td style=\"width: 50%;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">1<\/td>\n<td style=\"width: 50%;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">4<\/td>\n<td style=\"width: 50%;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111722&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"650\" data-mce-fragment=\"1\"><\/iframe><\/p>\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-39\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Plot Points Given as Ordered Pairs on the Coordinate Plane. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/p_MESleS3mw\">https:\/\/youtu.be\/p_MESleS3mw<\/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 class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Graph Basic Linear Equations by Completing a Table of Values. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/Graph%20the%20linear%20equation\">http:\/\/Graph%20the%20linear%20equation<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Determine If an Ordered Pair is a Solution to a Linear Equation. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9aWGxt7OnB8\">https:\/\/youtu.be\/9aWGxt7OnB8<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Intermediate Algebra: A Functional Approach. <strong>Authored by<\/strong>: Brenden Kelly, Emina Alibegovic, et al. <strong>License<\/strong>: <em>All Rights Reserved<\/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":395986,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Graph Basic Linear Equations by Completing a Table of Values\",\"author\":\"mathispower4u\",\"organization\":\"\",\"url\":\"Graph the linear equation\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine If an Ordered Pair is a Solution to a Linear Equation\",\"author\":\"mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/9aWGxt7OnB8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Plot Points Given as Ordered Pairs on the Coordinate Plane\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/p_MESleS3mw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Intermediate Algebra: A Functional Approach\",\"author\":\"Brenden Kelly, Emina Alibegovic, et al\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"843ccd06-dd21-4505-b847-171ce9bb555f","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-39","chapter","type-chapter","status-publish","hentry"],"part":24,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/39","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":49,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions"}],"predecessor-version":[{"id":2099,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions\/2099"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/39\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=39"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=39"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=39"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}