{"id":1988,"date":"2016-06-30T21:14:09","date_gmt":"2016-06-30T21:14:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1988"},"modified":"2019-07-24T21:06:54","modified_gmt":"2019-07-24T21:06:54","slug":"read-calculate-and-interpret-slope","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-calculate-and-interpret-slope\/","title":{"raw":"Calculate and Interpret Slope","rendered":"Calculate and Interpret Slope"},"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<\/ul>\r\n<\/div>\r\nOne well known\u00a0form for writing linear functions is known as\u00a0<strong>slope-intercept form<\/strong>, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change or slope, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable.\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 often need to calculate the <strong>slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows\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] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex] are [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write\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 graph below indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex] is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201021\/CNX_Precalc_Figure_02_01_005n2.jpg\" alt=\"Graph depicting how to calculate the slope of a line\" width=\"487\" height=\"569\" \/>\r\n\r\nThe slope of a function is calculated by the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex]. It does not matter which coordinate is used as the [latex]\\left({x}_{2,\\text{ }}{y}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }{y}_{1}\\right)[\/latex], as long as each calculation is started with the elements from the same coordinate pair.\r\n\r\nThe units for slope are always [latex]\\dfrac{\\text{units for the output}}{\\text{units for the input}}[\/latex]. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.\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\nWhen the slope of a linear function is positive, the line is moving in an uphill direction from left to right across the coordinate axes. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative. The graph of a decreasing linear function is a line moving in a downhill direction from left to right across the coordinate axes.\r\n\r\nIn mathematical terms,\r\n\r\nFor a linear function [latex]f(x)=mx+b[\/latex], if [latex]m&gt;0[\/latex], then [latex]f(x)[\/latex] is an increasing function.\r\n\r\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex], if [latex]m&lt;0[\/latex], then [latex]f(x)[\/latex] is a\u00a0decreasing\u00a0function.\r\n\r\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex], if [latex]m=0[\/latex], then [latex]f(x)[\/latex] is a constant\u00a0function. Sometimes we say this is neither increasing nor decreasing.\r\n\r\nIn the following example, we will first find the slope of a linear function through two points then determine whether the line is increasing, decreasing, or neither.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?\r\n\r\n[reveal-answer q=\"103343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"103343\"]\r\n\r\nThe coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{\\text{change in output}}{\\text{change in input}}=\\dfrac{1-\\left(-2\\right)}{8 - 3}=\\dfrac{3}{5}[\/latex]<\/p>\r\nWe could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m&gt;0[\/latex].\r\n\r\nAs noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or <em>y<\/em>-coordinate, used corresponds with the first input value, or <em>x<\/em>-coordinate, used.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show examples of 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<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe population of a city increased from\u00a0[latex]23,400[\/latex] to\u00a0[latex]27,800[\/latex] between\u00a0[latex]2008[\/latex] and\u00a0[latex]2012[\/latex]. Find the change of population per year if we assume the change was constant from\u00a0[latex]2008[\/latex] to\u00a0[latex]2012[\/latex].\r\n\r\n[reveal-answer q=\"246268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"246268\"]\r\n\r\nThe rate of change relates the change in population to the change in time. The population increased by [latex]27,800-23,400=4400[\/latex] people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\dfrac{\\text{people}}{\\text{year}}[\/latex]<\/p>\r\nSo the population increased by\u00a0[latex]1,100[\/latex] people per year.\r\n\r\nBecause we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, we show an example where we determine 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<\/ul>\n<\/div>\n<p>One well known\u00a0form for writing linear functions is known as\u00a0<strong>slope-intercept form<\/strong>, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change or slope, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable.<\/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 often need to calculate the <strong>slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows<\/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] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex] are [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write<\/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 graph below indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex] is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201021\/CNX_Precalc_Figure_02_01_005n2.jpg\" alt=\"Graph depicting how to calculate the slope of a line\" width=\"487\" height=\"569\" \/><\/p>\n<p>The slope of a function is calculated by the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex]. It does not matter which coordinate is used as the [latex]\\left({x}_{2,\\text{ }}{y}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }{y}_{1}\\right)[\/latex], as long as each calculation is started with the elements from the same coordinate pair.<\/p>\n<p>The units for slope are always [latex]\\dfrac{\\text{units for the output}}{\\text{units for the input}}[\/latex]. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.<\/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>When the slope of a linear function is positive, the line is moving in an uphill direction from left to right across the coordinate axes. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative. The graph of a decreasing linear function is a line moving in a downhill direction from left to right across the coordinate axes.<\/p>\n<p>In mathematical terms,<\/p>\n<p>For a linear function [latex]f(x)=mx+b[\/latex], if [latex]m>0[\/latex], then [latex]f(x)[\/latex] is an increasing function.<\/p>\n<p>For a linear function\u00a0[latex]f(x)=mx+b[\/latex], if [latex]m<0[\/latex], then [latex]f(x)[\/latex] is a\u00a0decreasing\u00a0function.\n\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex], if [latex]m=0[\/latex], then [latex]f(x)[\/latex] is a constant\u00a0function. Sometimes we say this is neither increasing nor decreasing.\n\nIn the following example, we will first find the slope of a linear function through two points then determine whether the line is increasing, decreasing, or neither.\n\n\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q103343\">Show Solution<\/span><\/p>\n<div id=\"q103343\" class=\"hidden-answer\" style=\"display: none\">\n<p>The coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{\\text{change in output}}{\\text{change in input}}=\\dfrac{1-\\left(-2\\right)}{8 - 3}=\\dfrac{3}{5}[\/latex]<\/p>\n<p>We could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m>0[\/latex].<\/p>\n<p>As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or <em>y<\/em>-coordinate, used corresponds with the first input value, or <em>x<\/em>-coordinate, used.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show examples of 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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The population of a city increased from\u00a0[latex]23,400[\/latex] to\u00a0[latex]27,800[\/latex] between\u00a0[latex]2008[\/latex] and\u00a0[latex]2012[\/latex]. Find the change of population per year if we assume the change was constant from\u00a0[latex]2008[\/latex] to\u00a0[latex]2012[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q246268\">Show Solution<\/span><\/p>\n<div id=\"q246268\" class=\"hidden-answer\" style=\"display: none\">\n<p>The rate of change relates the change in population to the change in time. The population increased by [latex]27,800-23,400=4400[\/latex] people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\dfrac{\\text{people}}{\\text{year}}[\/latex]<\/p>\n<p>So the population increased by\u00a0[latex]1,100[\/latex] people per year.<\/p>\n<p>Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video, we show an example where we determine 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-1988\">\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>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>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>Write and Graph a Linear Function by Making a Table of Values (Intro). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><\/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":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et 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