{"id":770,"date":"2025-07-15T16:08:00","date_gmt":"2025-07-15T16:08:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=770"},"modified":"2025-08-13T01:14:47","modified_gmt":"2025-08-13T01:14:47","slug":"linear-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-learn-it-3\/","title":{"raw":"Linear Functions: Learn It 3","rendered":"Linear Functions: Learn It 3"},"content":{"raw":"<h3>Graphing a Linear Function Using [latex]y[\/latex]-intercept and Slope<\/h3>\r\nAnother way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its [latex]y[\/latex]<em>-<\/em>intercept which is the point at which the input value is zero. To find the [latex]y[\/latex]<strong><em>-<\/em>intercept<\/strong>, we can set [latex]x=0[\/latex] in the equation. The other characteristic of the linear function is its slope [latex]m[\/latex].\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Keep in mind that if a function has a [latex]y[\/latex]-intercept, we can always find it by setting [latex]x=0[\/latex] and then solving for [latex]y[\/latex].<\/section><section class=\"textbox example\" aria-label=\"Example\">Let\u2019s consider the following function.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{1}{2}x+1[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>The slope is [latex]\\frac{1}{2}[\/latex]. Because the slope is positive, we know the graph will slant upward from left to right.<\/li>\r\n \t<li>The [latex]y[\/latex]<em>-<\/em>intercept is the point on the graph when [latex]x\u00a0= 0[\/latex]. The graph crosses the [latex]y[\/latex]-axis at [latex](0, 1)[\/latex].<\/li>\r\n<\/ul>\r\nNow we know the slope and the [latex]y[\/latex]-intercept. We can begin graphing by plotting the point [latex](0, 1)[\/latex] We know that the slope is rise over run, [latex]m=\\frac{\\text{rise}}{\\text{run}}[\/latex].\r\n\r\nFrom our example, we have [latex]m=\\frac{1}{2}[\/latex], which means that the rise is [latex]1[\/latex] and the run is [latex]2[\/latex]. Starting from our [latex]y[\/latex]-intercept [latex](0, 1)[\/latex], we can rise [latex]1[\/latex] and then run [latex]2[\/latex] or run [latex]2[\/latex] and then rise [latex]1[\/latex]. We repeat until we have multiple points, and then we draw a line through the points as shown 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\/21184323\/CNX_Precalc_Figure_02_02_0032.jpg\" alt=\"graph of the line y = (1\/2)x +1 showing the &quot;rise&quot;, or change in the y direction as 1 and the &quot;run&quot;, or change in x direction as 2, and the y-intercept at (0,1)\" width=\"617\" height=\"340\" \/>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>graphical interpretation of a linear function<\/h3>\r\nIn the equation [latex]f\\left(x\\right)=mx+b[\/latex]\r\n<ul>\r\n \t<li>[latex]b[\/latex]\u00a0is the [latex]y[\/latex]-intercept of the graph and indicates the point [latex](0, b)[\/latex] at which the graph crosses the [latex]y[\/latex]-axis.<\/li>\r\n \t<li>[latex]m[\/latex]\u00a0is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the equation for a linear function, graph the function using the [latex]y[\/latex]-intercept and slope.<\/strong>\r\n<ol>\r\n \t<li>Evaluate the function at an input value of zero to find the [latex]y[\/latex]<em>-<\/em>intercept.<\/li>\r\n \t<li>Identify the slope.<\/li>\r\n \t<li>Plot the point represented by the <em>y-<\/em>intercept.<\/li>\r\n \t<li>Use [latex]\\frac{\\text{rise}}{\\text{run}}[\/latex] to determine at least two more points on the line.<\/li>\r\n \t<li>Draw a line which passes through the points.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex] using the [latex]y[\/latex]<em>-<\/em>intercept and slope.[reveal-answer q=\"507667\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507667\"]Evaluate the function at [latex]x\u00a0= 0[\/latex] to find the [latex]y-[\/latex]intercept. The output value when [latex]x\u00a0= 0[\/latex] is [latex]5[\/latex], so the graph will cross the [latex]y[\/latex]-axis at [latex](0, 5)[\/latex].\r\n[latex]\\\\[\/latex]\r\nAccording to the equation for the function, the slope of the line is [latex]-\\frac{2}{3}[\/latex]. This tells us that for each vertical decrease in the \"rise\" of [latex]\u20132[\/latex] units, the \"run\" increases by [latex]3[\/latex] units in the horizontal direction.\r\n[latex]\\\\[\/latex]\r\nWe can now graph the function by first plotting the [latex]y[\/latex]-intercept. From the initial value [latex](0, 5)[\/latex] we move down [latex]2[\/latex] units and to the right [latex]3[\/latex] units. We can extend the line to the left and right by repeating, and then draw a line through the points.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184325\/CNX_Precalc_Figure_02_02_0042.jpg\" alt=\"graph of the line y = (-2\/3)x + 5 showing the change of -2 in y and change of 3 in x.\" width=\"487\" height=\"318\" \/>\r\n[latex]\\\\[\/latex]\r\n<strong>Analysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nThe graph slants downward from left to right which means it has a negative slope as expected.[\/hidden-answer]<\/section><section aria-label=\"Example\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Using a Graphing Utility to Plot Lines<\/strong>Graphing utilities are powerful tools that allow you to visualize mathematical concepts and plot lines quickly and accurately. Whether you're checking your work, exploring different equations, or just trying to understand how changes in variables affect a graph, these online tools can help. Below are some popular graphing utilities you can use to plot lines and analyze functions. Simply enter your equation, and the utility will generate the graph for you.\r\n\r\n<a href=\"https:\/\/www.geogebra.org\/graphing\">https:\/\/www.geogebra.org\/graphing<\/a>\r\n\r\n<a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>\r\n\r\n<a href=\"https:\/\/www.mathway.com\/Graph\">https:\/\/www.mathway.com\/Graph<\/a>\r\n\r\n<a href=\"https:\/\/www.symbolab.com\/graphing-calculator\">https:\/\/www.symbolab.com\/graphing-calculator<\/a>\r\n\r\n<strong>Try it now<\/strong>\r\n\r\nThese graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:\r\n<ol>\r\n \t<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\r\n \t<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says \"add slider: a\", click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\r\n \t<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\r\n<\/ol>\r\n<\/section><\/section>","rendered":"<h3>Graphing a Linear Function Using [latex]y[\/latex]-intercept and Slope<\/h3>\n<p>Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its [latex]y[\/latex]<em>&#8211;<\/em>intercept which is the point at which the input value is zero. To find the [latex]y[\/latex]<strong><em>&#8211;<\/em>intercept<\/strong>, we can set [latex]x=0[\/latex] in the equation. The other characteristic of the linear function is its slope [latex]m[\/latex].<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Keep in mind that if a function has a [latex]y[\/latex]-intercept, we can always find it by setting [latex]x=0[\/latex] and then solving for [latex]y[\/latex].<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let\u2019s consider the following function.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{1}{2}x+1[\/latex]<\/p>\n<ul>\n<li>The slope is [latex]\\frac{1}{2}[\/latex]. Because the slope is positive, we know the graph will slant upward from left to right.<\/li>\n<li>The [latex]y[\/latex]<em>&#8211;<\/em>intercept is the point on the graph when [latex]x\u00a0= 0[\/latex]. The graph crosses the [latex]y[\/latex]-axis at [latex](0, 1)[\/latex].<\/li>\n<\/ul>\n<p>Now we know the slope and the [latex]y[\/latex]-intercept. We can begin graphing by plotting the point [latex](0, 1)[\/latex] We know that the slope is rise over run, [latex]m=\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/p>\n<p>From our example, we have [latex]m=\\frac{1}{2}[\/latex], which means that the rise is [latex]1[\/latex] and the run is [latex]2[\/latex]. Starting from our [latex]y[\/latex]-intercept [latex](0, 1)[\/latex], we can rise [latex]1[\/latex] and then run [latex]2[\/latex] or run [latex]2[\/latex] and then rise [latex]1[\/latex]. We repeat until we have multiple points, and then we draw a line through the points as shown 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\/21184323\/CNX_Precalc_Figure_02_02_0032.jpg\" alt=\"graph of the line y = (1\/2)x +1 showing the &quot;rise&quot;, or change in the y direction as 1 and the &quot;run&quot;, or change in x direction as 2, and the y-intercept at (0,1)\" width=\"617\" height=\"340\" \/><\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>graphical interpretation of a linear function<\/h3>\n<p>In the equation [latex]f\\left(x\\right)=mx+b[\/latex]<\/p>\n<ul>\n<li>[latex]b[\/latex]\u00a0is the [latex]y[\/latex]-intercept of the graph and indicates the point [latex](0, b)[\/latex] at which the graph crosses the [latex]y[\/latex]-axis.<\/li>\n<li>[latex]m[\/latex]\u00a0is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the equation for a linear function, graph the function using the [latex]y[\/latex]-intercept and slope.<\/strong><\/p>\n<ol>\n<li>Evaluate the function at an input value of zero to find the [latex]y[\/latex]<em>&#8211;<\/em>intercept.<\/li>\n<li>Identify the slope.<\/li>\n<li>Plot the point represented by the <em>y-<\/em>intercept.<\/li>\n<li>Use [latex]\\frac{\\text{rise}}{\\text{run}}[\/latex] to determine at least two more points on the line.<\/li>\n<li>Draw a line which passes through the points.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex] using the [latex]y[\/latex]<em>&#8211;<\/em>intercept and slope.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q507667\">Show Solution<\/button><\/p>\n<div id=\"q507667\" class=\"hidden-answer\" style=\"display: none\">Evaluate the function at [latex]x\u00a0= 0[\/latex] to find the [latex]y-[\/latex]intercept. The output value when [latex]x\u00a0= 0[\/latex] is [latex]5[\/latex], so the graph will cross the [latex]y[\/latex]-axis at [latex](0, 5)[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nAccording to the equation for the function, the slope of the line is [latex]-\\frac{2}{3}[\/latex]. This tells us that for each vertical decrease in the &#8220;rise&#8221; of [latex]\u20132[\/latex] units, the &#8220;run&#8221; increases by [latex]3[\/latex] units in the horizontal direction.<br \/>\n[latex]\\\\[\/latex]<br \/>\nWe can now graph the function by first plotting the [latex]y[\/latex]-intercept. From the initial value [latex](0, 5)[\/latex] we move down [latex]2[\/latex] units and to the right [latex]3[\/latex] units. We can extend the line to the left and right by repeating, and then draw a line through the points.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21184325\/CNX_Precalc_Figure_02_02_0042.jpg\" alt=\"graph of the line y = (-2\/3)x + 5 showing the change of -2 in y and change of 3 in x.\" width=\"487\" height=\"318\" \/><br \/>\n[latex]\\\\[\/latex]<br \/>\n<strong>Analysis of the Solution<\/strong><br \/>\n[latex]\\\\[\/latex]<br \/>\nThe graph slants downward from left to right which means it has a negative slope as expected.<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Using a Graphing Utility to Plot Lines<\/strong>Graphing utilities are powerful tools that allow you to visualize mathematical concepts and plot lines quickly and accurately. Whether you&#8217;re checking your work, exploring different equations, or just trying to understand how changes in variables affect a graph, these online tools can help. Below are some popular graphing utilities you can use to plot lines and analyze functions. Simply enter your equation, and the utility will generate the graph for you.<\/p>\n<p><a href=\"https:\/\/www.geogebra.org\/graphing\">https:\/\/www.geogebra.org\/graphing<\/a><\/p>\n<p><a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a><\/p>\n<p><a href=\"https:\/\/www.mathway.com\/Graph\">https:\/\/www.mathway.com\/Graph<\/a><\/p>\n<p><a href=\"https:\/\/www.symbolab.com\/graphing-calculator\">https:\/\/www.symbolab.com\/graphing-calculator<\/a><\/p>\n<p><strong>Try it now<\/strong><\/p>\n<p>These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:<\/p>\n<ol>\n<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\n<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says &#8220;add slider: a&#8221;, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\n<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\n<\/ol>\n<\/section>\n<\/section>\n","protected":false},"author":13,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/770"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/770\/revisions"}],"predecessor-version":[{"id":2387,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/770\/revisions\/2387"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/770\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=770"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=770"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=770"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=770"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}