{"id":1016,"date":"2024-05-01T22:09:33","date_gmt":"2024-05-01T22:09:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1016"},"modified":"2025-09-05T15:36:37","modified_gmt":"2025-09-05T15:36:37","slug":"graphing-and-analyzing-linear-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphing-and-analyzing-linear-equations-fresh-take\/","title":{"raw":"Graphing and Analyzing Linear Equations: Fresh Take","rendered":"Graphing and Analyzing Linear Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Draw the graph of linear equations by plotting points.<\/li>\r\n \t<li>Determine the slope based on the steepness and direction of a line.<\/li>\r\n \t<li>Find the x-intercept and y-intercept of linear equations and graphs.<\/li>\r\n \t<li>Use formulas to calculate the distances and midpoints between points.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Plotting Points<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Origin and Axes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Two perpendicular lines: [latex]x[\/latex]-axis (horizontal) and y-axis (vertical)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Intersection point of axes: origin, denoted as [latex]0,0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadrants:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Plane divided into four quadrants, numbered counterclockwise\r\n\r\n[caption id=\"attachment_3322\" align=\"alignnone\" width=\"418\"]<img class=\"wp-image-3322 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"418\" height=\"370\" \/> x,y plane with quadrants labeled[\/caption]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ordered Pairs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Points represented as[latex](x,y)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinate: horizontal displacement from origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinate: vertical displacement from origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plotting Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Move horizontally by x units, then vertically by y units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Positive [latex]x[\/latex]: move right; Negative [latex]x[\/latex]: move left<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Positive [latex]y[\/latex]: move up; Negative [latex]y[\/latex]: move down<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special Cases:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Points on x-axis: y-coordinate is zero, <a href=\"a,0\">latex<\/a>[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Points on y-axis: x-coordinate is zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Plot the following points on the same coordinate plane:\r\n<p style=\"text-align: center;\">[latex]A(2,3), B(-4,1), C(0,-2), D(-3,-3)[\/latex]<\/p>\r\n[reveal-answer q=\"95164\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"95164\"]\r\n\r\n[caption id=\"attachment_8014\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-8014\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1.png\" alt=\"The coordinate plane displays four distinct points: point A in purple at position (3, 3), point B also in purple at (-4, 1), point C in green at (0, -2), and point D in blue at (-2, 3). The graph shows a standard grid with both x and y axes ranging from -6 to 6, with gridlines marked at each unit interval.\" width=\"500\" height=\"502\" \/> Coordinate plane with points A, B, C, and D plotted[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbabhaba-7JMXi_FxA2o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/7JMXi_FxA2o?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbabhaba-7JMXi_FxA2o\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844214&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbabhaba-7JMXi_FxA2o&vembed=0&video_id=7JMXi_FxA2o&video_target=tpm-plugin-hbabhaba-7JMXi_FxA2o'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Plotting+Points+on+the+Coordinate+Plane_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Plotting Points on the Coordinate Plane\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Graphing Equations by Plotting Points<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Equations in Two Variables:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Contain both x and y variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represented as graphs in a two-dimensional plane<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plotting Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Create a table with x, y, and (x,y) columns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose strategic x-values for easy calculation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate corresponding y-values<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the resulting ordered pairs (x,y)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Connecting Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If points form a straight line, connect them<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not all equations result in straight lines<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n[reveal-answer q=\"811886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811886\"]\r\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/> Graph with a line through the given points[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hdegcbcb-mYYGQrRQeUw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mYYGQrRQeUw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hdegcbcb-mYYGQrRQeUw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12779081&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hdegcbcb-mYYGQrRQeUw&vembed=0&video_id=mYYGQrRQeUw&video_target=tpm-plugin-hdegcbcb-mYYGQrRQeUw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+Equations+by+Plotting+Points+-+Example+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Equations by Plotting Points - Example 1\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Intercepts Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Points where a graph crosses the axes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">x-intercept: crosses x-axis (y = 0)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">y-intercept: crosses y-axis (x = 0)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding x-intercept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set y = 0 in the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for x<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Express as ordered pair (x, 0)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding y-intercept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set x = 0 in the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for y<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Express as ordered pair (0, y)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphing Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Calculate both intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the intercept points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw a line through these points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Efficiency:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only two points needed to define a line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quicker than plotting multiple points<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\">Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/> Graph of y = -3\/4x +3[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agdffddh-6m642-2D3V4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6m642-2D3V4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-agdffddh-6m642-2D3V4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844215&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-agdffddh-6m642-2D3V4&vembed=0&video_id=6m642-2D3V4&video_target=tpm-plugin-agdffddh-6m642-2D3V4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+using+x-+and+y-intercepts+%7C+Graphing+lines+and+slope+%7C+Algebra+Basics+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing using x- and y-intercepts | Graphing lines and slope | Algebra Basics | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2><strong>Slope of a Linear Equation<\/strong><\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Slope:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Ratio of vertical change (rise) to horizontal change (run)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Measures steepness and direction of a line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpretation of Slope:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Positive slope: Line rises from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative slope: Line falls from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero slope: Horizontal line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Undefined slope: Vertical line<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope in Different Contexts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">In economics: Rate of change (e.g., marginal cost)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">In physics: Velocity in distance-time graphs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">In statistics: Correlation between variables<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculating Slope:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose any two points on the line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the slope formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the fraction if possible<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].[reveal-answer q=\"196055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"196055\"]slope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex][\/hidden-answer]<\/section>\r\n<h2><\/h2>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chcbdghh-wvzBH46D6ho\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wvzBH46D6ho?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-chcbdghh-wvzBH46D6ho\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12779082&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-chcbdghh-wvzBH46D6ho&vembed=0&video_id=wvzBH46D6ho&video_target=tpm-plugin-chcbdghh-wvzBH46D6ho'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+find+the+slope+between+two+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to find the slope between two points\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Distance Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Origin and Concept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Derived from the Pythagorean Theorem<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculates the straight-line distance between two points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: [latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex] Where [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] are the coordinates of two points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Forms a right triangle with the two points and their projections<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Hypotenuse of this triangle is the distance between points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Navigation and GPS systems<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Computer graphics and game development<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Spatial analysis in geography<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].\r\n[reveal-answer q=\"934526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934526\"][latex]\\sqrt{125}=5\\sqrt{5}[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\">In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gahacfcb-Vj7twkiUgf0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vj7twkiUgf0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gahacfcb-Vj7twkiUgf0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844216&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gahacfcb-Vj7twkiUgf0&vembed=0&video_id=Vj7twkiUgf0&video_target=tpm-plugin-gahacfcb-Vj7twkiUgf0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Example+-++Determine+the+Distance+Between+Two+Points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine the Distance Between Two Points\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Midpoint Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The midpoint is the point that divides a line segment into two equal parts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">It's located exactly halfway between the endpoints<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: For endpoints [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex], the midpoint [latex]M[\/latex] is: [latex]M = \\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The x-coordinate of the midpoint is the average of the x-coordinates of the endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The y-coordinate of the midpoint is the average of the y-coordinates of the endpoints<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Finding centers of objects in computer graphics<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculating average positions in physics<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determining midpoints of ranges in statistics<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Find the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].[reveal-answer q=\"788934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"788934\"]Use the formula to find the midpoint of the line segment.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&amp;=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&amp;=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].[reveal-answer q=\"964077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964077\"][latex]\\left(-5,\\frac{5}{2}\\right)[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgcfdcbb-X_-iUuEiCqk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X_-iUuEiCqk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cgcfdcbb-X_-iUuEiCqk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12779083&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cgcfdcbb-X_-iUuEiCqk&vembed=0&video_id=X_-iUuEiCqk&video_target=tpm-plugin-cgcfdcbb-X_-iUuEiCqk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Given+two+Endpoints+Find+the+Midpoint+of+a+Segment_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGiven two Endpoints Find the Midpoint of a Segment\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Draw the graph of linear equations by plotting points.<\/li>\n<li>Determine the slope based on the steepness and direction of a line.<\/li>\n<li>Find the x-intercept and y-intercept of linear equations and graphs.<\/li>\n<li>Use formulas to calculate the distances and midpoints between points.<\/li>\n<\/ul>\n<\/section>\n<h2>Plotting Points<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Origin and Axes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Two perpendicular lines: [latex]x[\/latex]-axis (horizontal) and y-axis (vertical)<\/li>\n<li class=\"whitespace-normal break-words\">Intersection point of axes: origin, denoted as [latex]0,0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Quadrants:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Plane divided into four quadrants, numbered counterclockwise<br \/>\n<figure id=\"attachment_3322\" aria-describedby=\"caption-attachment-3322\" style=\"width: 418px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3322 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"418\" height=\"370\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83.jpg 418w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-300x266.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-65x58.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-225x199.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/03140115\/b60cd925724939299f52716cbf50b67c9a8ebf83-350x310.jpg 350w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><figcaption id=\"caption-attachment-3322\" class=\"wp-caption-text\">x,y plane with quadrants labeled<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Ordered Pairs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points represented as[latex](x,y)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinate: horizontal displacement from origin<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinate: vertical displacement from origin<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Plotting Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Move horizontally by x units, then vertically by y units<\/li>\n<li class=\"whitespace-normal break-words\">Positive [latex]x[\/latex]: move right; Negative [latex]x[\/latex]: move left<\/li>\n<li class=\"whitespace-normal break-words\">Positive [latex]y[\/latex]: move up; Negative [latex]y[\/latex]: move down<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Special Cases:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points on x-axis: y-coordinate is zero, <a href=\"a,0\">latex<\/a>[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Points on y-axis: x-coordinate is zero<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Plot the following points on the same coordinate plane:<\/p>\n<p style=\"text-align: center;\">[latex]A(2,3), B(-4,1), C(0,-2), D(-3,-3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q95164\">Show Answer<\/button><\/p>\n<div id=\"q95164\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_8014\" aria-describedby=\"caption-attachment-8014\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-8014\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1.png\" alt=\"The coordinate plane displays four distinct points: point A in purple at position (3, 3), point B also in purple at (-4, 1), point C in green at (0, -2), and point D in blue at (-2, 3). The graph shows a standard grid with both x and y axes ranging from -6 to 6, with gridlines marked at each unit interval.\" width=\"500\" height=\"502\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1.png 689w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1-225x226.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/05153616\/large-Graphing1-2-1-350x352.png 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-8014\" class=\"wp-caption-text\">Coordinate plane with points A, B, C, and D plotted<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbabhaba-7JMXi_FxA2o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/7JMXi_FxA2o?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbabhaba-7JMXi_FxA2o\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844214&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hbabhaba-7JMXi_FxA2o&#38;vembed=0&#38;video_id=7JMXi_FxA2o&#38;video_target=tpm-plugin-hbabhaba-7JMXi_FxA2o\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Plotting+Points+on+the+Coordinate+Plane_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Plotting Points on the Coordinate Plane\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Graphing Equations by Plotting Points<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Equations in Two Variables:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Contain both x and y variables<\/li>\n<li class=\"whitespace-normal break-words\">Represented as graphs in a two-dimensional plane<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Plotting Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Create a table with x, y, and (x,y) columns<\/li>\n<li class=\"whitespace-normal break-words\">Choose strategic x-values for easy calculation<\/li>\n<li class=\"whitespace-normal break-words\">Calculate corresponding y-values<\/li>\n<li class=\"whitespace-normal break-words\">Plot the resulting ordered pairs (x,y)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Connecting Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If points form a straight line, connect them<\/li>\n<li class=\"whitespace-normal break-words\">Not all equations result in straight lines<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q811886\">Show Solution<\/button><\/p>\n<div id=\"q811886\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\">Graph with a line through the given points<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hdegcbcb-mYYGQrRQeUw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mYYGQrRQeUw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hdegcbcb-mYYGQrRQeUw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12779081&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hdegcbcb-mYYGQrRQeUw&#38;vembed=0&#38;video_id=mYYGQrRQeUw&#38;video_target=tpm-plugin-hdegcbcb-mYYGQrRQeUw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+Equations+by+Plotting+Points+-+Example+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Equations by Plotting Points &#8211; Example 1\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Intercepts Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points where a graph crosses the axes<\/li>\n<li class=\"whitespace-normal break-words\">x-intercept: crosses x-axis (y = 0)<\/li>\n<li class=\"whitespace-normal break-words\">y-intercept: crosses y-axis (x = 0)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding x-intercept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set y = 0 in the equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve for x<\/li>\n<li class=\"whitespace-normal break-words\">Express as ordered pair (x, 0)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding y-intercept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set x = 0 in the equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve for y<\/li>\n<li class=\"whitespace-normal break-words\">Express as ordered pair (0, y)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphing Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculate both intercepts<\/li>\n<li class=\"whitespace-normal break-words\">Plot the intercept points<\/li>\n<li class=\"whitespace-normal break-words\">Draw a line through these points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Efficiency:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only two points needed to define a line<\/li>\n<li class=\"whitespace-normal break-words\">Quicker than plotting multiple points<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\">Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q80464\">Show Solution<\/button><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\"><em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><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\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/><figcaption class=\"wp-caption-text\">Graph of y = -3\/4x +3<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agdffddh-6m642-2D3V4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6m642-2D3V4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-agdffddh-6m642-2D3V4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844215&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-agdffddh-6m642-2D3V4&#38;vembed=0&#38;video_id=6m642-2D3V4&#38;video_target=tpm-plugin-agdffddh-6m642-2D3V4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+using+x-+and+y-intercepts+%7C+Graphing+lines+and+slope+%7C+Algebra+Basics+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing using x- and y-intercepts | Graphing lines and slope | Algebra Basics | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2><strong>Slope of a Linear Equation<\/strong><\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Slope:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ratio of vertical change (rise) to horizontal change (run)<\/li>\n<li class=\"whitespace-normal break-words\">Measures steepness and direction of a line<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation of Slope:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Positive slope: Line rises from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Negative slope: Line falls from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Zero slope: Horizontal line<\/li>\n<li class=\"whitespace-normal break-words\">Undefined slope: Vertical line<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Slope in Different Contexts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">In economics: Rate of change (e.g., marginal cost)<\/li>\n<li class=\"whitespace-normal break-words\">In physics: Velocity in distance-time graphs<\/li>\n<li class=\"whitespace-normal break-words\">In statistics: Correlation between variables<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Calculating Slope:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose any two points on the line<\/li>\n<li class=\"whitespace-normal break-words\">Apply the slope formula<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the fraction if possible<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q196055\">Show Solution<\/button><\/p>\n<div id=\"q196055\" class=\"hidden-answer\" style=\"display: none\">slope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2><\/h2>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chcbdghh-wvzBH46D6ho\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/wvzBH46D6ho?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-chcbdghh-wvzBH46D6ho\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12779082&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-chcbdghh-wvzBH46D6ho&#38;vembed=0&#38;video_id=wvzBH46D6ho&#38;video_target=tpm-plugin-chcbdghh-wvzBH46D6ho\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+find+the+slope+between+two+points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to find the slope between two points\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Distance Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Origin and Concept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Derived from the Pythagorean Theorem<\/li>\n<li class=\"whitespace-normal break-words\">Calculates the straight-line distance between two points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex] Where [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] are the coordinates of two points<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Forms a right triangle with the two points and their projections<\/li>\n<li class=\"whitespace-normal break-words\">Hypotenuse of this triangle is the distance between points<\/li>\n<\/ul>\n<\/li>\n<li>Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Navigation and GPS systems<\/li>\n<li class=\"whitespace-normal break-words\">Computer graphics and game development<\/li>\n<li class=\"whitespace-normal break-words\">Spatial analysis in geography<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q934526\">Show Solution<\/button><\/p>\n<div id=\"q934526\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sqrt{125}=5\\sqrt{5}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\">In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gahacfcb-Vj7twkiUgf0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vj7twkiUgf0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gahacfcb-Vj7twkiUgf0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844216&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gahacfcb-Vj7twkiUgf0&#38;vembed=0&#38;video_id=Vj7twkiUgf0&#38;video_target=tpm-plugin-gahacfcb-Vj7twkiUgf0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Example+-++Determine+the+Distance+Between+Two+Points_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine the Distance Between Two Points\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Midpoint Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The midpoint is the point that divides a line segment into two equal parts<\/li>\n<li class=\"whitespace-normal break-words\">It&#8217;s located exactly halfway between the endpoints<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formula: For endpoints [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex], the midpoint [latex]M[\/latex] is: [latex]M = \\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The x-coordinate of the midpoint is the average of the x-coordinates of the endpoints<\/li>\n<li class=\"whitespace-normal break-words\">The y-coordinate of the midpoint is the average of the y-coordinates of the endpoints<\/li>\n<\/ul>\n<\/li>\n<li>Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Finding centers of objects in computer graphics<\/li>\n<li class=\"whitespace-normal break-words\">Calculating average positions in physics<\/li>\n<li class=\"whitespace-normal break-words\">Determining midpoints of ranges in statistics<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Find the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q788934\">Show Solution<\/button><\/p>\n<div id=\"q788934\" class=\"hidden-answer\" style=\"display: none\">Use the formula to find the midpoint of the line segment.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q964077\">Show Solution<\/button><\/p>\n<div id=\"q964077\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cgcfdcbb-X_-iUuEiCqk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X_-iUuEiCqk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cgcfdcbb-X_-iUuEiCqk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12779083&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cgcfdcbb-X_-iUuEiCqk&#38;vembed=0&#38;video_id=X_-iUuEiCqk&#38;video_target=tpm-plugin-cgcfdcbb-X_-iUuEiCqk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Given+two+Endpoints+Find+the+Midpoint+of+a+Segment_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGiven two Endpoints Find the Midpoint of a Segment\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Plotting Points on the Coordinate Plane 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