{"id":16021,"date":"2019-09-26T17:24:00","date_gmt":"2019-09-26T17:24:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-calculate-and-interpret-slope\/"},"modified":"2024-05-02T15:54:08","modified_gmt":"2024-05-02T15:54:08","slug":"read-calculate-and-interpret-slope","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-calculate-and-interpret-slope\/","title":{"raw":"Characteristics of Linear Functions","rendered":"Characteristics of Linear Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify important features of graphs of linear functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have previously learned about linear equations and their line graphs.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-slope-given-two-points-on-a-line\/\">We saw how to find the slope of a line given two points<\/a>.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/outcome-equations-of-lines\/\">We also learned some ways to find the equations for a line graph<\/a>.\u00a0 We will now consider linear functions and their graphs. We previously learned about linear equations written in slope-intercept form, which is the form [latex]y=mx+b[\/latex] where\u00a0[latex]m[\/latex] is the rate of change or slope and [latex]b[\/latex] is the y-intercept.\r\n\r\nThe\u00a0<strong>slope-intercept form<\/strong>\u00a0is also a well-known form for writing linear functions, where [latex]f(x)[\/latex] is the output value, or dependant variable, [latex]x[\/latex] is the input value, or independant variable, [latex]m[\/latex] is the rate of change or slope, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable, that is [latex]b=f(0)[\/latex].\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 previously learned how 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.\r\n\r\nIn function notation, the corresponding values for the outputs [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 when using function notation we 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 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{ }}f({x}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }f({x}_{1}\\right))[\/latex], as long as each calculation is started with the elements from the same coordinate pair.\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}=\\dfrac{f\\left({x}_{1}\\right)-f\\left({x}_{2}\\right)}{{x}_{1}-{x}_{2}}[\/latex]<\/p>\r\n\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}}=\\dfrac{f(x_{2})-f(x_{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\nThe slope provides us with important information about the graph of a linear function.\u00a0 The greater the absolute value of the slope, the steeper the line is.\u00a0 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.\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]f(3)=-2[\/latex] and [latex]f(8)=1[\/latex], 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\nWe have been given two input and output values for [latex]f(x)[\/latex].\u00a0 [latex]f(3)=-2[\/latex] corresponds to the coordinate pair [latex]\\left(3,-2\\right)[\/latex]\u00a0and\u00a0[latex]f(8)=1[\/latex] corresponds to the coordinate pair [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{f\\left(8\\right)-f\\left(3\\right)}{8-3}=\\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<h2>Vertical and Horizontal Lines<\/h2>\r\nVertical and horizontal lines are two special cases of linear equations.\u00a0 Recall that the equation of a <strong>vertical line<\/strong> is given as\r\n<p style=\"text-align: center;\">[latex]x=c[\/latex]<\/p>\r\nwhere <em>c <\/em>is a constant. Regardless of the <em>y-<\/em>value of any point on the line, the <em>x-<\/em>coordinate of the point will be <em>c<\/em>.\r\n\r\nConsider a line containing the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex].\u00a0\u00a0First, we will try to find the slope.\r\n<p style=\"text-align: center;\">[latex]m=\\dfrac{5 - 3}{-3-\\left(-3\\right)}=\\dfrac{2}{0}[\/latex]<\/p>\r\nZero in the denominator means that the slope is undefined.\u00a0 As a result, it is impossible to write the vertical line as an equation in the form [latex]y=mx+b[\/latex].\r\n\r\nAlso, recall that a function must have exactly one output for every input.\u00a0 As a result,\u00a0a vertical line is NOT a function.\u00a0These points do not define a function because for our input, [latex]x=-3[\/latex], we have an infinite number of outputs!\r\n\r\nAlthough these points cannot be defined as a function, we can nevertheless plot the points.\u00a0 You can see the line in the graph below. Notice that all of the <em>x-<\/em>coordinates are the same and we find a vertical line through [latex]x=-3[\/latex].\r\n\r\nThe equation of a <strong>horizontal line<\/strong> is given as\r\n<p style=\"text-align: center;\">[latex]y=c[\/latex]<\/p>\r\nwhere <em>c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em>x-<\/em>value of a point on the line, the corresponding\u00a0<em>y-<\/em>coordinate will be <em>c<\/em>.\r\n\r\nSuppose we want to find the equation of a line that contains the following set of points: [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex]. We can use point-slope form. First, we find the slope using any two points on the line.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}m &amp; =\\dfrac{-2-\\left(-2\\right)}{0-\\left(-2\\right)}\\hfill \\\\ &amp; =\\dfrac{0}{2}\\hfill \\\\ &amp; =0\\hfill \\end{array}[\/latex]<\/p>\r\nThe graph is a horizontal line through [latex]y=-2[\/latex]. Notice that all of the <em>y-<\/em>coordinates are the same.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200322\/CNX_CAT_Figure_02_02_003.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"487\" height=\"367\" \/> The line [latex]x=\u22123[\/latex] is a vertical line. The line [latex]y=\u22122[\/latex] is a horizontal line.[\/caption]\r\nWhile we learned above that a vertical line is <em>not<\/em> a function, a horizontal line <em>does<\/em> define a function.\u00a0 Recall that for a function, every input must have exactly one output.\u00a0 In our example, for every input of x, the output is [latex]y=-2[\/latex].\u00a0 The fact that the output is always the same for every input is okay!\u00a0 Notice that our horizontal line shown in the graph above passes the vertical line test because any vertical line drawn on the coordinate system will pass through the line [latex]y=-2[\/latex] exactly once.\r\n\r\nThe function for our horizontal line is defined as [latex]f(x)=-2[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the equation of the line passing through the given points: [latex]\\left(1,-3\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"346281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"346281\"]\r\n\r\nThe <em>x-<\/em>coordinate of both points is\u00a0[latex]1[\/latex]. Therefore, we have a vertical line, [latex]x=1[\/latex].\u00a0 This line cannot be defined as a function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify important features of graphs of linear functions<\/li>\n<\/ul>\n<\/div>\n<p>We have previously learned about linear equations and their line graphs.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-slope-given-two-points-on-a-line\/\">We saw how to find the slope of a line given two points<\/a>.\u00a0 <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/outcome-equations-of-lines\/\">We also learned some ways to find the equations for a line graph<\/a>.\u00a0 We will now consider linear functions and their graphs. We previously learned about linear equations written in slope-intercept form, which is the form [latex]y=mx+b[\/latex] where\u00a0[latex]m[\/latex] is the rate of change or slope and [latex]b[\/latex] is the y-intercept.<\/p>\n<p>The\u00a0<strong>slope-intercept form<\/strong>\u00a0is also a well-known form for writing linear functions, where [latex]f(x)[\/latex] is the output value, or dependant variable, [latex]x[\/latex] is the input value, or independant variable, [latex]m[\/latex] is the rate of change or slope, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable, that is [latex]b=f(0)[\/latex].<\/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 previously learned how 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.<\/p>\n<p>In function notation, the corresponding values for the outputs [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 when using function notation we 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 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{ }}f({x}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }f({x}_{1}\\right))[\/latex], as long as each calculation is started with the elements from the same coordinate pair.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}=\\dfrac{f\\left({x}_{1}\\right)-f\\left({x}_{2}\\right)}{{x}_{1}-{x}_{2}}[\/latex]<\/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}}=\\dfrac{f(x_{2})-f(x_{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>The slope provides us with important information about the graph of a linear function.\u00a0 The greater the absolute value of the slope, the steeper the line is.\u00a0 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]f(3)=-2[\/latex] and [latex]f(8)=1[\/latex], 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>We have been given two input and output values for [latex]f(x)[\/latex].\u00a0 [latex]f(3)=-2[\/latex] corresponds to the coordinate pair [latex]\\left(3,-2\\right)[\/latex]\u00a0and\u00a0[latex]f(8)=1[\/latex] corresponds to the coordinate pair [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{f\\left(8\\right)-f\\left(3\\right)}{8-3}=\\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<h2>Vertical and Horizontal Lines<\/h2>\n<p>Vertical and horizontal lines are two special cases of linear equations.\u00a0 Recall that the equation of a <strong>vertical line<\/strong> is given as<\/p>\n<p style=\"text-align: center;\">[latex]x=c[\/latex]<\/p>\n<p>where <em>c <\/em>is a constant. Regardless of the <em>y-<\/em>value of any point on the line, the <em>x-<\/em>coordinate of the point will be <em>c<\/em>.<\/p>\n<p>Consider a line containing the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex].\u00a0\u00a0First, we will try to find the slope.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\dfrac{5 - 3}{-3-\\left(-3\\right)}=\\dfrac{2}{0}[\/latex]<\/p>\n<p>Zero in the denominator means that the slope is undefined.\u00a0 As a result, it is impossible to write the vertical line as an equation in the form [latex]y=mx+b[\/latex].<\/p>\n<p>Also, recall that a function must have exactly one output for every input.\u00a0 As a result,\u00a0a vertical line is NOT a function.\u00a0These points do not define a function because for our input, [latex]x=-3[\/latex], we have an infinite number of outputs!<\/p>\n<p>Although these points cannot be defined as a function, we can nevertheless plot the points.\u00a0 You can see the line in the graph below. Notice that all of the <em>x-<\/em>coordinates are the same and we find a vertical line through [latex]x=-3[\/latex].<\/p>\n<p>The equation of a <strong>horizontal line<\/strong> is given as<\/p>\n<p style=\"text-align: center;\">[latex]y=c[\/latex]<\/p>\n<p>where <em>c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em>x-<\/em>value of a point on the line, the corresponding\u00a0<em>y-<\/em>coordinate will be <em>c<\/em>.<\/p>\n<p>Suppose we want to find the equation of a line that contains the following set of points: [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex]. We can use point-slope form. First, we find the slope using any two points on the line.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}m & =\\dfrac{-2-\\left(-2\\right)}{0-\\left(-2\\right)}\\hfill \\\\ & =\\dfrac{0}{2}\\hfill \\\\ & =0\\hfill \\end{array}[\/latex]<\/p>\n<p>The graph is a horizontal line through [latex]y=-2[\/latex]. Notice that all of the <em>y-<\/em>coordinates are the same.<\/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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200322\/CNX_CAT_Figure_02_02_003.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\">The line [latex]x=\u22123[\/latex] is a vertical line. The line [latex]y=\u22122[\/latex] is a horizontal line.<\/p>\n<\/div>\n<p>While we learned above that a vertical line is <em>not<\/em> a function, a horizontal line <em>does<\/em> define a function.\u00a0 Recall that for a function, every input must have exactly one output.\u00a0 In our example, for every input of x, the output is [latex]y=-2[\/latex].\u00a0 The fact that the output is always the same for every input is okay!\u00a0 Notice that our horizontal line shown in the graph above passes the vertical line test because any vertical line drawn on the coordinate system will pass through the line [latex]y=-2[\/latex] exactly once.<\/p>\n<p>The function for our horizontal line is defined as [latex]f(x)=-2[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the equation of the line passing through the given points: [latex]\\left(1,-3\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q346281\">Show Solution<\/span><\/p>\n<div id=\"q346281\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>x-<\/em>coordinate of both points is\u00a0[latex]1[\/latex]. Therefore, we have a vertical line, [latex]x=1[\/latex].\u00a0 This line cannot be defined as a function.<\/p>\n<\/div>\n<\/div>\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-16021\">\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":169554,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"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 (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/in3NTcx11I8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Slope Application Involving Production Costs\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/4RbniDgEGE4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Write and Graph a Linear Function by Making a Table of Values (Intro)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/X3Sx2TxH-J0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"59bc4e998d814b60802c43c965682ffc, 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