{"id":164,"date":"2023-06-21T13:22:39","date_gmt":"2023-06-21T13:22:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/the-slope-of-a-linear-function\/"},"modified":"2023-07-03T19:58:03","modified_gmt":"2023-07-03T19:58:03","slug":"the-slope-of-a-linear-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/the-slope-of-a-linear-function\/","title":{"raw":"R3.2   The Slope of a Linear Function","rendered":"R3.2   The Slope of a Linear Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define slope for a linear function<\/li>\r\n \t<li>Calculate slope given two points<\/li>\r\n \t<li>Interpret the slope of a linear function that models a real-world situation<\/li>\r\n<\/ul>\r\n<\/div>\r\nWriting the equation that describes a linear function follows the same procedure for writing the equation of a line. You may be familiar with the\u00a0<strong>slope-intercept form <\/strong>of a linear equation, where\u00a0[latex]m[\/latex] stands for the slope of the line and [latex]b[\/latex] represents the y-intercept, the place where the line crosses the y-axis of the graph.\r\n\r\nWhen discussing a linear\u00a0function, ordered pairs of the form [latex]\\left(x, y\\right)[\/latex] are written using\u00a0<strong>function notation<\/strong> as\u00a0[latex]\\left(x, f(x)\\right)[\/latex]. The slope can be thought of as a rate of change in function output over a corresponding interval of input.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\text{Equation form}\\hfill &amp; y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill &amp; f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\r\nWe can calculate the <strong>slope<\/strong>\u00a0[latex]m[\/latex]\u00a0of a line given two points on the line, [latex]\\left(x_1, y_1\\right)[\/latex] and [latex]\\left(x_2, y_2\\right)[\/latex] by using the formula\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex],<\/p>\r\nwhere [latex]\\Delta y[\/latex] represents the difference between the [latex]y[\/latex] coordinates and [latex]\\Delta x[\/latex] represents the corresponding difference between the [latex]x[\/latex] coordinates.\r\n\r\nIn a linear function, the coordinates of the points on the graph of a function represent input and output values for the function. The ordered pairs are written\u00a0[latex]\\left(x_1, f(x_1)\\right)[\/latex] and [latex]\\left(x_2, f(x_2)\\right)[\/latex], and the formula for slope may be written equivalently as the formula for average rate of change\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex].<\/p>\r\nThe units for slope or for a rate of change are expressed as a ratio of output units over input units in the form\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{units for the output}}{\\text{units for the input}}[\/latex].<\/p>\r\nRead these units as \"change in units of the output value <em>for each<\/em> unit of change in input value.\" For example, a rate may be given in <em>miles per hour<\/em> or\u00a0a wage given in\u00a0<em>dollars<\/em><em>\u00a0per day.<\/em>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: left;\">Calculating Slope<\/h3>\r\nThe slope, or rate of change, of a function [latex]m[\/latex] can be calculated using the following formula:\r\n\r\n[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]\r\n\r\nwhere [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex] are input values, [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] are output values.\r\n\r\n<\/div>\r\nGiven the equation of a line or a linear function in slope-intercept form, certain behavior of the graph can be seen by examining the value of [latex]m[\/latex] in the equation [latex]y=mx+b[\/latex] or the function [latex]f(x)=mx+b[\/latex].\r\n<ul>\r\n \t<li>When the slope of a line or the rate of change of a linear function is positive, [latex]m \\gt 0,[\/latex] it will describe an \"uphill\" line in the plane, rising from left to right as the input increases.<\/li>\r\n \t<li>When the slope of a line or the rate of change of a linear function is negative, [latex]m \\lt 0,[\/latex] it will describe a \"downhill\" line in the plan, falling from left to right as the input increases.<\/li>\r\n \t<li>When the slope of a line or the rate of change of a linear function is 0, [latex]m = 0,[\/latex] it will describe a flat, horizontal line in the plane, neither rising nor falling as the input increases.<\/li>\r\n<\/ul>\r\nWatch the video below to see how to find the slope of a line passing through two points and then determine whether the line is increasing, decreasing or neither.\r\n\r\nhttps:\/\/youtu.be\/in3NTcx11I8\r\n\r\nWatch the next video to see an example of an application of slope in determining the increase in cost for producing solar panels given two data points.\r\n\r\nhttps:\/\/youtu.be\/4RbniDgEGE4\r\n\r\nThe following video provides an example of how to write a function that will give the cost in dollars for a given number of credit hours taken, x.\r\n\r\nhttps:\/\/youtu.be\/X3Sx2TxH-J0","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define slope for a linear function<\/li>\n<li>Calculate slope given two points<\/li>\n<li>Interpret the slope of a linear function that models a real-world situation<\/li>\n<\/ul>\n<\/div>\n<p>Writing the equation that describes a linear function follows the same procedure for writing the equation of a line. You may be familiar with the\u00a0<strong>slope-intercept form <\/strong>of a linear equation, where\u00a0[latex]m[\/latex] stands for the slope of the line and [latex]b[\/latex] represents the y-intercept, the place where the line crosses the y-axis of the graph.<\/p>\n<p>When discussing a linear\u00a0function, ordered pairs of the form [latex]\\left(x, y\\right)[\/latex] are written using\u00a0<strong>function notation<\/strong> as\u00a0[latex]\\left(x, f(x)\\right)[\/latex]. The slope can be thought of as a rate of change in function output over a corresponding interval of input.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\text{Equation form}\\hfill & y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill & f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\n<p>We can calculate the <strong>slope<\/strong>\u00a0[latex]m[\/latex]\u00a0of a line given two points on the line, [latex]\\left(x_1, y_1\\right)[\/latex] and [latex]\\left(x_2, y_2\\right)[\/latex] by using the formula<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex],<\/p>\n<p>where [latex]\\Delta y[\/latex] represents the difference between the [latex]y[\/latex] coordinates and [latex]\\Delta x[\/latex] represents the corresponding difference between the [latex]x[\/latex] coordinates.<\/p>\n<p>In a linear function, the coordinates of the points on the graph of a function represent input and output values for the function. The ordered pairs are written\u00a0[latex]\\left(x_1, f(x_1)\\right)[\/latex] and [latex]\\left(x_2, f(x_2)\\right)[\/latex], and the formula for slope may be written equivalently as the formula for average rate of change<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex].<\/p>\n<p>The units for slope or for a rate of change are expressed as a ratio of output units over input units in the form<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{units for the output}}{\\text{units for the input}}[\/latex].<\/p>\n<p>Read these units as &#8220;change in units of the output value <em>for each<\/em> unit of change in input value.&#8221; For example, a rate may be given in <em>miles per hour<\/em> or\u00a0a wage given in\u00a0<em>dollars<\/em><em>\u00a0per day.<\/em><\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: left;\">Calculating Slope<\/h3>\n<p>The slope, or rate of change, of a function [latex]m[\/latex] can be calculated using the following formula:<\/p>\n<p>[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n<p>where [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex] are input values, [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] are output values.<\/p>\n<\/div>\n<p>Given the equation of a line or a linear function in slope-intercept form, certain behavior of the graph can be seen by examining the value of [latex]m[\/latex] in the equation [latex]y=mx+b[\/latex] or the function [latex]f(x)=mx+b[\/latex].<\/p>\n<ul>\n<li>When the slope of a line or the rate of change of a linear function is positive, [latex]m \\gt 0,[\/latex] it will describe an &#8220;uphill&#8221; line in the plane, rising from left to right as the input increases.<\/li>\n<li>When the slope of a line or the rate of change of a linear function is negative, [latex]m \\lt 0,[\/latex] it will describe a &#8220;downhill&#8221; line in the plan, falling from left to right as the input increases.<\/li>\n<li>When the slope of a line or the rate of change of a linear function is 0, [latex]m = 0,[\/latex] it will describe a flat, horizontal line in the plane, neither rising nor falling as the input increases.<\/li>\n<\/ul>\n<p>Watch the video below to see how to find the slope of a line passing through two points and then determine whether the line is increasing, decreasing or neither.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Find the Slope Given Two Points and Describe the Line\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/in3NTcx11I8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Watch the next video to see an example of an application of slope in determining the increase in cost for producing solar panels given two data points.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Slope Application Involving Production Costs\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4RbniDgEGE4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following video provides an example of how to write a function that will give the cost in dollars for a given number of credit hours taken, x.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Write and Graph a Linear Function by Making a Table of Values (Intro)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/X3Sx2TxH-J0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-164\">\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>Ex: Find the Slope Given Two Points and Describe the Line. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/in3NTcx11I8\">https:\/\/youtu.be\/in3NTcx11I8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Slope Application Involving Production Costs. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/4RbniDgEGE4\">https:\/\/youtu.be\/4RbniDgEGE4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Write and Graph a Linear Function by Making a Table of Values (Intro). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/X3Sx2TxH-J0\">https:\/\/youtu.be\/X3Sx2TxH-J0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":4,"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\":\"cc\",\"description\":\"Ex: Find the Slope Given Two Points and Describe the Line\",\"author\":\"James Sousa 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