{"id":1709,"date":"2024-06-05T16:30:29","date_gmt":"2024-06-05T16:30:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1709"},"modified":"2025-08-13T23:23:49","modified_gmt":"2025-08-13T23:23:49","slug":"graphs-of-linear-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-linear-functions-learn-it-1\/","title":{"raw":"Graphs of Linear Functions: Learn It 1","rendered":"Graphs of Linear Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Create and interpret equations of linear functions<\/li>\r\n \t<li>Identify and graph lines that are vertical or horizontal<\/li>\r\n \t<li>Graph straight lines by plotting points, using slope and y-intercept, and make changes like shifts to graphs<\/li>\r\n \t<li>Write equations for lines that run parallel or at a right angle to another line<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\r\n[caption id=\"attachment_3996\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3996 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c.jpg\" alt=\"This graph shows a linear function graphed on an x y coordinate plane. The x axis is labeled from negative 2 to 8 and the y axis is labeled from negative 1 to 8. The function f is graph along the points (0, 7) and (4, 4).\" width=\"487\" height=\"347\" \/> Graph of a linear function with two points labeled[\/caption]\r\n\r\nPreviously we wrote equations in both the\u00a0<span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">slope-intercept form<\/span>\u00a0and the\u00a0<span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">point-slope form<\/span>. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function [latex]f[\/latex] given below:\r\n\r\nWe are not given the slope of the line, but we can choose any two points on the line to find the slope. Let\u2019s choose [latex](0, 7)[\/latex] and [latex](4, 4)[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m &amp; = &amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\ m &amp; = &amp; \\frac{4 - 7}{4 - 0} \\\\ m &amp; = &amp; -\\frac{3}{4} \\end{array} [\/latex]<\/p>\r\nNow we can substitute the slope and the coordinates of one of the points into the point-slope form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} y - y_1 &amp; = &amp; m(x - x_1) \\\\ y - 4 &amp; = &amp; -\\frac{3}{4}(x - 4) \\end{array} [\/latex]<\/p>\r\nIf we want to rewrite the equation in the slope-intercept form, we would find\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} y - 4 &amp; = &amp; -\\frac{3}{4}(x - 4) \\\\ y - 4 &amp; = &amp; -\\frac{3}{4}x + 3 \\\\ y &amp; = &amp; -\\frac{3}{4}x + 7 \\end{array} [\/latex]<\/p>\r\nIf we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore, [latex]b = 7[\/latex]. We now have the initial value [latex]b[\/latex] and the slope [latex]m[\/latex], so we can substitute [latex]m[\/latex] and [latex]b[\/latex] into the slope-intercept form of a line.\r\n\r\n[caption id=\"attachment_3997\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3997 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc.jpg\" alt=\"This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.\" width=\"487\" height=\"155\" \/> Substituting values into slope-intercept form[\/caption]\r\n\r\nSo the function is [latex]f(x) = -\\frac{3}{4}x + 7[\/latex], and the linear equation would be [latex]y = -\\frac{3}{4}x + 7[\/latex].\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p id=\"fs-id1791064\"><strong>How to: Given the graph of a linear function, write an equation to represent the function.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1575825\" type=\"1\">\r\n \t<li>Identify two points on the line.<\/li>\r\n \t<li>Use the two points to calculate the slope.<\/li>\r\n \t<li>Determine where the line crosses the\u00a0<em data-effect=\"italics\">y<\/em>-axis to identify the\u00a0<em data-effect=\"italics\">y<\/em>-intercept by visual inspection.<\/li>\r\n \t<li>Substitute the slope and\u00a0<em data-effect=\"italics\">y<\/em>-intercept into the slope-intercept form of a line equation.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the equations of the linear function for the following graphs.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201110\/CNX_Precalc_Figure_02_02_019n2.jpg\" alt=\"Graph of two functions where the blue line is y = -2\/3x + 1, and the baby blue line is y = -2\/3x +7. Notice that they are parallel lines.\" width=\"325\" height=\"274\" \/> Graph of two linear functions[\/caption]\r\n\r\n[reveal-answer q=\"258972\"]Darker Line[\/reveal-answer]\r\n[hidden-answer a=\"258972\"]\r\n<div class=\"page\" title=\"Page 405\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nWe can see right away that the graph crosses the [latex]y[\/latex]-axis at the point [latex](0,1)[\/latex], so, this is the [latex]y[\/latex]-intercept. Thus, [latex]b = 1[\/latex].\r\n<div class=\"page\" title=\"Page 405\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nThen we can calculate the slope by finding the rise and run. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">We can choose any two points:<\/span>\r\n\r\n[caption id=\"attachment_1735\" align=\"aligncenter\" width=\"299\"]<img class=\"wp-image-1735\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2.jpg\" alt=\"\" width=\"299\" height=\"252\" \/> Graph of two linear functions[\/caption]\r\n\r\nThus, the slope is [latex]-\\dfrac{3}{2}[\/latex].\r\n\r\n<strong>The equation of the line is [latex]f(x) = -\\dfrac{3}{2}x+1[\/latex].<\/strong>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"12499\"]Lighter Line[\/reveal-answer]\r\n[hidden-answer a=\"12499\"]We can see right away that the graph crosses the [latex]y[\/latex]-axis at the point [latex](0,7)[\/latex], so, this is the [latex]y[\/latex]-intercept. Thus, [latex]b = 7[\/latex].\r\n<div class=\"page\" title=\"Page 411\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nTwo two lines have exactly the same steepness, which means their slopes are identical. Thus, the slope is also [latex]-\\dfrac{3}{2}[\/latex].\r\n\r\n<strong>The equation of the line is [latex]f(x) = -\\dfrac{3}{2}x+7[\/latex].<\/strong>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]19211[\/ohm2_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]f[\/latex] is a linear function, with [latex]f(3) = -2[\/latex], and [latex]f(8) = 1[\/latex], find an equation for the function in slope-intercept form.[reveal-answer q=\"917490\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"917490\"]We can write the given points using coordinates.\r\n<center>[latex] f(3) = -2 \\rightarrow (3, -2) \\ f(8) = 1 \\rightarrow (8, 1) [\/latex]<\/center>\r\nWe can then use the points to calculate the slope.\r\n<center>[latex] \\begin{array}{rcl} m &amp; = &amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\ m &amp; = &amp; \\frac{1 - (-2)}{8 - 3} \\\\ m &amp; = &amp; \\frac{3}{5} \\end{array} [\/latex]<\/center>\r\nSubstitute the slope and the coordinates of one of the points into the point-slope form.\r\n<center>[latex] \\begin{array}{rcl} y - y_1 &amp; = &amp; m(x - x_1) \\\\ y - (-2) &amp; = &amp; \\frac{3}{5}(x - 3) \\end{array} [\/latex]<\/center>\r\nWe can use algebra to rewrite the equation in the slope-intercept form.\r\n<center>[latex] \\begin{array}{rcl} y + 2 &amp; = &amp; \\frac{3}{5}(x - 3) \\\\ y + 2 &amp; = &amp; \\frac{3}{5}x - \\frac{9}{5} \\\\ y &amp; = &amp; \\frac{3}{5}x - \\frac{19}{5} \\end{array} [\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Suppose Ben starts a company in which he incurs a fixed cost of [latex]$1,250[\/latex] per month for the overhead, which includes his office rent. His production costs are [latex]$37.50[\/latex] per item.\r\n[latex]\\\\[\/latex]\r\nWrite a linear function [latex]C(x)[\/latex] where [latex]C(x)[\/latex] is the cost for [latex]x[\/latex] items produced in a given month.[reveal-answer q=\"251216\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"251216\"]We can analyze Ben's cost problem using the slope-intercept form [latex]y = mx + b[\/latex], where the slope [latex]m[\/latex] represents the variable cost per item, and the [latex]y[\/latex]-intercept [latex]b[\/latex] represents the fixed monthly cost.In this case:\r\n<ul>\r\n \t<li>The <strong>slope<\/strong> is [latex]37.50[\/latex], meaning the cost increases by [latex]$37.50[\/latex] for each additional item produced.<\/li>\r\n \t<li>The <strong>y-intercept<\/strong> is [latex]1250[\/latex], which is the fixed cost Ben incurs even when no items are produced.<\/li>\r\n<\/ul>\r\nThus, the linear function for Ben's total cost [latex]C(x)[\/latex] can be written as:\r\n<p style=\"text-align: center;\">[latex]C(x) = 37.50x + 1250[\/latex]<\/p>\r\nThis equation represents the total monthly cost for producing [latex]x[\/latex] items, where the slope reflects the rate of change in cost and the [latex]y[\/latex]-intercept reflects the fixed cost.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]290348[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Create and interpret equations of linear functions<\/li>\n<li>Identify and graph lines that are vertical or horizontal<\/li>\n<li>Graph straight lines by plotting points, using slope and y-intercept, and make changes like shifts to graphs<\/li>\n<li>Write equations for lines that run parallel or at a right angle to another line<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\n<figure id=\"attachment_3996\" aria-describedby=\"caption-attachment-3996\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3996 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c.jpg\" alt=\"This graph shows a linear function graphed on an x y coordinate plane. The x axis is labeled from negative 2 to 8 and the y axis is labeled from negative 1 to 8. The function f is graph along the points (0, 7) and (4, 4).\" width=\"487\" height=\"347\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c-300x214.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c-65x46.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c-225x160.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164838\/ad267d8ce45e7410c4a1abb402789284ed15ea9c-350x249.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><figcaption id=\"caption-attachment-3996\" class=\"wp-caption-text\">Graph of a linear function with two points labeled<\/figcaption><\/figure>\n<p>Previously we wrote equations in both the\u00a0<span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">slope-intercept form<\/span>\u00a0and the\u00a0<span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">point-slope form<\/span>. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function [latex]f[\/latex] given below:<\/p>\n<p>We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let\u2019s choose [latex](0, 7)[\/latex] and [latex](4, 4)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m & = & \\frac{y_2 - y_1}{x_2 - x_1} \\\\ m & = & \\frac{4 - 7}{4 - 0} \\\\ m & = & -\\frac{3}{4} \\end{array}[\/latex]<\/p>\n<p>Now we can substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} y - y_1 & = & m(x - x_1) \\\\ y - 4 & = & -\\frac{3}{4}(x - 4) \\end{array}[\/latex]<\/p>\n<p>If we want to rewrite the equation in the slope-intercept form, we would find<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} y - 4 & = & -\\frac{3}{4}(x - 4) \\\\ y - 4 & = & -\\frac{3}{4}x + 3 \\\\ y & = & -\\frac{3}{4}x + 7 \\end{array}[\/latex]<\/p>\n<p>If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore, [latex]b = 7[\/latex]. We now have the initial value [latex]b[\/latex] and the slope [latex]m[\/latex], so we can substitute [latex]m[\/latex] and [latex]b[\/latex] into the slope-intercept form of a line.<\/p>\n<figure id=\"attachment_3997\" aria-describedby=\"caption-attachment-3997\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3997 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc.jpg\" alt=\"This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.\" width=\"487\" height=\"155\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc-300x95.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc-65x21.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc-225x72.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16164923\/d0721a58dc75094ba9ea4cf8d804d63dbb9cf6fc-350x111.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><figcaption id=\"caption-attachment-3997\" class=\"wp-caption-text\">Substituting values into slope-intercept form<\/figcaption><\/figure>\n<p>So the function is [latex]f(x) = -\\frac{3}{4}x + 7[\/latex], and the linear equation would be [latex]y = -\\frac{3}{4}x + 7[\/latex].<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p id=\"fs-id1791064\"><strong>How to: Given the graph of a linear function, write an equation to represent the function.<\/strong><\/p>\n<ol id=\"fs-id1575825\" type=\"1\">\n<li>Identify two points on the line.<\/li>\n<li>Use the two points to calculate the slope.<\/li>\n<li>Determine where the line crosses the\u00a0<em data-effect=\"italics\">y<\/em>-axis to identify the\u00a0<em data-effect=\"italics\">y<\/em>-intercept by visual inspection.<\/li>\n<li>Substitute the slope and\u00a0<em data-effect=\"italics\">y<\/em>-intercept into the slope-intercept form of a line equation.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the equations of the linear function for the following graphs.<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201110\/CNX_Precalc_Figure_02_02_019n2.jpg\" alt=\"Graph of two functions where the blue line is y = -2\/3x + 1, and the baby blue line is y = -2\/3x +7. Notice that they are parallel lines.\" width=\"325\" height=\"274\" \/><figcaption class=\"wp-caption-text\">Graph of two linear functions<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q258972\">Darker Line<\/button><\/p>\n<div id=\"q258972\" class=\"hidden-answer\" style=\"display: none\">\n<div class=\"page\" title=\"Page 405\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>We can see right away that the graph crosses the [latex]y[\/latex]-axis at the point [latex](0,1)[\/latex], so, this is the [latex]y[\/latex]-intercept. Thus, [latex]b = 1[\/latex].<\/p>\n<div class=\"page\" title=\"Page 405\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Then we can calculate the slope by finding the rise and run. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">We can choose any two points:<\/span><\/p>\n<figure id=\"attachment_1735\" aria-describedby=\"caption-attachment-1735\" style=\"width: 299px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1735\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2.jpg\" alt=\"\" width=\"299\" height=\"252\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2-300x253.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2-65x55.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2-225x189.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/05211954\/CNX_Precalc_Figure_02_02_019n2-350x295.jpg 350w\" sizes=\"(max-width: 299px) 100vw, 299px\" \/><figcaption id=\"caption-attachment-1735\" class=\"wp-caption-text\">Graph of two linear functions<\/figcaption><\/figure>\n<p>Thus, the slope is [latex]-\\dfrac{3}{2}[\/latex].<\/p>\n<p><strong>The equation of the line is [latex]f(x) = -\\dfrac{3}{2}x+1[\/latex].<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q12499\">Lighter Line<\/button><\/p>\n<div id=\"q12499\" class=\"hidden-answer\" style=\"display: none\">We can see right away that the graph crosses the [latex]y[\/latex]-axis at the point [latex](0,7)[\/latex], so, this is the [latex]y[\/latex]-intercept. Thus, [latex]b = 7[\/latex].<\/p>\n<div class=\"page\" title=\"Page 411\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Two two lines have exactly the same steepness, which means their slopes are identical. Thus, the slope is also [latex]-\\dfrac{3}{2}[\/latex].<\/p>\n<p><strong>The equation of the line is [latex]f(x) = -\\dfrac{3}{2}x+7[\/latex].<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm19211\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=19211&theme=lumen&iframe_resize_id=ohm19211&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]f[\/latex] is a linear function, with [latex]f(3) = -2[\/latex], and [latex]f(8) = 1[\/latex], find an equation for the function in slope-intercept form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q917490\">Show Answer<\/button><\/p>\n<div id=\"q917490\" class=\"hidden-answer\" style=\"display: none\">We can write the given points using coordinates.<\/p>\n<div style=\"text-align: center;\">[latex]f(3) = -2 \\rightarrow (3, -2) \\ f(8) = 1 \\rightarrow (8, 1)[\/latex]<\/div>\n<p>We can then use the points to calculate the slope.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} m & = & \\frac{y_2 - y_1}{x_2 - x_1} \\\\ m & = & \\frac{1 - (-2)}{8 - 3} \\\\ m & = & \\frac{3}{5} \\end{array}[\/latex]<\/div>\n<p>Substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} y - y_1 & = & m(x - x_1) \\\\ y - (-2) & = & \\frac{3}{5}(x - 3) \\end{array}[\/latex]<\/div>\n<p>We can use algebra to rewrite the equation in the slope-intercept form.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} y + 2 & = & \\frac{3}{5}(x - 3) \\\\ y + 2 & = & \\frac{3}{5}x - \\frac{9}{5} \\\\ y & = & \\frac{3}{5}x - \\frac{19}{5} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose Ben starts a company in which he incurs a fixed cost of [latex]$1,250[\/latex] per month for the overhead, which includes his office rent. His production costs are [latex]$37.50[\/latex] per item.<br \/>\n[latex]\\\\[\/latex]<br \/>\nWrite a linear function [latex]C(x)[\/latex] where [latex]C(x)[\/latex] is the cost for [latex]x[\/latex] items produced in a given month.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q251216\">Show Answer<\/button><\/p>\n<div id=\"q251216\" class=\"hidden-answer\" style=\"display: none\">We can analyze Ben&#8217;s cost problem using the slope-intercept form [latex]y = mx + b[\/latex], where the slope [latex]m[\/latex] represents the variable cost per item, and the [latex]y[\/latex]-intercept [latex]b[\/latex] represents the fixed monthly cost.In this case:<\/p>\n<ul>\n<li>The <strong>slope<\/strong> is [latex]37.50[\/latex], meaning the cost increases by [latex]$37.50[\/latex] for each additional item produced.<\/li>\n<li>The <strong>y-intercept<\/strong> is [latex]1250[\/latex], which is the fixed cost Ben incurs even when no items are produced.<\/li>\n<\/ul>\n<p>Thus, the linear function for Ben&#8217;s total cost [latex]C(x)[\/latex] can be written as:<\/p>\n<p style=\"text-align: center;\">[latex]C(x) = 37.50x + 1250[\/latex]<\/p>\n<p>This equation represents the total monthly cost for producing [latex]x[\/latex] items, where the slope reflects the rate of change in cost and the [latex]y[\/latex]-intercept reflects the fixed cost.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm290348\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=290348&theme=lumen&iframe_resize_id=ohm290348&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":164,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1709"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":23,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1709\/revisions"}],"predecessor-version":[{"id":7693,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1709\/revisions\/7693"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/164"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1709\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1709"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1709"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1709"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}