{"id":172,"date":"2023-06-21T13:22:40","date_gmt":"2023-06-21T13:22:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/write-equations-of-linear-functions\/"},"modified":"2023-09-21T08:32:04","modified_gmt":"2023-09-21T08:32:04","slug":"write-equations-of-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/write-equations-of-linear-functions\/","title":{"raw":"\u25aa   Writing Equations of Linear Functions","rendered":"\u25aa   Writing Equations of Linear Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write the equation of a linear function given its graph<\/li>\r\n \t<li>Match linear functions with their graphs<\/li>\r\n \t<li>Find the x-intercept of a function given its equation<\/li>\r\n \t<li>Find the equations of vertical and horizontal lines<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe previously wrote\u00a0the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at the graph below. We can see right away that the graph crosses the <em>y<\/em>-axis at the point (0, 4), so this is the <em>y<\/em>-intercept.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184341\/CNX_Precalc_Figure_02_02_0102.jpg\" width=\"369\" height=\"378\" \/>\r\n\r\nThen we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point (\u20132, 0). To get from this point to the <em>y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be:\r\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex]<\/p>\r\nSubstituting the slope and <em>y-<\/em>intercept into slope-intercept form of a line gives:\r\n<p style=\"text-align: center;\">[latex]y=2x+4[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given THE graph of A linear function, find the equation to describe the function.<\/h3>\r\n<ol>\r\n \t<li>Identify the <em>y-<\/em>intercept from the graph.<\/li>\r\n \t<li>Choose two points to determine the slope.<\/li>\r\n \t<li>Substitute the <em>y-<\/em>intercept and slope into slope-intercept form of a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Matching Linear Functions to Their Graphs<\/h3>\r\nMatch each equation of a linear function with one of the lines in the graph below.\r\n<ol>\r\n \t<li>[latex]f\\left(x\\right)=2x+3[\/latex]<\/li>\r\n \t<li>[latex]g\\left(x\\right)=2x - 3[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)=-2x+3[\/latex]<\/li>\r\n \t<li>[latex]j\\left(x\\right)=\\frac{1}{2}x+3[\/latex]<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184343\/CNX_Precalc_Figure_02_02_0112.jpg\" alt=\"Graph of three lines, line 1) passes through (0,3) and (-2, -1), line 2) passes through (0,3) and (-6,0), line 3) passes through (0,-3) and (2,1)\" width=\"393\" height=\"305\" \/>\r\n[reveal-answer q=\"659573\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"659573\"]\r\n\r\nAnalyze the information for each function.\r\n<ol>\r\n \t<li>This function has a slope of 2 and a <em>y<\/em>-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function <em>g<\/em>\u00a0has the same slope, but a different <em>y-<\/em>intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through (0, 3) so <em>f<\/em>\u00a0must be represented by line I.<\/li>\r\n \t<li>This function also has a slope of 2, but a <em>y<\/em>-intercept of \u20133. It must pass through the point (0, \u20133) and slant upward from left to right. It must be represented by line III.<\/li>\r\n \t<li>This function has a slope of \u20132 and a <em>y-<\/em>intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.<\/li>\r\n \t<li>This function has a slope of [latex]\\frac{1}{2}[\/latex] and a <em>y-<\/em>intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through (0, 3), but the slope of <em>j<\/em>\u00a0is less than the slope of <em>f<\/em>\u00a0so the line for <em>j<\/em>\u00a0must be flatter. This function is represented by Line II.<\/li>\r\n<\/ol>\r\nNow we can re-label the lines.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184346\/CNX_Precalc_Figure_02_02_0122.jpg\" width=\"489\" height=\"374\" \/>\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\n[ohm_question height=\"550\"]1440[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding the <em>x<\/em>-intercept of a Line<\/h2>\r\nSo far we have been finding the <em>y-<\/em>intercepts of functions: the point at which the graph of a function crosses the <em>y<\/em>-axis. A function may also have an <strong><em>x<\/em><\/strong><strong>-intercept,<\/strong> which is the <em>x<\/em>-coordinate of the point where the graph of a function crosses the <em>x<\/em>-axis. In other words, it is the input value when the output value is zero.\r\n\r\nTo find the <em>x<\/em>-intercept, set the function <em>f<\/em>(<em>x<\/em>) equal to zero and solve for the value of <em>x<\/em>. For example, consider the function shown:\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=3x - 6[\/latex]<\/p>\r\nSet the function equal to 0 and solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=3x - 6\\hfill \\\\ 6=3x\\hfill \\\\ 2=x\\hfill \\\\ x=2\\hfill \\end{array}[\/latex]<\/p>\r\nThe graph of the function crosses the <em>x<\/em>-axis at the point (2, 0).\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Do all linear functions have <em>x<\/em>-intercepts?<\/strong>\r\n\r\n<em>No. However, linear functions of the form <\/em>y\u00a0<em>= <\/em>c<em>, where <\/em>c<em> is a nonzero real number are the only examples of linear functions with no <\/em>x<em>-intercept. For example, <\/em>y\u00a0<em>= 5 is a horizontal line 5 units above the <\/em>x<em>-axis. This function has no <\/em>x<em>-intercepts<\/em>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184348\/CNX_Precalc_Figure_02_02_0262.jpg\" alt=\"Graph of y = 5.\" width=\"421\" height=\"231\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: <em>x<\/em>-intercept<\/h3>\r\nThe <strong><em>x<\/em>-intercept<\/strong> of a function is the value of <em>x<\/em>\u00a0where\u00a0<em>f<\/em>(<em>x<\/em>) = 0. It can be found by solving the equation 0 = <em>mx\u00a0<\/em>+ <em>b<\/em>.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding an <em>x<\/em>-intercept<\/h3>\r\nFind the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].\r\n\r\n[reveal-answer q=\"400055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"400055\"]\r\n\r\nSet the function equal to zero to solve for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=\\frac{1}{2}x - 3\\\\ 3=\\frac{1}{2}x\\\\ 6=x\\\\ x=6\\end{array}[\/latex]<\/p>\r\nThe graph crosses the <em>x<\/em>-axis at the point (6, 0).\r\n<h4>Analysis of the Solution<\/h4>\r\nA graph of the function is shown below. We can see that the <em>x<\/em>-intercept is (6, 0) as expected.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"369\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184351\/CNX_Precalc_Figure_02_02_0132.jpg\" width=\"369\" height=\"378\" \/> The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].[\/caption]<strong>\u00a0<\/strong>[\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{4}x - 4[\/latex].\r\n\r\n[reveal-answer q=\"406982\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"406982\"]\r\n\r\n[latex]\\left(16,\\text{ 0}\\right)[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"140\"]79757[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Describing Horizontal and Vertical Lines<\/h2>\r\nThere are two special cases of lines on a graph\u2014horizontal and vertical lines. A <strong>horizontal line<\/strong> indicates a constant output or <em>y<\/em>-value. In the graph below, we see that the output has a value of 2 for every input value. The change in outputs between any two points is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use <em>m\u00a0<\/em>= 0 in the equation [latex]f\\left(x\\right)=mx+b[\/latex], the equation simplifies to [latex]f\\left(x\\right)=b[\/latex]. In other words, the value of the function is a constant. This graph represents the function [latex]f\\left(x\\right)=2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184353\/CNX_Precalc_Figure_02_02_0142.jpg\" width=\"487\" height=\"473\" \/> A horizontal line representing the function [latex]f\\left(x\\right)=2[\/latex].[\/caption]&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184355\/CNX_Precalc_Figure_02_02_0152.jpg\" alt=\"M equals change of output divided by change of input. The numerator is a non-zero real number, and the change of input is zero.\" width=\"487\" height=\"99\" \/>\r\n\r\nA <strong>vertical line<\/strong> indicates a constant input or <em>x<\/em>-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.\r\n\r\nNotice that a vertical line has an <em>x<\/em>-intercept but no <em>y-<\/em>intercept unless it\u2019s the line <em>x<\/em> = 0. This graph represents the line <em>x<\/em> = 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184358\/CNX_Precalc_Figure_02_02_0162.jpg\" width=\"487\" height=\"473\" \/> The vertical line [latex]x=2[\/latex] which does not represent a function.[\/caption]\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal and Vertical Lines<\/h3>\r\nLines can be horizontal or vertical.\r\n\r\nA <strong>horizontal line<\/strong> is a line defined by an equation of the form [latex]f\\left(x\\right)=b[\/latex] where\u00a0[latex]b[\/latex] is a constant.\r\n\r\nA <strong>vertical line<\/strong> is a line defined by an equation of the form [latex]x=a[\/latex]\u00a0where\u00a0[latex]a[\/latex] is a constant.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing the Equation of a Horizontal Line<\/h3>\r\nWrite the equation of the line 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\/21184401\/CNX_Precalc_Figure_02_02_0172.jpg\" alt=\"Graph of x = 7.\" width=\"369\" height=\"378\" \/>\r\n[reveal-answer q=\"891444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"891444\"]\r\n\r\nFor any <em>x<\/em>-value, the <em>y<\/em>-value is [latex]\u20134[\/latex], so the equation is [latex]y=\u20134[\/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\n[ohm_question height=\"140\"]15599[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing the Equation of a Vertical Line<\/h3>\r\nWrite the equation of the line 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\/21184404\/CNX_Precalc_Figure_02_02_0182.jpg\" alt=\"Graph of two functions where the baby blue line is y = -2\/3x + 7, and the blue line is y = -x + 1.\" width=\"369\" height=\"378\" \/>\r\n\r\n[reveal-answer q=\"178822\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"178822\"]\r\n\r\nThe constant <em>x<\/em>-value is 7, so the equation is [latex]x=7[\/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\n[ohm_question height=\"850\"]114592[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFrom the information given below, write the related equations. Then walk through the slides to experiment with slope changes<em>.<\/em>\r\n<ul>\r\n \t<li>Write the equation of the function passing through the points [latex](2,6)[\/latex] and [latex](4,4)[\/latex] in slope-intercept form.<\/li>\r\n \t<li>Write the equation of a function whose slope is 2 and passes through the point [latex](-1,0)[\/latex]<\/li>\r\n \t<li>Write the equation of a function whose slope is undefined.<\/li>\r\n<\/ul>\r\n<strong>INTERACTIVE, move the points to see what happens to the slope<\/strong>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write the equation of a linear function given its graph<\/li>\n<li>Match linear functions with their graphs<\/li>\n<li>Find the x-intercept of a function given its equation<\/li>\n<li>Find the equations of vertical and horizontal lines<\/li>\n<\/ul>\n<\/div>\n<p>We previously wrote\u00a0the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at the graph below. We can see right away that the graph crosses the <em>y<\/em>-axis at the point (0, 4), so this is the <em>y<\/em>-intercept.<\/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\/21184341\/CNX_Precalc_Figure_02_02_0102.jpg\" width=\"369\" height=\"378\" alt=\"image\" \/><\/p>\n<p>Then we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point (\u20132, 0). To get from this point to the <em>y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be:<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex]<\/p>\n<p>Substituting the slope and <em>y-<\/em>intercept into slope-intercept form of a line gives:<\/p>\n<p style=\"text-align: center;\">[latex]y=2x+4[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given THE graph of A linear function, find the equation to describe the function.<\/h3>\n<ol>\n<li>Identify the <em>y-<\/em>intercept from the graph.<\/li>\n<li>Choose two points to determine the slope.<\/li>\n<li>Substitute the <em>y-<\/em>intercept and slope into slope-intercept form of a line.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Matching Linear Functions to Their Graphs<\/h3>\n<p>Match each equation of a linear function with one of the lines in the graph below.<\/p>\n<ol>\n<li>[latex]f\\left(x\\right)=2x+3[\/latex]<\/li>\n<li>[latex]g\\left(x\\right)=2x - 3[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=-2x+3[\/latex]<\/li>\n<li>[latex]j\\left(x\\right)=\\frac{1}{2}x+3[\/latex]<\/li>\n<\/ol>\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\/21184343\/CNX_Precalc_Figure_02_02_0112.jpg\" alt=\"Graph of three lines, line 1) passes through (0,3) and (-2, -1), line 2) passes through (0,3) and (-6,0), line 3) passes through (0,-3) and (2,1)\" width=\"393\" height=\"305\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q659573\">Show Solution<\/span><\/p>\n<div id=\"q659573\" class=\"hidden-answer\" style=\"display: none\">\n<p>Analyze the information for each function.<\/p>\n<ol>\n<li>This function has a slope of 2 and a <em>y<\/em>-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function <em>g<\/em>\u00a0has the same slope, but a different <em>y-<\/em>intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through (0, 3) so <em>f<\/em>\u00a0must be represented by line I.<\/li>\n<li>This function also has a slope of 2, but a <em>y<\/em>-intercept of \u20133. It must pass through the point (0, \u20133) and slant upward from left to right. It must be represented by line III.<\/li>\n<li>This function has a slope of \u20132 and a <em>y-<\/em>intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.<\/li>\n<li>This function has a slope of [latex]\\frac{1}{2}[\/latex] and a <em>y-<\/em>intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through (0, 3), but the slope of <em>j<\/em>\u00a0is less than the slope of <em>f<\/em>\u00a0so the line for <em>j<\/em>\u00a0must be flatter. This function is represented by Line II.<\/li>\n<\/ol>\n<p>Now we can re-label the lines.<\/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\/21184346\/CNX_Precalc_Figure_02_02_0122.jpg\" width=\"489\" height=\"374\" alt=\"image\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1440\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1440&theme=oea&iframe_resize_id=ohm1440&show_question_numbers\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the <em>x<\/em>-intercept of a Line<\/h2>\n<p>So far we have been finding the <em>y-<\/em>intercepts of functions: the point at which the graph of a function crosses the <em>y<\/em>-axis. A function may also have an <strong><em>x<\/em><\/strong><strong>-intercept,<\/strong> which is the <em>x<\/em>-coordinate of the point where the graph of a function crosses the <em>x<\/em>-axis. In other words, it is the input value when the output value is zero.<\/p>\n<p>To find the <em>x<\/em>-intercept, set the function <em>f<\/em>(<em>x<\/em>) equal to zero and solve for the value of <em>x<\/em>. For example, consider the function shown:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=3x - 6[\/latex]<\/p>\n<p>Set the function equal to 0 and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=3x - 6\\hfill \\\\ 6=3x\\hfill \\\\ 2=x\\hfill \\\\ x=2\\hfill \\end{array}[\/latex]<\/p>\n<p>The graph of the function crosses the <em>x<\/em>-axis at the point (2, 0).<\/p>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do all linear functions have <em>x<\/em>-intercepts?<\/strong><\/p>\n<p><em>No. However, linear functions of the form <\/em>y\u00a0<em>= <\/em>c<em>, where <\/em>c<em> is a nonzero real number are the only examples of linear functions with no <\/em>x<em>-intercept. For example, <\/em>y\u00a0<em>= 5 is a horizontal line 5 units above the <\/em>x<em>-axis. This function has no <\/em>x<em>-intercepts<\/em>.<\/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\/21184348\/CNX_Precalc_Figure_02_02_0262.jpg\" alt=\"Graph of y = 5.\" width=\"421\" height=\"231\" \/><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: <em>x<\/em>-intercept<\/h3>\n<p>The <strong><em>x<\/em>-intercept<\/strong> of a function is the value of <em>x<\/em>\u00a0where\u00a0<em>f<\/em>(<em>x<\/em>) = 0. It can be found by solving the equation 0 = <em>mx\u00a0<\/em>+ <em>b<\/em>.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding an <em>x<\/em>-intercept<\/h3>\n<p>Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400055\">Show Solution<\/span><\/p>\n<div id=\"q400055\" class=\"hidden-answer\" style=\"display: none\">\n<p>Set the function equal to zero to solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0=\\frac{1}{2}x - 3\\\\ 3=\\frac{1}{2}x\\\\ 6=x\\\\ x=6\\end{array}[\/latex]<\/p>\n<p>The graph crosses the <em>x<\/em>-axis at the point (6, 0).<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>A graph of the function is shown below. We can see that the <em>x<\/em>-intercept is (6, 0) as expected.<\/p>\n<div style=\"width: 379px\" 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\/21184351\/CNX_Precalc_Figure_02_02_0132.jpg\" width=\"369\" height=\"378\" alt=\"image\" \/><\/p>\n<p class=\"wp-caption-text\">The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\n<\/div>\n<p><strong>\u00a0<\/strong><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{4}x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q406982\">Show Solution<\/span><\/p>\n<div id=\"q406982\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(16,\\text{ 0}\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm79757\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79757&theme=oea&iframe_resize_id=ohm79757&show_question_numbers\" width=\"100%\" height=\"140\"><\/iframe><\/p>\n<\/div>\n<h2>Describing Horizontal and Vertical Lines<\/h2>\n<p>There are two special cases of lines on a graph\u2014horizontal and vertical lines. A <strong>horizontal line<\/strong> indicates a constant output or <em>y<\/em>-value. In the graph below, we see that the output has a value of 2 for every input value. The change in outputs between any two points is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use <em>m\u00a0<\/em>= 0 in the equation [latex]f\\left(x\\right)=mx+b[\/latex], the equation simplifies to [latex]f\\left(x\\right)=b[\/latex]. In other words, the value of the function is a constant. This graph represents the function [latex]f\\left(x\\right)=2[\/latex].<\/p>\n<div style=\"width: 497px\" 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\/21184353\/CNX_Precalc_Figure_02_02_0142.jpg\" width=\"487\" height=\"473\" alt=\"image\" \/><\/p>\n<p class=\"wp-caption-text\">A horizontal line representing the function [latex]f\\left(x\\right)=2[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/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\/21184355\/CNX_Precalc_Figure_02_02_0152.jpg\" alt=\"M equals change of output divided by change of input. The numerator is a non-zero real number, and the change of input is zero.\" width=\"487\" height=\"99\" \/><\/p>\n<p>A <strong>vertical line<\/strong> indicates a constant input or <em>x<\/em>-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.<\/p>\n<p>Notice that a vertical line has an <em>x<\/em>-intercept but no <em>y-<\/em>intercept unless it\u2019s the line <em>x<\/em> = 0. This graph represents the line <em>x<\/em> = 2.<\/p>\n<div style=\"width: 497px\" 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\/21184358\/CNX_Precalc_Figure_02_02_0162.jpg\" width=\"487\" height=\"473\" alt=\"image\" \/><\/p>\n<p class=\"wp-caption-text\">The vertical line [latex]x=2[\/latex] which does not represent a function.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal and Vertical Lines<\/h3>\n<p>Lines can be horizontal or vertical.<\/p>\n<p>A <strong>horizontal line<\/strong> is a line defined by an equation of the form [latex]f\\left(x\\right)=b[\/latex] where\u00a0[latex]b[\/latex] is a constant.<\/p>\n<p>A <strong>vertical line<\/strong> is a line defined by an equation of the form [latex]x=a[\/latex]\u00a0where\u00a0[latex]a[\/latex] is a constant.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Equation of a Horizontal Line<\/h3>\n<p>Write the equation of the line 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\/21184401\/CNX_Precalc_Figure_02_02_0172.jpg\" alt=\"Graph of x = 7.\" width=\"369\" height=\"378\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q891444\">Show Solution<\/span><\/p>\n<div id=\"q891444\" class=\"hidden-answer\" style=\"display: none\">\n<p>For any <em>x<\/em>-value, the <em>y<\/em>-value is [latex]\u20134[\/latex], so the equation is [latex]y=\u20134[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm15599\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15599&theme=oea&iframe_resize_id=ohm15599&show_question_numbers\" width=\"100%\" height=\"140\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Equation of a Vertical Line<\/h3>\n<p>Write the equation of the line 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\/21184404\/CNX_Precalc_Figure_02_02_0182.jpg\" alt=\"Graph of two functions where the baby blue line is y = -2\/3x + 7, and the blue line is y = -x + 1.\" width=\"369\" height=\"378\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q178822\">Show Solution<\/span><\/p>\n<div id=\"q178822\" class=\"hidden-answer\" style=\"display: none\">\n<p>The constant <em>x<\/em>-value is 7, so the equation is [latex]x=7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm114592\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=114592&theme=oea&iframe_resize_id=ohm114592&show_question_numbers\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>From the information given below, write the related equations. Then walk through the slides to experiment with slope changes<em>.<\/em><\/p>\n<ul>\n<li>Write the equation of the function passing through the points [latex](2,6)[\/latex] and [latex](4,4)[\/latex] in slope-intercept form.<\/li>\n<li>Write the equation of a function whose slope is 2 and passes through the point [latex](-1,0)[\/latex]<\/li>\n<li>Write the equation of a function whose slope is undefined.<\/li>\n<\/ul>\n<p><strong>INTERACTIVE, move the points to see what happens to the slope<\/strong><\/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-172\">\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><li>Slope Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/7ighwgcjyi\">https:\/\/www.desmos.com\/calculator\/7ighwgcjyi<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/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 1440. <strong>Authored by<\/strong>: unknown, 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 79757. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15599. <strong>Authored by<\/strong>: Johns,Bryan. <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 114592. <strong>Authored 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>. <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":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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