{"id":4316,"date":"2013-06-29T03:53:09","date_gmt":"2013-06-29T03:53:09","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=4316"},"modified":"2026-03-26T12:55:35","modified_gmt":"2026-03-26T17:55:35","slug":"changing-constants-in-graphs-of-functions-linear-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/changing-constants-in-graphs-of-functions-linear-functions\/","title":{"rendered":"Graphing Linear Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_SVQUVLNNIzk\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_SVQUVLNNIzk\" data-source-videoID=\"SVQUVLNNIzk\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Graphing Linear Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Graphing Linear Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_SVQUVLNNIzk:hover {cursor:pointer;} img#videoThumbnailImage_SVQUVLNNIzk {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-graphing-linear-functions-64c13ed51706c-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_SVQUVLNNIzk\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_SVQUVLNNIzk\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Graphing Linear Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_SVQUVLNNIzk\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_SVQUVLNNIzk\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_SVQUVLNNIzk\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction b5I_Function() {\n  var x = document.getElementById(\"b5I\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"b5I_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"b5I\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Reviewing_Terms\" class=\"smooth-scroll\">Reviewing Terms<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_Linear_Functions\" class=\"smooth-scroll\">Graphing Linear Functions<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_1_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_3\" class=\"smooth-scroll\">Example #3<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_4\" class=\"smooth-scroll\">Example #4<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_5\" class=\"smooth-scroll\">Example #5<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_Linear_Function_Practice_Questions\" class=\"smooth-scroll\">Graphing Linear Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello, and welcome to this video about graphs of linear functions!<\/p>\n<p>Today we\u2019ll explore what happens to a graph when the slope or <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept is changed. Specifically, we\u2019ll examine what happens when these constants are positive or negative values, as well as when the slope is a fractional value. <\/p>\n<h2><span id=\"Reviewing_Terms\" class=\"m-toc-anchor\"><\/span>Reviewing Terms<\/h2>\n<p>\nBefore we get started, let\u2019s review a few things.<\/p>\n<p>A <strong>linear function<\/strong> is a function that is a straight line when graphed. Its equation can be written in <strong>slope-intercept form<\/strong>, <span style=\"font-style:normal; font-size:90%\">\\(y = mx + b\\)<\/span>.<\/p>\n<p>The variable <span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span> represents the <strong>slope<\/strong>, which measures the direction and steepness of the line graphed. The slope is found by calculating the rise over run, which is the change in <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-coordinates divided by the change in <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-coordinates.<\/p>\n<p>The variable <span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span> represents the <span style=\"font-style:normal; font-size:90%\"><strong>\\(\\mathbf{y}\\)<\/strong><\/span><strong>-intercept<\/strong>, the point where the graph of a line intersects the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis. <\/p>\n<h2><span id=\"Graphing_Linear_Functions\" class=\"m-toc-anchor\"><\/span>Graphing Linear Functions<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s take a look at an example together. Consider the equation <span style=\"font-style:normal; font-size:90%\">\\(y = 2x + 1\\)<\/span>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89953\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png\" alt=\"The linear equation y= 2x + 1 graphed.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Let\u2019s start by finding the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept. Looking at the graph, we see that the line crosses through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>. Our equation reflects this because the value of <span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span> is <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>. <\/p>\n<p>Now let\u2019s examine the slope. From the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept, the second point is found by moving in a vertical direction, the rise, and then a horizontal direction, the run. From the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept, move two units up and one unit to the right.<\/p>\n<p>Therefore, our slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) equals <span style=\"font-style:normal; font-size:90%\">\\(\\frac{2}{1}\\)<\/span>, which equals <span style=\"font-style:normal; font-size:90%\">\\(2\\)<\/span>. Our equation reflects this because the value for <span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span> is <span style=\"font-style:normal; font-size:90%\">\\(2\\)<\/span>. <\/p>\n<p>Now let\u2019s consider how the graph changes if we change the slope. Consider the equation <span style=\"font-style:normal; font-size:90%\">\\(y = -2x + 1\\)<\/span>. It is the same as our last equation, except now our value for the slope is a negative number, <span style=\"font-style:normal; font-size:90%\">\\(-\\frac{2}{1}\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\(-2\\)<\/span>. Let\u2019s examine the new graph for this equation and compare it to the previous graph: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-2.png\" alt=\"A graph of linear equation y=-2x+1\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89956\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-2.png 1915w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-2-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-2-768x433.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-2-1536x866.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>As you can see, the line in this graph moves in an opposite direction as compared to the first graph. Since the slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) is negative, the line moves in a negative direction. <\/p>\n<p>The <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept (<span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span>) is <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, which is the same as our previous graph. The line crosses through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>. Our equation reflects this because the value for <span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span> is also 1. <\/p>\n<p>From the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>, plot the second point on the line by moving in a vertical direction (rise) and then a horizontal direction (run). Because the numerator of the slope is <span style=\"font-style:normal; font-size:90%\">\\(-2\\)<\/span>, move <span style=\"font-style:normal; font-size:90%\">\\(2\\)<\/span> units down from the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept. From there, move <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span> unit to the right, as indicated by the slope\u2019s denominator, <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>.<\/p>\n<p>Note: A positive \u201crise\u201d moves <strong>up<\/strong>, and a negative \u201crise\u201d moves <strong>down<\/strong>; a positive \u201crun\u201d moves <strong>right<\/strong>, and a negative \u201crun\u201d moves <strong>left<\/strong>. <\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s examine another graph that changes the slope again. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3.png\" alt=\"Image of graphed y=-2 over 3x + 1\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89959\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-3-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>The equation for this graph is <span style=\"font-style:normal; font-size:90%\">\\(y=-\\frac{2}{3}x+1\\)<\/span>. This time, our slope is a fraction, <span style=\"font-style:normal; font-size:90%\">\\(-\\frac{2}{3}\\)<\/span>. Since the value of <span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span> is negative, this line moves in a negative direction. Our <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept value has not changed, so we still see that the line crosses through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>. Our equation reflects this because the value for <span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span> is also <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>. <\/p>\n<p>Notice how the steepness of this line is different. Compared to the last two graphs, this line is less steep. Let\u2019s understand why that is. From the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>, plot the second point on the line by moving in a vertical direction (rise) and then a horizontal direction (run). Since <span style=\"font-style:normal; font-size:90%\">\\(m=-\\frac{2}{3}\\)<\/span>, move two units down and three units to the right. From this example, we can see that the larger the slope\u2019s denominator is, the less steep the line will be. <\/p>\n<p>What if the value of the slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) was zero? Consider the equation <span style=\"font-style:normal; font-size:90%\">\\(y=0x + 1\\)<\/span>. In this case, there is no rise or run because the value of <span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span> equals <span style=\"font-style:normal; font-size:90%\">\\(0\\)<\/span>. There is a <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept at <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, 1)\\)<\/span>. When graphed, a line with a slope of zero is a horizontal line, as shown: <\/p>\n<p>Based on this information, what would the graph for <span style=\"font-style:normal; font-size:90%\">\\(y=0x + 5\\)<\/span> look like? It would look like a horizontal line passing through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept of <span style=\"font-style:normal; font-size:90%\">\\(5\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, 5)\\)<\/span>. <\/p>\n<p>What would the graph for <span style=\"font-style:normal; font-size:90%\">\\(y=0x + 0\\)<\/span> look like? &#8230;That\u2019s right, a horizontal line passing through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept of <span style=\"font-style:normal; font-size:90%\">\\(0\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0,0)\\)<\/span>. <\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nNow that we know what happens to the graph of a linear function when we change slope, let\u2019s examine what happens when we change the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept. Consider the graph for the equation <span style=\"font-style:normal; font-size:90%\">\\(y=2x &#8211; 1\\)<\/span>. <\/p>\n<p>Recall the first equation and graph we looked at, <span style=\"font-style:normal; font-size:90%\">\\(y=2x + 1\\)<\/span>. The only difference in this equation is that the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept <span style=\"font-style:normal; font-size:90%\">(\\(b\\))<\/span> is a negative value, <span style=\"font-style:normal; font-size:90%\">\\(-1\\)<\/span>. As a result, we see on our graph that the line intersects the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at <span style=\"font-style:normal; font-size:90%\">\\(-1\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, -1)\\)<\/span>. <\/p>\n<p>From the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept <span style=\"font-style:normal; font-size:90%\">\\((0, -1)\\)<\/span>, the second point on the line is plotted by moving in a vertical direction (rise) and then a horizontal direction (run). Since <span style=\"font-style:normal; font-size:90%\">\\(m=\\frac{2}{1}\\)<\/span>, move two units up and one unit over to the right. Therefore, our slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) equals <span style=\"font-style:normal; font-size:90%\">\\(\\frac{2}{1}\\)<\/span>, which equals <span style=\"font-style:normal; font-size:90%\">\\(2\\)<\/span>. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4.png\" alt=\"image of linear equation y= 2x -1 graphed \" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89962\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-4-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>What do you think the graph would look like for a linear equation with a <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept value of zero? Let\u2019s take a look. Consider the equation <span style=\"font-style:normal; font-size:90%\">\\(y=2x + 0\\)<\/span>, which can also be written as <span style=\"font-style:normal; font-size:90%\">\\(y = 2x\\)<\/span>: <\/p>\n<p>As you can see, the line passes through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at the origin, or zero. Since the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept (\\(b\\)) is <span style=\"font-style:normal; font-size:90%\">\\(0\\)<\/span>, this makes sense. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89977\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-corrected-5.png\" alt=\"Image of linear equation y=2x+0\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-corrected-5-300x168.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-corrected-5-1024x574.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-corrected-5-1536x861.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>The slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) is <span style=\"font-style:normal; font-size:90%\">\\(\\frac{2}{1}\\)<\/span>. From the origin, move two units up (rise) and one unit over (run) to reach the next point on the line. <\/p>\n<h3><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h3>\n<p>\nWhat if the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept is a fraction? Consider the equation <span style=\"font-style:normal; font-size:90%\">\\(y=2x+\\frac{1}{2}\\)<\/span>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6.png\" alt=\"linear equation y=2x+1 over 2.\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89968\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-6-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>In this case, we see the line passes through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis halfway between <span style=\"font-style:normal; font-size:90%\">\\(0\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span>, at <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}\\)<\/span> or <span style=\"font-style:normal; font-size:90%\">\\((0, \\frac{1}{2})\\)<\/span>. If the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept is a fractional value, then it will pass through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at the fractional value it represents. <\/p>\n<p>The slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) is <span style=\"font-style:normal; font-size:90%\">\\(\\frac{2}{1}\\)<\/span>. From <span style=\"font-style:normal; font-size:90%\">\\((0,\\frac{1}{2})\\)<\/span>, move two units up (rise) and one unit over (run) to reach the next point, <span style=\"font-style:normal; font-size:90%\">\\((1,2\\frac{1}{2})\\)<\/span>. <\/p>\n<h3><span id=\"Example_5\" class=\"m-toc-anchor\"><\/span>Example #5<\/h3>\n<p>\nWe\u2019re going to take a look at one final example. This time, you are going to try it on your own. The next graph will combine everything we\u2019ve talked about so far. It\u2019s a little more challenging, but I know you can handle it. <\/p>\n<p>I\u2019m going to give you the equation. Once you see the equation, pause the video, draw a coordinate plane, and see if you can graph the equation yourself. When you\u2019re done, resume and we will go over the graph together. <\/p>\n<p>The equation I want you to graph is <span style=\"font-style:normal; font-size:90%\">\\(y=-\\frac{1}{4}x-3\\)<\/span>: <\/p>\n<p>Now that you\u2019re ready to check your work, let\u2019s take a look at the graph together. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7.png\" alt=\"Image of linear equation graphed as y= -1 over 4x-3\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89974\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Changing-Constants-in-Graphs-of-linear-functions.png-7-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>First, let\u2019s take a look at the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept (\\(b\\)). Recall that the value for <span style=\"font-style:normal; font-size:90%\">\\(b\\)<\/span> in our formula was <span style=\"font-style:normal; font-size:90%\">\\(-3\\)<\/span>. That means that the line passes through the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis at <span style=\"font-style:normal; font-size:90%\">\\(-3\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\((0, -3)\\)<\/span>. <\/p>\n<p>Now that we\u2019ve graphed our <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept point, let\u2019s consider the slope. The value for the slope (<span style=\"font-style:normal; font-size:90%\">\\(m\\)<\/span>) in the formula is <span style=\"font-style:normal; font-size:90%\">\\(-\\frac{1}{4}\\)<\/span>. So, from the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-intercept point, we need to move down <span style=\"font-style:normal; font-size:90%\">\\(1\\)<\/span> unit and right <span style=\"font-style:normal; font-size:90%\">\\(4\\)<\/span> units. This brings us to the next point on the graph, which is <span style=\"font-style:normal; font-size:90%\">\\((4, -4)\\)<\/span>. <\/p>\n<p>Understanding how constants work helps mathematicians recognize patterns in graphs of linear functions. I hope that this video about changing constants in graphs of linear functions was helpful. Thanks for watching, and happy studying! <\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Graphing_Linear_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Graphing Linear Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph below shows the linear function \\(y=3x+1\\). This equation is in the form \\(y=mx+b\\). What would happen to the line if \\(b\\) was changed to 8? <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-1.svg\" alt=\"A red line passes through points (0,1) and (\u22121,\u22122) on a Cartesian plane, extending beyond both points with arrows indicating direction.\" width=\"502.55\" height=\"577.3\" class=\"aligncenter size-full wp-image-287728\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">The line would intersect the \\(y\\)-axis at 8. <\/div><div class=\"PQ\"  id=\"PQ-1-2\">The line would intersect the \\(x\\)-axis at 8. <\/div><div class=\"PQ\"  id=\"PQ-1-3\">The line would have a slope of 8, increasing its steepness. <\/div><div class=\"PQ\"  id=\"PQ-1-4\">The line would have a slope of \u22128, changing its direction and increasing its steepness. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The variable \\(b\\) stands for the \\(y\\)-intercept in the slope-intercept form of the equation, \\(y=mx+b\\).<\/p>\n<p>If the \\(y\\)-intercept was changed from 1 to 8, then the resulting line would intersect the \\(y\\)-axis at 8.<\/p>\n<p>In the graph below, the original function (in red) shows the line intersecting the \\(y\\)-axis at 1. The new function (in blue) shows the line intersecting the \\(y\\)-axis at 8.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-1.2.svg\" alt=\"Graph with two lines: a blue line passing through points (0,8) and (-1,5), and a red line passing through (0,1) and (-1,-2), both lines shown on a labeled grid.\" width=\"436.25\" height=\"436.25\" class=\"aligncenter size-full wp-image-287731\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph below shows the linear function \\(y=\\frac{1}{2}x+3\\). This equation is in the form \\(y=mx+b\\). What would happen to the line if \\(m\\) was changed to \\(-\\frac{1}{2}\\)? <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-2.svg\" alt=\"A red line passes through points (0, 3) and (2, 4) on a Cartesian plane with labeled axes and gridlines.\" width=\"306.25\" height=\"285\" class=\"aligncenter size-full wp-image-287734\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">The line would have a slope of \\(-\\frac{1}{2}\\), changing its direction from negative to positive. <\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">The line would have a slope of \\(-\\frac{1}{2}\\), changing its direction from positive to negative. <\/div><div class=\"PQ\"  id=\"PQ-2-3\">The line would intersect the \\(y\\)-axis at \\(-\\frac{1}{2}\\). <\/div><div class=\"PQ\"  id=\"PQ-2-4\">The line would intersect the \\(x\\)-axis at \\(-\\frac{1}{2}\\). <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The variable \\(m\\) stands for the slope in the slope-intercept form of the equation, \\(y=mx+b\\). If the slope was changed from \\(\\frac{1}{2}\\) to \\(-\\frac{1}{2}\\), then the direction of the line would change from positive to negative.<\/p>\n<p>In the graph below, the original function (in red) shows a line moving in a positive direction. The new function (in blue) shows a line moving in a negative direction.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-2.2.svg\" alt=\"Two intersecting lines on a graph cross at (0, 3); one passes through (2, 4), and the other through (\u22122, 4). Points are marked at their intersections.\" width=\"602.6\" height=\"535.9\" class=\"aligncenter size-full wp-image-287737\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph below shows the linear function \\(y=2x-4\\). This equation is in the form \\(y=mx+b\\). What would happen to the line if m was changed to \\(\\frac{3}{4}\\)?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-3.svg\" alt=\"A red line passes through points (0, -4) and (1, -2) on a grid with labeled axes and gridlines.\" width=\"401.35\" height=\"596.85\" class=\"aligncenter size-full wp-image-287740\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">The line would have a slope of \\(\\frac{3}{4}\\), increasing its steepness.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">The line would have a slope of \\(\\frac{3}{4}\\), decreasing its steepness. <\/div><div class=\"PQ\"  id=\"PQ-3-3\">The line would intersect the \\(y\\)-axis at \\(\\frac{3}{4}\\). <\/div><div class=\"PQ\"  id=\"PQ-3-4\">The line would intersect the \\(x\\)-axis at \\(\\frac{3}{4}\\). <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The variable \\(m\\) stands for the slope in the slope-intercept form of the equation, \\(y=mx+b\\). If the slope was changed from 2 to \\(\\frac{3}{4}\\), then the line\u2019s slope would become less steep.<\/p>\n<p>In the graph below, the original function (in red) shows a line with a slope of 2. The new function (in blue) shows a line with a slope of \\(\\frac{3}{4}\\), which is less steep than the original line. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-3.2.svg\" alt=\"A graph shows two lines intersecting at point (1, -2); the red line passes through (0, -4), and the blue line passes through (4, -1).\" width=\"662.4\" height=\"596.85\" class=\"aligncenter size-full wp-image-287743\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMaria graphed the linear function \\(y=6x+2\\) onto the coordinate plane, as shown below. She wants to adjust her equation to make her line less steep. She also wants to move the \\(y\\)-intercept further down. Which equation should Maria use to reflect these changes? <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-4.svg\" alt=\"A red line passes through the points (-1, -4) and (0, 2) on a Cartesian plane with labeled axes and gridlines.\" width=\"257.25\" height=\"457.17\" class=\"aligncenter size-full wp-image-287749\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">\\(y=\\frac{1}{2}x-3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(y=\\frac{1}{2}x+3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-3\">\\(y=6x-3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(y=6x+3\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>In the slope-intercept equation \\(y=mx+b\\), \\(m\\) stands for the slope and \\(b\\) stands for the \\(y\\)-intercept. For the slope to be less steep than the original line, \\(m\\) must have a value that is less than 6. To move the \\(y\\)-intercept further down on the coordinate plane, \\(b\\) must be less than 2. The equation that satisfies both these requirements is \\(y=\\frac{1}{2}x-3\\).<\/p>\n<p>In the graph below, the original function, \\(y=6x+2\\), is shown in red, and the new function, \\(y=\\frac{1}{2}x-3\\), is shown in blue. The blue line has a less steep slope and a lower \\(y\\)-intercept than the red line. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Changing-Constants-Graph-Example-4.2.svg\" alt=\"A graph showing two intersecting lines, one red and one blue, with labeled points at (-1,-4), (0,-3), (2,-2), and (0,2) on a grid with x- and y-axes.\" width=\"360.15\" height=\"386.61\" class=\"aligncenter size-full wp-image-287746\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":187046,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-4316","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-how-graphs-change-when-the-constants-change-videos","7":"page_category-math-advertising-group","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"content_type-practice-questions","11":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4316","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=4316"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4316\/revisions"}],"predecessor-version":[{"id":247618,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/4316\/revisions\/247618"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/187046"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=4316"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}