{"id":16627,"date":"2019-10-03T20:16:52","date_gmt":"2019-10-03T20:16:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/radical-functions\/"},"modified":"2024-05-02T15:56:19","modified_gmt":"2024-05-02T15:56:19","slug":"radical-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/radical-functions\/","title":{"raw":"Graphing Radical Functions","rendered":"Graphing Radical Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Graph radical functions using tables and transformations<\/li>\r\n<\/ul>\r\n<\/div>\r\nJust like linear functions and quadratic functions, we can graph radical functions (such as square root functions) by choosing values for <i>x<\/i> and finding points that will be on the graph. Again, it is helpful to have some idea about what the graph will look like.\r\n\r\nThink about the basic square root function, [latex]f(x)=\\sqrt{x}[\/latex]. Take a look at a table of values for <i>x<\/i> and <i>y<\/i> and then graph the function. Notice that all the values for <i>x<\/i> in the table are perfect squares. Since you are taking the square root of <i>x<\/i>, using perfect squares makes more sense than just finding the square roots of\u00a0[latex]0, 1, 2, 3, 4[\/latex], etc.\r\n<div align=\"center\">\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\"><i>x<\/i><\/th>\r\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRecall that <i>x<\/i> can never be negative because when you square a real number, the result is always positive. For example, [latex]\\sqrt{49}[\/latex], this\u00a0means \"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 function [latex]f(x)=\\sqrt{x}[\/latex] will be defined for [latex]x\\ge0[\/latex].\r\n\r\nTake a look at the graph.\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\nDoes the shape of the graph look a little familiar?\u00a0 It should.\u00a0 The graph of a radical function is half of a horizontal parabola.\u00a0 You could also think of it as half of a parabola lying on its side.\u00a0 As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.<i>\u00a0<\/i>\r\n\r\nMultiplying [latex]\\sqrt{x}[\/latex] by a positive value changes the width of the half-parabola. Multiplying [latex]\\sqrt{x}[\/latex] by a negative number gives you the other half of a horizontal parabola.\r\n\r\nIn the following example, we will show how multiplying a radical function by a constant can change the shape of the graph.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]\r\n\r\nb)[latex]f(x)=2\\sqrt{x}[\/latex]\r\n\r\nc)\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex]\r\n\r\n1)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/>\r\n\r\n2)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/>\r\n\r\n3)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/>\r\n[reveal-answer q=\"651190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"651190\"]\r\n\r\nFunction a) goes with graph 2)\r\n\r\nFunction b) goes with graph 1)\r\n\r\nFunction c) goes with graph 3)\r\n\r\n&nbsp;\r\n\r\nFunction\u00a0a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex] means that all the outputs will be negative - the function is the negative of the square roots of the input. This will give the other half of the parabola on its side. Therefore, graph 2)\u00a0goes with the function\u00a0[latex]f(x)=-\\sqrt{x}[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/>\r\n\r\n&nbsp;\r\n\r\nFunction b) [latex]f(x)=2\\sqrt{x}[\/latex] means take the square root of all the inputs, then multiply by two, so the outputs will be larger than the outputs for [latex]\\sqrt{x}[\/latex]. Graph 1)\u00a0goes with the function\u00a0[latex]f(x)=2\\sqrt{x}[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/>\r\n\r\n&nbsp;\r\n\r\nFunction\u00a0c)\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex] means take the square root of the inputs then multiply by [latex]\\dfrac{1}{2}[\/latex]. The outputs will be smaller than the outputs for\u00a0[latex]\\sqrt{x}[\/latex]. Graph 3)\u00a0goes with the function\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAdding a value <i>outside <\/i>the radical shifts the graph up or down. Think about it as adding the value to the basic <i>y<\/i> value of [latex] \\sqrt{x}[\/latex], so a positive value added moves the graph up.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex] f(x)=\\sqrt{x}+3[\/latex]\r\n\r\nb)[latex] f(x)=\\sqrt{x}-2[\/latex]\r\n\r\n1)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/>\r\n\r\n2)\r\n\r\n<img class=\"wp-image-2062 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/>\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"928501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"928501\"]\r\n\r\nFunction a) goes with graph 2)\r\n\r\nFunction b) goes with graph 1)\r\n\r\n&nbsp;\r\n\r\nFunction a)\u00a0[latex] f(x)=\\sqrt{x}+3[\/latex] means take the square root of all the inputs and add three, so the out puts will be greater than those for [latex]\\sqrt{x}[\/latex], therefore graph 2) goes with this function.\r\n\r\n<img class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nFunction\u00a0b) [latex]f(x)=\\sqrt{x}-2[\/latex] means take the square root of the input then subtract two. Every output will be [latex]2[\/latex] less than those for\u00a0[latex]\\sqrt{x}[\/latex].\u00a0 This shifts the entire function down two units. Therefore, graph 1) goes with this function.\r\n<img class=\"alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAdding a value <i>inside <\/i>the radical moves the graph left or right. Think about it as adding a value to <i>x<\/i> before you take the square root\u2014so the <i>y<\/i> value gets moved to a different <i>x<\/i> value. For example, for [latex]f(x)=\\sqrt{x}[\/latex], the square root is\u00a0[latex]3[\/latex] if [latex]x=9[\/latex]. For [latex]f(x)=\\sqrt{x+1}[\/latex], the square root is\u00a0[latex]3[\/latex] when [latex]x+1[\/latex] is\u00a0[latex]9[\/latex], so when [latex]x[\/latex] is\u00a0[latex]8[\/latex]. Changing [latex]x[\/latex]\u00a0to [latex]x+1[\/latex] shifts the graph to the left by\u00a0[latex]1[\/latex] unit (for example, from\u00a0[latex]9[\/latex] to [latex]8[\/latex]). Changing [latex]x[\/latex]\u00a0to [latex]x\u22122[\/latex] shifts the graph to the right by\u00a0[latex]2[\/latex] units.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex]\r\n\r\nb)[latex]f(x)=\\sqrt{x-2}[\/latex]\r\n\r\n1)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/>\r\n\r\n2)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/>\r\n[reveal-answer q=\"602483\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"602483\"]\r\n\r\nFunction a) matches graph 2)\r\n\r\nFunction b) matches graph 1)\r\n\r\n&nbsp;\r\n\r\nFunction a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex] adds one to the inputs before the square root is taken. \u00a0The outputs will be greater, so it ends up looking like a shift to the left. Graph 2) matches this function.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/>\r\n\r\nFunction b)\u00a0[latex]f(x)=\\sqrt{x-2}[\/latex] means subtract before the square root is taken. \u00a0This makes the outputs less than they would be for the standard [latex]\\sqrt{x}[\/latex], and looks like a shift to the right. Graph 1) matches this function.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that as <i>x<\/i> gets greater, adding or subtracting a number inside the square root has less of an effect on the value of <i>y.<\/i>\r\n\r\nIn the next example, we will combine some of the changes that we have seen into one function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph [latex] f(x)=-2+\\sqrt{x-1}[\/latex].\r\n\r\n[reveal-answer q=\"493141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"493141\"]\r\n\r\nBefore making a table of values, look at the function equation to get a general idea about what the graph should look like.\r\n\r\nInside the square root, you are subtracting\u00a0[latex]1[\/latex], so the graph will move to the right\u00a0[latex]1[\/latex] from the basic [latex] f(x)=\\sqrt{x}[\/latex] graph.\u00a0 This agrees with what we know about the domain of our function, which is that all of our x values will be greater than or equal to 1.\r\n\r\nYou are also adding [latex]\u22122[\/latex] outside the square root, so the graph will move down two from the basic [latex] f(x)=\\sqrt{x}[\/latex] graph.\r\n\r\nCreate a table of values. Choose values that will make your calculations easy. You want [latex]x\u20131[\/latex] to be a perfect square ([latex]0, 1, 4, 9[\/latex], and so on) so you can take the square root.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">x<\/th>\r\n<th style=\"text-align: center;\">f(x)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince values of <em>x<\/em> less than\u00a0[latex]1[\/latex] makes the value inside the square root negative, there will be no points on the coordinate graph to the left of [latex]x=1[\/latex]. There is no need to choose x values less than\u00a0[latex]1[\/latex] for your table!\r\n\r\nUse ordered pairs from each row of the table to plot points.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/22224437\/Radical_Function_Points_Correct.jpg\" alt=\"The x y coordinate planewith 4 plotted points. The x axis spans from negative 2 to 10 and the y from negative 4 to 4. The 4 points are (1, -2), (2, -1), (5, 0), and (10, 1).\" width=\"296\" height=\"210\" \/>\r\n\r\nConnect the points as best you can using a smooth curve.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/22224433\/Radical_Function_Graph_Correct.jpg\" alt=\"The x y coordinate planewith 4 plotted points. The x axis spans from negative 2 to 10 and the y from negative 4 to 4. The 4 points are (1, -2), (2, -1), (5, 0), and (10, 1). A curved line is drawn through the points to fit the shape of the square root function.\" width=\"292\" height=\"211\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Graph radical functions using tables and transformations<\/li>\n<\/ul>\n<\/div>\n<p>Just like linear functions and quadratic functions, we can graph radical functions (such as square root functions) by choosing values for <i>x<\/i> and finding points that will be on the graph. Again, it is helpful to have some idea about what the graph will look like.<\/p>\n<p>Think about the basic square root function, [latex]f(x)=\\sqrt{x}[\/latex]. Take a look at a table of values for <i>x<\/i> and <i>y<\/i> and then graph the function. Notice that all the values for <i>x<\/i> in the table are perfect squares. Since you are taking the square root of <i>x<\/i>, using perfect squares makes more sense than just finding the square roots of\u00a0[latex]0, 1, 2, 3, 4[\/latex], etc.<\/p>\n<div style=\"margin: auto;\">\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\"><i>x<\/i><\/th>\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Recall that <i>x<\/i> can never be negative because when you square a real number, the result is always positive. For example, [latex]\\sqrt{49}[\/latex], this\u00a0means &#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 function [latex]f(x)=\\sqrt{x}[\/latex] will be defined for [latex]x\\ge0[\/latex].<\/p>\n<p>Take a look at the graph.<\/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<p>Does the shape of the graph look a little familiar?\u00a0 It should.\u00a0 The graph of a radical function is half of a horizontal parabola.\u00a0 You could also think of it as half of a parabola lying on its side.\u00a0 As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.<i>\u00a0<\/i><\/p>\n<p>Multiplying [latex]\\sqrt{x}[\/latex] by a positive value changes the width of the half-parabola. Multiplying [latex]\\sqrt{x}[\/latex] by a negative number gives you the other half of a horizontal parabola.<\/p>\n<p>In the following example, we will show how multiplying a radical function by a constant can change the shape of the graph.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]<\/p>\n<p>b)[latex]f(x)=2\\sqrt{x}[\/latex]<\/p>\n<p>c)\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex]<\/p>\n<p>1)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<p>2)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/><\/p>\n<p>3)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q651190\">Show Solution<\/span><\/p>\n<div id=\"q651190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a) goes with graph 2)<\/p>\n<p>Function b) goes with graph 1)<\/p>\n<p>Function c) goes with graph 3)<\/p>\n<p>&nbsp;<\/p>\n<p>Function\u00a0a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex] means that all the outputs will be negative &#8211; the function is the negative of the square roots of the input. This will give the other half of the parabola on its side. Therefore, graph 2)\u00a0goes with the function\u00a0[latex]f(x)=-\\sqrt{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Function b) [latex]f(x)=2\\sqrt{x}[\/latex] means take the square root of all the inputs, then multiply by two, so the outputs will be larger than the outputs for [latex]\\sqrt{x}[\/latex]. Graph 1)\u00a0goes with the function\u00a0[latex]f(x)=2\\sqrt{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Function\u00a0c)\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex] means take the square root of the inputs then multiply by [latex]\\dfrac{1}{2}[\/latex]. The outputs will be smaller than the outputs for\u00a0[latex]\\sqrt{x}[\/latex]. Graph 3)\u00a0goes with the function\u00a0[latex]f(x)=\\dfrac{1}{2}\\sqrt{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Adding a value <i>outside <\/i>the radical shifts the graph up or down. Think about it as adding the value to the basic <i>y<\/i> value of [latex]\\sqrt{x}[\/latex], so a positive value added moves the graph up.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=\\sqrt{x}+3[\/latex]<\/p>\n<p>b)[latex]f(x)=\\sqrt{x}-2[\/latex]<\/p>\n<p>1)<\/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\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/><\/p>\n<p>2)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2062 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/><\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q928501\">Show Solution<\/span><\/p>\n<div id=\"q928501\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a) goes with graph 2)<\/p>\n<p>Function b) goes with graph 1)<\/p>\n<p>&nbsp;<\/p>\n<p>Function a)\u00a0[latex]f(x)=\\sqrt{x}+3[\/latex] means take the square root of all the inputs and add three, so the out puts will be greater than those for [latex]\\sqrt{x}[\/latex], therefore graph 2) goes with this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Function\u00a0b) [latex]f(x)=\\sqrt{x}-2[\/latex] means take the square root of the input then subtract two. Every output will be [latex]2[\/latex] less than those for\u00a0[latex]\\sqrt{x}[\/latex].\u00a0 This shifts the entire function down two units. Therefore, graph 1) goes with this function.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Adding a value <i>inside <\/i>the radical moves the graph left or right. Think about it as adding a value to <i>x<\/i> before you take the square root\u2014so the <i>y<\/i> value gets moved to a different <i>x<\/i> value. For example, for [latex]f(x)=\\sqrt{x}[\/latex], the square root is\u00a0[latex]3[\/latex] if [latex]x=9[\/latex]. For [latex]f(x)=\\sqrt{x+1}[\/latex], the square root is\u00a0[latex]3[\/latex] when [latex]x+1[\/latex] is\u00a0[latex]9[\/latex], so when [latex]x[\/latex] is\u00a0[latex]8[\/latex]. Changing [latex]x[\/latex]\u00a0to [latex]x+1[\/latex] shifts the graph to the left by\u00a0[latex]1[\/latex] unit (for example, from\u00a0[latex]9[\/latex] to [latex]8[\/latex]). Changing [latex]x[\/latex]\u00a0to [latex]x\u22122[\/latex] shifts the graph to the right by\u00a0[latex]2[\/latex] units.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex]<\/p>\n<p>b)[latex]f(x)=\\sqrt{x-2}[\/latex]<\/p>\n<p>1)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/><\/p>\n<p>2)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602483\">Show Solution<\/span><\/p>\n<div id=\"q602483\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a) matches graph 2)<\/p>\n<p>Function b) matches graph 1)<\/p>\n<p>&nbsp;<\/p>\n<p>Function a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex] adds one to the inputs before the square root is taken. \u00a0The outputs will be greater, so it ends up looking like a shift to the left. Graph 2) matches this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/><\/p>\n<p>Function b)\u00a0[latex]f(x)=\\sqrt{x-2}[\/latex] means subtract before the square root is taken. \u00a0This makes the outputs less than they would be for the standard [latex]\\sqrt{x}[\/latex], and looks like a shift to the right. Graph 1) matches this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that as <i>x<\/i> gets greater, adding or subtracting a number inside the square root has less of an effect on the value of <i>y.<\/i><\/p>\n<p>In the next example, we will combine some of the changes that we have seen into one function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph [latex]f(x)=-2+\\sqrt{x-1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q493141\">Show Solution<\/span><\/p>\n<div id=\"q493141\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before making a table of values, look at the function equation to get a general idea about what the graph should look like.<\/p>\n<p>Inside the square root, you are subtracting\u00a0[latex]1[\/latex], so the graph will move to the right\u00a0[latex]1[\/latex] from the basic [latex]f(x)=\\sqrt{x}[\/latex] graph.\u00a0 This agrees with what we know about the domain of our function, which is that all of our x values will be greater than or equal to 1.<\/p>\n<p>You are also adding [latex]\u22122[\/latex] outside the square root, so the graph will move down two from the basic [latex]f(x)=\\sqrt{x}[\/latex] graph.<\/p>\n<p>Create a table of values. Choose values that will make your calculations easy. You want [latex]x\u20131[\/latex] to be a perfect square ([latex]0, 1, 4, 9[\/latex], and so on) so you can take the square root.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\">x<\/th>\n<th style=\"text-align: center;\">f(x)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since values of <em>x<\/em> less than\u00a0[latex]1[\/latex] makes the value inside the square root negative, there will be no points on the coordinate graph to the left of [latex]x=1[\/latex]. There is no need to choose x values less than\u00a0[latex]1[\/latex] for your table!<\/p>\n<p>Use ordered pairs from each row of the table to plot points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/22224437\/Radical_Function_Points_Correct.jpg\" alt=\"The x y coordinate planewith 4 plotted points. The x axis spans from negative 2 to 10 and the y from negative 4 to 4. The 4 points are (1, -2), (2, -1), (5, 0), and (10, 1).\" width=\"296\" height=\"210\" \/><\/p>\n<p>Connect the points as best you can using a smooth curve.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3698\/2016\/06\/22224433\/Radical_Function_Graph_Correct.jpg\" alt=\"The x y coordinate planewith 4 plotted points. The x axis spans from negative 2 to 10 and the y from negative 4 to 4. The 4 points are (1, -2), (2, -1), (5, 0), and (10, 1). A curved line is drawn through the points to fit the shape of the square root function.\" width=\"292\" height=\"211\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-16627\">\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>Chapter 17. <strong>Authored by<\/strong>: Monterey Institute for Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Chapter 17\",\"author\":\"Monterey Institute for Technology and Education\",\"organization\":\"\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"f969fc936dbb43a7a67bbcd0f062befa, 8d99817e020743b29281d04abfe31c66","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16627","chapter","type-chapter","status-publish","hentry"],"part":16017,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16627","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16627\/revisions"}],"predecessor-version":[{"id":20522,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16627\/revisions\/20522"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16017"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16627\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16627"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16627"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16627"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16627"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}