{"id":174,"date":"2023-06-21T13:22:40","date_gmt":"2023-06-21T13:22:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/absolute-value-functions\/"},"modified":"2023-09-21T08:40:46","modified_gmt":"2023-09-21T08:40:46","slug":"absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/absolute-value-functions\/","title":{"raw":"\u25aa   Absolute Value Functions*","rendered":"\u25aa   Absolute Value Functions*"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Find the intercepts of an absolute value function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015138\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" \/> Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: \"s58y\"\/Flickr)[\/caption]\r\n\r\nUntil the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate <strong>absolute value functions<\/strong>.\r\n<h2>Understanding Absolute Value<\/h2>\r\nRecall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance a number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Absolute Value Function<\/h3>\r\nThe absolute value function can be defined as a piecewise function\r\n<p style=\"text-align: center;\">$latex f(x) =\\begin{cases}x ,\\ x \\geq 0 \\\\ -x , x &lt; 0\\\\ \\end{cases} $<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nIt can help to visualize the graph of an absolute value function as the graph of the identity function, [latex]f(x) = x[\/latex] where, for all negative input, the function value is forced to be positive.\r\n\r\nWhen describing absolute value in words, visualize it as a distance. We can see, for example, that the two values that are [latex]3[\/latex] units away from the number [latex]1[\/latex] on the number line are [latex]4[\/latex] and [latex]-2[\/latex].\r\n\r\nWe can use an absolute value function statement to describe this by asking the question\u00a0<em>what values are 3 units away from 1?<\/em> Let [latex]x[\/latex] be the numbers. That is,\u00a0<em>the distance from some number and 1 is 3.<\/em>\r\n<p style=\"text-align: center;\">[latex]|x-1| = 3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] x-1 = 3 \\quad \\Rightarrow \\quad x = 4 \\qquad \\text{ and } \\qquad x-1 = -3 \\quad \\Rightarrow \\quad x = -2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determine a Number within a Prescribed Distance<\/h3>\r\nDescribe all values [latex]x[\/latex] within or including a distance of 4 from the number 5.\r\n\r\n[reveal-answer q=\"68130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68130\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015140\/CNX_Precalc_Figure_01_06_0022.jpg\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"487\" height=\"81\" \/>\r\n\r\nWe want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.\r\n\r\nThe distance from [latex]x[\/latex] to 5 can be represented using [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that\r\n\r\n[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]\r\n[latex]\\displaystyle{1}\\le{x}[\/latex]\r\nAnd:\r\n[latex]\\displaystyle{x-5}\\le{4}[\/latex]\r\n[latex]\\displaystyle{x}\\le{9}[\/latex]\r\n\r\nSo [latex]|x - 5|\\le 4[\/latex] is equal to [latex]1\\le x\\le 9[\/latex].\r\n\r\nHowever, mathematicians generally prefer absolute value notation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe all values [latex]x[\/latex] within a distance of 3 from the number 2.\r\n\r\n[reveal-answer q=\"98953\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"98953\"]\r\n\r\n[latex]|x - 2|\\le 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"300\"]469[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Resistance of a Resistor<\/h3>\r\nElectrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\pm 1\\%,\\pm5\\%,[\/latex] or [latex]\\displaystyle\\pm10\\%[\/latex].\r\n\r\nSuppose we have a resistor rated at 680 ohms, [latex]\\pm 5\\%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.\r\n\r\n[reveal-answer q=\"845830\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"845830\"]\r\n\r\n5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,\r\n<p style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nStudents who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.\r\n\r\n[reveal-answer q=\"584159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"584159\"]\r\n\r\nUsing the variable [latex]p[\/latex] for passing, [latex]|p - 80|\\le 20[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <strong>origin<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015143\/CNX_Precalc_Figure_01_06_0032.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\" \/>\r\n\r\nThe graph below is of [latex]y=2\\left|x - 3\\right|+4[\/latex]. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)[\/latex] for this transformed function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015145\/CNX_Precalc_Figure_01_06_0042.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing an Equation for an Absolute Value Function<\/h3>\r\nWrite an equation for the function graphed below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015148\/CNX_Precalc_Figure_01_06_0052.jpg\" alt=\"Graph of an absolute function. Two rays stem from the point 3, negative 2. One ray crosses the point 0, 4. The other ray crosses the point 5, 2.\" width=\"487\" height=\"363\" \/>\r\n\r\n[reveal-answer q=\"203899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"203899\"]\r\n\r\nThe basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015150\/CNX_Precalc_Figure_01_06_0062.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\" \/>\r\n\r\nWe also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015152\/CNX_Precalc_Figure_01_06_0072.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.\" width=\"487\" height=\"363\" \/>\r\n\r\nFrom this information we can write the equation\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)=2\\left|x - 3\\right|-2,\\hfill &amp; \\text{treating the stretch as a vertical stretch, or}\\hfill \\\\ f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2,\\hfill &amp; \\text{treating the stretch as a horizontal compression}.\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that these equations are algebraically the same\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n<strong>Q &amp; A<\/strong>\r\n\r\n<strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong>\r\n\r\n<em>Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x[\/latex] and [latex]f\\left(x\\right)[\/latex].<\/em>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a|x - 3|-2[\/latex]<\/p>\r\n<em>Now substituting in the point <\/em>(1, 2)\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2=a|1 - 3|-2\\hfill \\\\ 4=2a\\hfill \\\\ a=2\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the equation for the absolute value function that is horizontally shifted left 2 units, vertically flipped, and vertically shifted up 3 units.\r\n\r\n[reveal-answer q=\"364286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"364286\"]\r\n\r\n[latex]f\\left(x\\right)=-|x+2|+3\\\\[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"200\"]60791[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n<strong>Q &amp; A<\/strong>\r\n\r\n<strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\r\n<\/strong>\r\n\r\n<em>Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.\r\n<\/em>\r\n\r\n<em>No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015154\/CNX_Precalc_Figure_01_06_008abc2.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\" \/> (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.[\/caption]\r\n\r\n<\/div>\r\n<h2>Find the Intercepts of an Absolute Value Function<\/h2>\r\nKnowing how to solve problems involving <strong>absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\r\n<ol>\r\n \t<li>Isolate the absolute value term.<\/li>\r\n \t<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B&gt;0[\/latex].<\/li>\r\n \t<li>Solve for [latex]x[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nThe places where a graph crosses the horizontal axis are sometimes called\u00a0<em>horizontal intercepts<\/em>,\u00a0<em>x-intercepts<\/em>, or\u00a0<em>zeros.<\/em>\r\n\r\nWe call them\u00a0<em>zeros<\/em> because those are the places where the function value equals zero. To find them, set the function value equal to zero and solve for [latex]x[\/latex]. We can do this for many functions using the techniques of college algebra.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Zeros of an Absolute Value Function<\/h3>\r\nFor the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .\r\n\r\n[reveal-answer q=\"594739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594739\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=|4x+1|-7\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ 7=|4x+1|\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ 7=4x+1\\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; -7=4x+1\\hfill &amp; \\text{Break into two separate equations and solve}.\\hfill \\\\ 6=4x\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; -8=4x\\hfill &amp; \\hfill \\\\ \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ x=\\frac{6}{4}=1.5\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{ }x=\\frac{-8}{4}=-2\\hfill &amp; \\hfill \\end{array}[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015157\/CNX_Precalc_Figure_01_06_011F2.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\" \/>\r\n\r\nThe function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFor the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].\r\n\r\n[reveal-answer q=\"206693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"206693\"]\r\n\r\n[latex]x=-1[\/latex] or [latex]x=2[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"565\"]40657[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Find the intercepts of an absolute value function.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015138\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" \/><\/p>\n<p class=\"wp-caption-text\">Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: &#8220;s58y&#8221;\/Flickr)<\/p>\n<\/div>\n<p>Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate <strong>absolute value functions<\/strong>.<\/p>\n<h2>Understanding Absolute Value<\/h2>\n<p>Recall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance a number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Absolute Value Function<\/h3>\n<p>The absolute value function can be defined as a piecewise function<\/p>\n<p style=\"text-align: center;\">[latex]f(x) =\\begin{cases}x ,\\ x \\geq 0 \\\\ -x , x < 0\\\\ \\end{cases}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>It can help to visualize the graph of an absolute value function as the graph of the identity function, [latex]f(x) = x[\/latex] where, for all negative input, the function value is forced to be positive.<\/p>\n<p>When describing absolute value in words, visualize it as a distance. We can see, for example, that the two values that are [latex]3[\/latex] units away from the number [latex]1[\/latex] on the number line are [latex]4[\/latex] and [latex]-2[\/latex].<\/p>\n<p>We can use an absolute value function statement to describe this by asking the question\u00a0<em>what values are 3 units away from 1?<\/em> Let [latex]x[\/latex] be the numbers. That is,\u00a0<em>the distance from some number and 1 is 3.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]|x-1| = 3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x-1 = 3 \\quad \\Rightarrow \\quad x = 4 \\qquad \\text{ and } \\qquad x-1 = -3 \\quad \\Rightarrow \\quad x = -2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determine a Number within a Prescribed Distance<\/h3>\n<p>Describe all values [latex]x[\/latex] within or including a distance of 4 from the number 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68130\">Show Solution<\/span><\/p>\n<div id=\"q68130\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015140\/CNX_Precalc_Figure_01_06_0022.jpg\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"487\" height=\"81\" \/><\/p>\n<p>We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.<\/p>\n<p>The distance from [latex]x[\/latex] to 5 can be represented using [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that<\/p>\n<p>[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<br \/>\n[latex]\\displaystyle{1}\\le{x}[\/latex]<br \/>\nAnd:<br \/>\n[latex]\\displaystyle{x-5}\\le{4}[\/latex]<br \/>\n[latex]\\displaystyle{x}\\le{9}[\/latex]<\/p>\n<p>So [latex]|x - 5|\\le 4[\/latex] is equal to [latex]1\\le x\\le 9[\/latex].<\/p>\n<p>However, mathematicians generally prefer absolute value notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe all values [latex]x[\/latex] within a distance of 3 from the number 2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q98953\">Show Solution<\/span><\/p>\n<div id=\"q98953\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]|x - 2|\\le 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm469\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=469&theme=oea&iframe_resize_id=ohm469&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Resistance of a Resistor<\/h3>\n<p>Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\pm 1\\%,\\pm5\\%,[\/latex] or [latex]\\displaystyle\\pm10\\%[\/latex].<\/p>\n<p>Suppose we have a resistor rated at 680 ohms, [latex]\\pm 5\\%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q845830\">Show Solution<\/span><\/p>\n<div id=\"q845830\" class=\"hidden-answer\" style=\"display: none\">\n<p>5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,<\/p>\n<p style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584159\">Show Solution<\/span><\/p>\n<div id=\"q584159\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the variable [latex]p[\/latex] for passing, [latex]|p - 80|\\le 20[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <strong>origin<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015143\/CNX_Precalc_Figure_01_06_0032.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\" \/><\/p>\n<p>The graph below is of [latex]y=2\\left|x - 3\\right|+4[\/latex]. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)[\/latex] for this transformed function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015145\/CNX_Precalc_Figure_01_06_0042.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing an Equation for an Absolute Value Function<\/h3>\n<p>Write an equation for the function graphed 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\/896\/2016\/10\/19015148\/CNX_Precalc_Figure_01_06_0052.jpg\" alt=\"Graph of an absolute function. Two rays stem from the point 3, negative 2. One ray crosses the point 0, 4. The other ray crosses the point 5, 2.\" width=\"487\" height=\"363\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q203899\">Show Solution<\/span><\/p>\n<div id=\"q203899\" class=\"hidden-answer\" style=\"display: none\">\n<p>The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015150\/CNX_Precalc_Figure_01_06_0062.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\" \/><\/p>\n<p>We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015152\/CNX_Precalc_Figure_01_06_0072.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.\" width=\"487\" height=\"363\" \/><\/p>\n<p>From this information we can write the equation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)=2\\left|x - 3\\right|-2,\\hfill & \\text{treating the stretch as a vertical stretch, or}\\hfill \\\\ f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2,\\hfill & \\text{treating the stretch as a horizontal compression}.\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that these equations are algebraically the same\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<p><strong>Q &amp; A<\/strong><\/p>\n<p><strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong><\/p>\n<p><em>Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x[\/latex] and [latex]f\\left(x\\right)[\/latex].<\/em><\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a|x - 3|-2[\/latex]<\/p>\n<p><em>Now substituting in the point <\/em>(1, 2)<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2=a|1 - 3|-2\\hfill \\\\ 4=2a\\hfill \\\\ a=2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the equation for the absolute value function that is horizontally shifted left 2 units, vertically flipped, and vertically shifted up 3 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q364286\">Show Solution<\/span><\/p>\n<div id=\"q364286\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-|x+2|+3\\\\[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm60791\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60791&theme=oea&iframe_resize_id=ohm60791&show_question_numbers\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<p><strong>Q &amp; A<\/strong><\/p>\n<p><strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?<br \/>\n<\/strong><\/p>\n<p><em>Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.<br \/>\n<\/em><\/p>\n<p><em>No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em><\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19015154\/CNX_Precalc_Figure_01_06_008abc2.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\" \/><\/p>\n<p class=\"wp-caption-text\">(a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.<\/p>\n<\/div>\n<\/div>\n<h2>Find the Intercepts of an Absolute Value Function<\/h2>\n<p>Knowing how to solve problems involving <strong>absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\n<ol>\n<li>Isolate the absolute value term.<\/li>\n<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B>0[\/latex].<\/li>\n<li>Solve for [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>The places where a graph crosses the horizontal axis are sometimes called\u00a0<em>horizontal intercepts<\/em>,\u00a0<em>x-intercepts<\/em>, or\u00a0<em>zeros.<\/em><\/p>\n<p>We call them\u00a0<em>zeros<\/em> because those are the places where the function value equals zero. To find them, set the function value equal to zero and solve for [latex]x[\/latex]. We can do this for many functions using the techniques of college algebra.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Zeros of an Absolute Value Function<\/h3>\n<p>For the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594739\">Show Solution<\/span><\/p>\n<div id=\"q594739\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=|4x+1|-7\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ 7=|4x+1|\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ 7=4x+1\\hfill & \\text{or}\\hfill & \\hfill & \\hfill & \\hfill & -7=4x+1\\hfill & \\text{Break into two separate equations and solve}.\\hfill \\\\ 6=4x\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & -8=4x\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\\\ x=\\frac{6}{4}=1.5\\hfill & \\hfill & \\hfill & \\hfill & \\hfill & \\text{ }x=\\frac{-8}{4}=-2\\hfill & \\hfill \\end{array}[\/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\/896\/2016\/10\/19015157\/CNX_Precalc_Figure_01_06_011F2.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\" \/><\/p>\n<p>The function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>For the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q206693\">Show Solution<\/span><\/p>\n<div id=\"q206693\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-1[\/latex] or [latex]x=2[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm40657\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40657&theme=oea&iframe_resize_id=ohm40657&show_question_numbers\" width=\"100%\" height=\"565\"><\/iframe><\/p>\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-174\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 469. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 60791. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 40657. <strong>Authored by<\/strong>:  Jenck,Michael. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et 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