{"id":13096,"date":"2014-02-07T16:35:48","date_gmt":"2014-02-07T16:35:48","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=13096"},"modified":"2026-03-25T10:52:29","modified_gmt":"2026-03-25T15:52:29","slug":"graphs-of-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/graphs-of-functions\/","title":{"rendered":"Graphs of Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_6rsjCBm5BQI\" 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_6rsjCBm5BQI\" data-source-videoID=\"6rsjCBm5BQI\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Graphs of Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Graphs of Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_6rsjCBm5BQI:hover {cursor:pointer;} img#videoThumbnailImage_6rsjCBm5BQI {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/691-how-to-graph-a-function-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_6rsjCBm5BQI\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_6rsjCBm5BQI\" 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\",\"Graphs of Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_6rsjCBm5BQI\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_6rsjCBm5BQI\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_6rsjCBm5BQI\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 4CF_Function() {\n  var x = document.getElementById(\"4CF\");\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=\"4CF_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=\"4CF\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Practice\" class=\"smooth-scroll\">Practice<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#The_yIntercept\" class=\"smooth-scroll\">The y-Intercept<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Special_Slope_Relationships\" class=\"smooth-scroll\">Special Slope Relationships<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_to_Graph_a_Function_Practice_Questions\" class=\"smooth-scroll\">How to Graph a 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>Hi, and welcome to this video on graphing functions! Today, I\u2019ll be showing you how to graph a function. Specifically, we\u2019ll focus on the concept of <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/finding-the-slope-of-a-line\/\">slope<\/a>, which determines how the function will take shape when we graph it. We will also discuss the relationships between the slopes of perpendicular and parallel lines. Let\u2019s get started!<\/p>\n<p>First, a quick reminder of linear functions. As you can probably guess by the name, a linear function graphs as a straight line on the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/cartesian-coordinate-plane-and-graphing\/\">Cartesian plane<\/a>. The line represents every point on the plane that satisfies the linear equation.<\/p>\n<p>Each point can be expressed as an ordered pair \\((x,y)\\), which represents the units on the \\(x\\)-axis and the units on the \\(y\\)-axis. The slope of the line is determined by measuring the vertical distance between the y-values of any two points on the line and dividing by the horizontal distance, as determined by the \\(x\\)-values of those points.<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s take a minute to apply this information with a few examples.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/graph-function-scaled.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\" role=\"img\" style=\"box-shadow: 1.5px 1.5px 3px gray;\" \/><\/p>\n<p>Determine the slope of the line that passes through the points \\((3,7)\\) and \\((6,3)\\).<\/p>\n<p>The vertical distance, sometimes referred to as the \u201cchange in \\(y\\)\u201d or \u201crise\u201d, is calculated by subtracting the \\(y\\)-values of these two points: \\(7-3=4\\).<\/p>\n<p>The horizontal distance is known as the \u201cchange in \\(x\\)\u201d or \u201crun\u201d, and is calculated by subtracting the \\(x\\)-values: \\(3-6= -3\\).<\/p>\n<p>Note that the subtraction must be consistent! We subtracted both of the coordinates of the point G from the coordinates of point F.<\/p>\n<p>The slope of this line is, therefore, \\(\\frac{change\\; in\\; y}{change \\;in\\; x}\\), or \\(\\frac{-4}{3}\\). It is important to note that a <strong>negative slope<\/strong> indicates that the line slopes <strong>downward<\/strong>, from left to right.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s look at another example.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/graph-function-2-scaled.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\" role=\"img\" style=\"box-shadow: 1.5px 1.5px 3px gray;\" \/><\/p>\n<p>Calculate the slope of the line that passes through the points \\((1,1)\\) and \\((4,3)\\).<\/p>\n<ol style=\"list-style-type: none;\">\n<li><strong>Step 1:<\/strong> Calculate the change in \\(y\\).<\/li>\n<\/ol>\n<div class=\"examplesentence\">\\(3-1=2\\)<\/div>\n<p>\n&nbsp;<\/p>\n<ol style=\"list-style-type: none;\">\n<li><strong>Step 2:<\/strong> Calculate the change in \\(x\\).<\/li>\n<\/ol>\n<div class=\"examplesentence\">\\(4-1=3\\)<\/div>\n<p>\n&nbsp;<\/p>\n<ol style=\"list-style-type: none;\">\n<li><strong>Step 3:<\/strong> Calculate slope by dividing the change in \\(y\\) by the change in \\(x\\).<\/li>\n<\/ol>\n<div class=\"examplesentence\">Slope=\\(\\frac{2}{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe line with a positive slope will slant upward, from left to right.<\/p>\n<p>The ability to analyze a graph and knowing the attributes of slope are important skills. <\/p>\n<p>There are two special types of lines that do not follow the rules noted above with regard to slanting up or down, according to the sign of their slope. <\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nHere\u2019s another example:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/graph-function-3-scaled.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\" role=\"img\" \/><\/p>\n<p>Calculate the slope of the line that travels through the points \\((5,3)\\) and \\((-2,3)\\). <\/p>\n<p>This time, when we determine the vertical change, or the change in \\(y\\), we get an answer of \\(3-3=0\\).<\/p>\n<p>Intuitively, this makes sense because there is no change in the \\(y\\)-coordinates from one point to the other. There is a change in \\(x\\); however, dividing 0 by any value results in 0.<\/p>\n<p>The slope of this line is, therefore, \\(\\frac{0}{-2-5}=\\frac{0}{-7}=0\\).<\/p>\n<p>This is true for any <em>horizontal line<\/em> that exists on the coordinate plane. These equations are written in the form \\(y=c\\), where \\(c\\) is any value on the \\(y\\)-axis. Accordingly, the line shown at right is \\(y=3\\).<\/p>\n<p>Some people find it helpful to visualize a horizontal line with a slope that does not \u201crise\u201d but \u201cruns\u201d forever.<\/p>\n<p>With similar reasoning, let\u2019s explore the nature of slope for points that lie on a vertical line. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/graph-function-3-scaled.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\" role=\"img\" style=\"box-shadow: 1.5px 1.5px 3px gray;\" \/><\/p>\n<p>Determine the slope for the line that passes through the points \\((2,5)\\) and \\((2,-1)\\).<\/p>\n<p>Just by looking at these points, we see that there is no change in \\(x\\)-values, so the denominator of the slope calculation will be 0. <\/p>\n<p>Slope for this line is be calculated as follows: \\(\\frac{-1-5}{2-2}=-60\\).<\/p>\n<p>You may remember that dividing by zero is \u201cundefined.\u201d Vertical lines are written in the form of \\(x=c\\), where \\(c\\) is any value on the \\(x\\)-axis. This line is written as \\(x=2\\).<\/p>\n<p>Note that there is no slope for vertical lines, as they do not \u201cslant\u201d in either direction. These lines \u201crise\u201d forever, but do not \u201crun.\u201d<\/p>\n<h2><span id=\"Practice\" class=\"m-toc-anchor\"><\/span>Practice<\/h2>\n<p>\nFor the following graphs, indicate whether the slope of the line is positive, negative, zero, or undefined.<\/p>\n<h3><span id=\"Example_1_1\" class=\"m-toc-anchor\"><\/span>Example 1<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-1-coordinate-plane.webp\" alt=\"A graph with a blue background shows a grid with labeled axes from -5 to 5. A red diagonal arrow points from bottom left to top right.\" width=\"1007\" height=\"679\" class=\"aligncenter size-full wp-image-251990\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-1-coordinate-plane.webp 1007w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-1-coordinate-plane-300x202.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-1-coordinate-plane-768x518.webp 768w\" sizes=\"auto, (max-width: 1007px) 100vw, 1007px\" \/><\/p>\n<p>Answer: Positive slope. The line slopes upward from left to right.<\/p>\n<h3><span id=\"Example_2_1\" class=\"m-toc-anchor\"><\/span>Example 2<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-2-coordinate-plane.webp\" alt=\"A grid on a blue background with white axes marked -5 to 5 horizontally and vertically. A vertical red arrow spans from -5 to 5.\" width=\"1016\" height=\"687\" class=\"aligncenter size-full wp-image-251999\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-2-coordinate-plane.webp 1016w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-2-coordinate-plane-300x203.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-2-coordinate-plane-768x519.webp 768w\" sizes=\"auto, (max-width: 1016px) 100vw, 1016px\" \/><\/p>\n<p>Answer: Slope is \u201cundefined.\u201d All vertical lines have undefined slope.<\/p>\n<h3><span id=\"Example_3_1\" class=\"m-toc-anchor\"><\/span>Example 3<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-3-coordinate-plane.webp\" alt=\"Graph with a blue grid, featuring a horizontal red line along the \\(x\\)-axis from -5 to 5 and white arrows on both axes indicating positive and negative directions.\" width=\"1002\" height=\"677\" class=\"aligncenter size-full wp-image-251993\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-3-coordinate-plane.webp 1002w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-3-coordinate-plane-300x203.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-3-coordinate-plane-768x519.webp 768w\" sizes=\"auto, (max-width: 1002px) 100vw, 1002px\" \/><\/p>\n<p>Answer: Slope is equal to 0. All horizontal lines have a slope equal to 0.<\/p>\n<h3><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example 4<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-4-coordinate-plane.webp\" alt=\"A graph on a blue grid depicts a red diagonal line from top left to bottom right, intersecting the x and y axes symmetrically at 0.5 and -0.5 intervals.\" width=\"1020\" height=\"682\" class=\"aligncenter size-full wp-image-251996\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-4-coordinate-plane.webp 1020w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-4-coordinate-plane-300x201.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-4-coordinate-plane-768x514.webp 768w\" sizes=\"auto, (max-width: 1020px) 100vw, 1020px\" \/><\/p>\n<p>Answer: Slope is negative. The line slopes downward from left to right.<\/p>\n<h2><span id=\"The_yIntercept\" class=\"m-toc-anchor\"><\/span>The y-Intercept<\/h2>\n<p>\nWhen graphing a linear function, the slope is extremely important. If we have one point and our slope, we can find every other point on the line and see what the line looks like. We just learned that in our slope-intercept form of our linear equation \\(y=mx+b\\), the letter \\(m\\) represents our slope.<\/p>\n<p>Remember, the letters \\(x\\) and \\(y\\) represent the \\(x\\)-coordinate and \\(y\\)-coordinate of any point that satisfies our equation. Now the only thing left we have to talk about in our equation is the letter \\(b\\).<\/p>\n<p>The letter \\(b\\) in our slope-intercept form represents the \\(y\\)-intercept of the equation. The \\(y\\)-intercept is the point where the graph hits the \\(y\\)-axis, or where \\(x=0\\). So the \\(y\\)-intercept gives us the point that we need to figure out our graph. The point will be \\((0,b)\\) for whatever number is in the \\(b\\) position.<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s try an example.<\/p>\n<p>What is the slope and \\(y\\)-intercept of the equation \\(y = x + 7\\)?<\/p>\n<p>Our slope is our \\(m\\)-value, so for this equation, slope is equal to \\(\\frac{1}{2}\\).<\/p>\n<p>We determine our \\(y\\)-intercept by looking at our \\(b\\)-value, which in this case is 7. Remember, this number gives us our \\(y\\)-coordinate for our point where \\(x = 0\\). So the \\(y\\)-intercept of this equation is \\((0,7)\\).<\/p>\n<p>Now that we know how to identify our slope and \\(y\\)-intercept, let\u2019s look at how we can use these two things to graph our equation.<\/p>\n<ol>\n<li style=\"margin-bottom: 1em;\">Our first step is to identify the \\(y\\)-intercept. For our equation above, we already did this. Our \\(y\\)-intercept is \\((0,7)\\).<\/li>\n<li style=\"margin-bottom: 1em;\">The second step is to plot that point on our graph. So we are going to place a point at \\((0,7)\\).<\/li>\n<li style=\"margin-bottom: 1em;\">Next, we want to look at our slope, which we found earlier to be \\(\\frac{1}{2}\\). Remember, the slope is the ratio of our change in \\(y\\) over our change in \\(x\\), or our rise over run.<\/li>\n<li style=\"margin-bottom: 1em;\">Fourth, we are going to use our slope to find another point on the line. Start at the \\(y\\)-intercept and move up (or rise) one and move right (or run) two, and place another point here.<\/li>\n<li>Since we have two points, we can make a straight line. So our final step is to draw a straight line between the two points, making sure to put arrowheads on either end of the line to indicate that it continues forever in either direction.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane.webp\" alt=\"A red diagonal arrowed line with various marked points on a blue grid, with x and y axes labeled with intervals of 5. The point (0, 7) is marked.\" width=\"1551\" height=\"993\" class=\"aligncenter size-full wp-image-252005\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane.webp 1551w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane-300x192.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane-1024x656.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane-768x492.webp 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/04\/Example-5-coordinate-plane-1536x983.webp 1536w\" sizes=\"auto, (max-width: 1551px) 100vw, 1551px\" \/><\/p>\n<p>It\u2019s as simple as that!<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s try another example following the steps we used before.<\/p>\n<p>Graph the equation \\(y = 2x -4\\).<\/p>\n<ol>\n<li style=\"margin-bottom: 1em;\">Identify the \\(y\\)-intercept. Our \\(b\\)-value in this equation is negative, so our \\(y\\)-intercept is also going to be \\((0, -4)\\).<\/li>\n<li style=\"margin-bottom: 1em;\">Plot the \\(y\\)-intercept on the graph.<\/li>\n<li style=\"margin-bottom: 1em;\">Identify the slope. Remember, the slope is our \\(m\\)-value so in this case it is 2.<\/li>\n<li style=\"margin-bottom: 1em;\">Use the slope to plot a second point. When we are given a whole number, we can turn it into a fraction by placing it over 1. This doesn\u2019t change the value of the number because any number divided by 1 is itself. So our slope now looks like \\(\\frac{2}{1}\\). This means that from our \\(y\\)-intercept we are going to rise 2 values and run 1.<\/li>\n<li>Draw a line between the two points.<\/li>\n<\/ol>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nI want you to try one more. This time, try to do it without my help.<\/p>\n<p>Graph the line \\(y = \\frac{-2}{3x} +1\\).<\/p>\n<p>Think you\u2019ve got it? Let\u2019s walk through it together.<\/p>\n<p>Our \\(y\\)-intercept is at the point \\((0,1)\\) so we are going to plot that first.<\/p>\n<p>Now we look at our slope. This tells us that we are going to go over 3 places and down 2. We go down instead of up because our slope is negative.<\/p>\n<p>Now that we have our two points, we draw our line through them and add our arrowheads.<\/p>\n<p>Before we go, I want to look at few more things about slope that are helpful to know. <\/p>\n<h2><span id=\"Special_Slope_Relationships\" class=\"m-toc-anchor\"><\/span>Special Slope Relationships<\/h2>\n<p>\nThere are two more very important types of lines that have special slope relationships. Consider the following graphs:<\/p>\n<p>These lines have the special relationship of being equidistant, or parallel. That means that they are the same distance apart for each value on both the positive and negative sides of the \\(x\\)-axis. As we have discussed, the slope determines the \u201cslant\u201d of the line and, clearly, equidistant lines have the same slant.<\/p>\n<p>In other words, parallel lines have the same slope. <\/p>\n<p>The equations in slope-intercept form for these lines are \\(y=\\frac{2}{3}x-1\\) and \\(y=\\frac{2}{3}x+2\\).<\/p>\n<p>The slope is clearly identifiable in both equations. If given two equations, the lines are parallel if the slopes are the same.<\/p>\n<p>This graph shows two lines that have the special feature of intersecting at a right angle. As a result, these lines are considered perpendicular. <\/p>\n<p>The equations for these lines are \\(y=3x-1\\) and \\(y= \\frac{-1}{3}x-2\\).<\/p>\n<p>Again, there seems to be a relationship between the slopes, \\(m=3\\) and \\(m= \\frac{-1}{3}\\). Any guesses?<\/p>\n<p>Well, there are actually two distinct differences between these slope values. <\/p>\n<p>The signs are opposite: the slope of one line is positive and the slope of the other is negative. The slope values are reciprocals, meaning that the numerator and denominator are \u201cflipped.\u201d<\/p>\n<p>If these two changes are recognized in the slopes of two linear equations, then the lines that they represent are perpendicular.<\/p>\n<p>Let\u2019s apply this information with some practice. See if you can answer these on your own:<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nWhat is the slope of a line that is parallel to \\(y=\\frac{5}{7}x-2\\)?\t<\/p>\n<p>Answer: The slope of this parallel line is equal to \\(\\frac{5}{7}\\). Remember, parallel lines have equal slope.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nWhat is the slope of a line that is perpendicular to \\(y=\\frac{2}{5}x+1\\)\t\t<\/p>\n<p>Answer: The slope of this perpendicular line is equal to \\(\\frac{-5}{2}\\). Remember, perpendicular lines have slopes that are opposite reciprocals.<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h3>\n<p>\nState whether the lines are parallel, perpendicular, or neither:<\/p>\n<p>\\(y=3x+4\\)   and   \\(y= \\frac{-1}{3}x-8\\)\t\t<\/p>\n<p>Answer: Perpendicular. The slopes are opposite reciprocals.<\/p>\n<h3><span id=\"Example_4_1\" class=\"m-toc-anchor\"><\/span>Example #4<\/h3>\n<p>\\(y= -x+5\\)   and   \\(y=\\frac{1}{2}x+2\\) <\/p>\n<p>Answer: Neither. <\/p>\n<p>Slopes are not the same and slopes are not opposite reciprocals. These lines intersect but NOT at a right angle.<\/p>\n<h3><span id=\"Example_5\" class=\"m-toc-anchor\"><\/span>Example #5<\/h3>\n<p>\nAnd one more:<\/p>\n<p>\\(y=x+2\\)    and   \\(y=x-5\\) <\/p>\n<p>Answer: Parallel. Slopes are both equal to 1.<\/p>\n<p>I hope I&#8217;ve cleared up any confusion that you may have had with regard to slope and graphing linear equations. As you can see, slope is the driving force in graphing and holds the key to identifying special relationships between lines.<\/p>\n<p>That\u2019s all for this review! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"How_to_Graph_a_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>How to Graph a 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 \/>\nWhich line is parallel to the line that passes through the points \\((3,-1)\\) and \\((-2,4)\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\(y=-x-5\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(y=x+5\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(y=-5x-1\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(y=5x+1\\)<\/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>Since parallel lines have the same slope, the first step is to find the slope of the line that goes through the points \\((3,-1)\\) and \\((-2,4)\\).<\/p>\n<p>The slope formula is:<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{\\text{Change in } y}{\\text{Change in } x}\\)<\/p>\n<p>Change in \\(y\\) is \\(4-(-1)=5\\), and the change in \\(x\\) is \\(-2-3=-5\\).<\/p>\n<p>Therefore, the slope of the line that passes through the points is \\(\\frac{5}{-5}=-1\\) and the slope of the line \\(y=-x\u20135\\) is also \u22121.<\/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 \/>\nWhich is true about the slope of the line that goes through the points \\((-2,3)\\) and \\((4,3)\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">Positive slope<\/div><div class=\"PQ\"  id=\"PQ-2-2\">Negative slope<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">Zero slope<\/div><div class=\"PQ\"  id=\"PQ-2-4\">Undefined slope<\/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 graph of a horizontal line has a zero slope.<\/p>\n<p>There are two ways to find the solution to this problem. The first way is to graph the line, which produces a horizontal line, which will conclude that the slope is zero. The second way is to use the slope formula, which is:<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{\\text{Change in } y}{\\text{Change in } x}\\)<\/p>\n<p>The change in \\(y\\) is \\(3-3=0\\) and the change in \\(x\\) is \\(4-(-2)=4+2=6\\).<\/p>\n<p>A fraction with a zero in the numerator is equal to zero, the slope of the line.<\/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 \/>\nWhat is the slope and \\(y\\)-intercept of a line represented by the equation \\(y=3x\u20139\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">Slope is 3, \\(y\\)-intercept is 9<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">Slope is 3, \\(y\\)-intercept is \u22129<\/div><div class=\"PQ\"  id=\"PQ-3-3\">Slope is 9, \\(y\\)-intercept is 3<\/div><div class=\"PQ\"  id=\"PQ-3-4\">Slope is \u22129, \\(y\\)-intercept is 3<\/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 equation of the line is written in slope-intercept form, which is \\(y=mx+b\\), where \\(m\\) is the slope and \\(b\\) is the \\(y\\)-intercept.<\/p>\n<p>Therefore, the slope of the line is 3 and the \\(y\\)-intercept is \u22129.<\/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 \/>\nWhich equation represents the graph of the line below?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-70757 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/Linear-line-passing-through-10-and-0-2.png\" alt=\"Linear line passing through (-1,0) and (0,-2)\" width=\"359\" height=\"356\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/Linear-line-passing-through-10-and-0-2.png 962w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/Linear-line-passing-through-10-and-0-2-300x297.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/Linear-line-passing-through-10-and-0-2-150x150.png 150w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/Linear-line-passing-through-10-and-0-2-768x761.png 768w\" sizes=\"auto, (max-width: 359px) 100vw, 359px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">\\(y=-2x-2\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(y=-2x+2\\)<\/div><div class=\"PQ\"  id=\"PQ-4-3\">\\(y=-\\frac{1}{2}x-2\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(y=-\\frac{1}{2}x+2\\)<\/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>To find the slope of the line, we can use two points on the line and apply the slope formula, or we can count the rise and run on the graph between two points as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-70754 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/graph-y-2x-2-with-slope-displayed-.png\" alt=\"graph y=-2x-2 with slope displayed\" width=\"360\" height=\"357\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/graph-y-2x-2-with-slope-displayed-.png 962w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/graph-y-2x-2-with-slope-displayed--300x297.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/graph-y-2x-2-with-slope-displayed--150x150.png 150w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/graph-y-2x-2-with-slope-displayed--768x761.png 768w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/> <\/p>\n<p>The line goes down 2 and right 1, meaning it has a slope of \u22122. To find the \\(y\\)-intercept of the line, we can either look at where the graph intersects the \\(y\\)-axis or substitute a point from the line into the slope-intercept form of the equation with the slope and solve for \\(b\\).<\/p>\n<p>The line hits the \\(y\\)-axis at \\(y=-2\\), so the \\(y\\)-intercept is \u22122.<\/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>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe slope of a line is \\(\\frac{3}{5}\\) and passes through the point \\((0,-4)\\). What is the equation of the line?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\(y=-\\frac{3}{5}x+4\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(y=-\\frac{3}{5}x-4\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(y=\\frac{3}{5}x+4\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">\\(y=\\frac{3}{5}x-4\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The equation of the line when written in slope-intercept form, \\(y=mx+b\\), shows the slope, \\(m\\), and \\(y\\)-intercept, \\(b\\).<\/p>\n<p>The problem states that the slope is \\(\\frac{3}{5}\\), and if we recall, a point where the \\(x\\)-value is 0 is on the \\(y\\)-axis. Therefore, if the line goes through the point \\((0,-4)\\), it must be the \\(y\\)-intercept of the line.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-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":99709,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-13096","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-functions-and-their-graphs-videos","7":"page_category-graph-videos","8":"page_category-math-advertising-group","9":"page_category-video-pages-for-study-course-sidebar-ad","10":"page_type-video","11":"content_type-practice-questions","12":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Slope is a major puzzle piece to completing your knowledge of graphing a function. 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