{"id":1752,"date":"2024-06-05T22:41:28","date_gmt":"2024-06-05T22:41:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1752"},"modified":"2025-08-13T23:39:06","modified_gmt":"2025-08-13T23:39:06","slug":"graphs-of-linear-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-linear-functions-fresh-take\/","title":{"raw":"Graphs of Linear Functions: Fresh Take","rendered":"Graphs of Linear Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Create and interpret equations of linear functions<\/li>\r\n \t<li>Identify and graph lines that are vertical or horizontal<\/li>\r\n \t<li>Graph straight lines by plotting points, using slope and y-intercept, and make changes like shifts to graphs<\/li>\r\n \t<li>Write equations for lines that run parallel or at a right angle to another line<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Forms of Linear Equations<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope-intercept form: [latex]y = mx + b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Essential Components<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope (m): Rate of change<\/li>\r\n \t<li class=\"whitespace-normal break-words\">y-intercept (b): Point where the line crosses the y-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Determining the Equation<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">From a graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">From two points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">From a point and slope<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Graphical Approach<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify two points on the line<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the slope: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the y-intercept visually or by calculation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formulate the equation: [latex]y = mx + b[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Algebraic Approach<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Given two points: [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate slope: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rearrange to slope-intercept form if needed<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Consider the following graph of a linear function given below. Find the equation of this line in slope-intercept form.<\/p>\r\n\r\n\r\n[caption id=\"attachment_4008\" align=\"aligncenter\" width=\"497\"]<img class=\"wp-image-4008\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409.png\" alt=\"Graph of a red line crossing the y-axis at (0, 1) with a positive slope. The line also passes through the point (4, 4)\" width=\"497\" height=\"504\" \/> Graph of a linear function[\/caption]\r\n\r\n[reveal-answer q=\"142381\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"142381\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify two points on the line: [latex](0, 1)[\/latex] and [latex](4, 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the slope:<center>[latex] \\begin{array}{rcl} m &amp;=&amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\ &amp;=&amp; \\frac{4 - 1}{4 - 0} \\\\ &amp;=&amp; \\frac{3}{4} \\end{array} [\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify the y-intercept: The line passes through [latex](0, 1)[\/latex], so [latex]b = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the equation in slope-intercept form: [latex]y = \\frac{3}{4}x + 1[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation of the line is [latex]y = \\frac{3}{4}x + 1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A car rental company charges a base fee of [latex]$30[\/latex] plus [latex]$0.25[\/latex] per mile driven. Write a linear equation to represent the total cost [latex]C[\/latex] of renting a car as a function of the number of miles [latex]m[\/latex] driven.[reveal-answer q=\"142317\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"142317\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the components of the linear equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept ([latex]b[\/latex]): This is the base fee, [latex]$30[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): This is the per-mile charge, [latex]$0.25[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the slope-intercept form [latex]y = mx + b[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]C[\/latex] represents the total cost (y)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]m[\/latex] represents the number of miles driven ([latex]x[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute the values: [latex]C = 0.25m + 30[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation representing the total cost [latex]C[\/latex] as a function of miles driven [latex]m[\/latex] is:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]C = 0.25m + 30[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Interpretation: For every mile driven, the cost increases by [latex]$0.25[\/latex], starting from a base cost of [latex]$30[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Describing Horizontal and Vertical Lines<\/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\"><strong>Horizontal Lines<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation: [latex]y = b[\/latex] or [latex]f(x) = b[\/latex], where [latex]b[\/latex] is constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope: [latex]m = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]Y[\/latex]-intercept = [latex](0, b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept (unless [latex]b = 0[\/latex])<\/li>\r\n \t<li>Represent constant output ([latex]y[\/latex]-value)<\/li>\r\n \t<li>Parallel to [latex]x[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vertical Lines<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Equation: [latex]x = a[\/latex], where [latex]a[\/latex] is constant<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope: Undefined<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No [latex]y[\/latex]-intercept (unless [latex]a = 0[\/latex])<\/li>\r\n \t<li>[latex]X[\/latex]-intercept = [latex](a, 0)[\/latex]<\/li>\r\n \t<li>Represent constant input ([latex]x[\/latex]-value)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not a function<\/li>\r\n \t<li>Parallel to [latex]y[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Graphing Linear Functions<\/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\">Three primary methods for graphing linear functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Plotting points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using y-intercept and slope<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applying transformations to [latex]f(x) = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Linear function equation: [latex]f(x) = mx + b[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]m[\/latex]: slope (rate of change)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: [latex]y[\/latex]-intercept (where the line crosses the [latex]y[\/latex]-axis)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope-intercept form provides direct access to key graph characteristics<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transformations of the identity function offer insights into function behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each method provides unique perspectives on the function's properties<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Method 1: Plotting Points<\/strong><\/p>\r\n<p class=\"font-600 text-lg font-bold\">Steps:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose at least two input values ([latex]x[\/latex]-coordinates)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate corresponding output values ([latex]y[\/latex]-coordinates)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the points on a coordinate plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw a line through the points<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Method 2: Using [latex]y[\/latex]-intercept and Slope<\/strong><\/p>\r\n<p class=\"font-600 text-lg font-bold\">Key Concepts:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">y-intercept ([latex]b[\/latex]): Point where the line crosses the y-axis [latex](0, b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): Rate of change, [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Steps:<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the y-intercept ([latex]b[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify the slope ([latex]m[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the [latex]y[\/latex]-intercept<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the slope to plot additional points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw the line through the points<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Method 3: Transformations of [latex]f(x) = x[\/latex]<\/strong><\/p>\r\n<p class=\"font-600 text-lg font-bold\">Key Transformations:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical stretch\/compression: [latex]f(x) = mx[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical shift: [latex]f(x) = x + b[\/latex]<\/li>\r\n<\/ol>\r\n<h3 class=\"font-600 text-lg font-bold\">Steps:<\/h3>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Start with [latex]f(x) = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply vertical stretch\/compression by factor [latex]m[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply vertical shift by [latex]b[\/latex] units<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex] by plotting points.[reveal-answer q=\"156351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"156351\"]\r\n\r\n[caption id=\"attachment_2803\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\"><img class=\"wp-image-2803 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\" alt=\"cnx_precalc_figure_02_02_0022\" width=\"487\" height=\"316\" \/><\/a> Graph of a linear function with three points labeled[\/caption]\r\n\r\n[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f(x) = -\\frac{3}{2}x + 4[\/latex][reveal-answer q=\"784642\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"784642\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">y-intercept: [latex](0, 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slope: [latex]-\\frac{3}{2}[\/latex]<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_4025\" align=\"aligncenter\" width=\"511\"]<img class=\"wp-image-4025 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005.png\" alt=\"The image shows a graph of the linear function f(x) = -3\/2x + 4. The graph is a straight line with a negative slope, crossing the y-axis at y = 4. The line descends from left to right due to the negative slope of -3\/2.\" width=\"511\" height=\"528\" \/> Graph of a linear function[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations.[reveal-answer q=\"350962\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"350962\"]\r\n\r\n[caption id=\"attachment_2804\" align=\"aligncenter\" width=\"510\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\"><img class=\"wp-image-2804 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"cnx_precalc_figure_02_02_0092\" width=\"510\" height=\"520\" \/><\/a> Graph of three linear functions[\/caption]\r\n\r\n[\/hidden-answer]<\/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-fgfcchbc-h9zn_ODlgbM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fgfcchbc-h9zn_ODlgbM\" 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=12844470&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fgfcchbc-h9zn_ODlgbM&vembed=0&video_id=h9zn_ODlgbM&video_target=tpm-plugin-fgfcchbc-h9zn_ODlgbM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Linear+Function+as+a+Transformation+of+f(x)%3Dx_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Linear Function as a Transformation of f(x)=x\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Determining Whether Lines are Parallel or Perpendicular<\/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\">Parallel lines never intersect and have the same slope.\r\n<ul>\r\n \t<li>\r\n<p class=\"whitespace-pre-wrap break-words\">Key characteristics:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Same slope ([latex]m[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Different y-intercept ([latex]b[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equation form: [latex]y = mx + b_1[\/latex] and [latex]y = mx + b_2[\/latex], where [latex]b_1 \\neq b_2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perpendicular lines intersect at a 90-degree angle and have slopes that are negative reciprocals of each other.\r\n<ul>\r\n \t<li>\r\n<p class=\"whitespace-pre-wrap break-words\">Key characteristics:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slopes are negative reciprocals: [latex]m_1 = -\\frac{1}{m_2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Product of slopes equals [latex]-1[\/latex]: [latex]m_1 \\cdot m_2 = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equation form: [latex]y = m_1x + b_1[\/latex] and [latex]y = -\\frac{1}{m_1}x + b_2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The slope-intercept form of a line ([latex]y = mx + b[\/latex]) is key to identifying parallel and perpendicular lines.<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = 3x + 2[\/latex] and [latex]g(x) = -\\frac{1}{3}x + 5[\/latex], are these lines perpendicular?[reveal-answer q=\"882910\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"882910\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Slopes: [latex]m_1 = 3[\/latex], [latex]m_2 = -\\frac{1}{3}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Product of slopes: [latex]3 \\cdot (-\\frac{1}{3}) = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conclusion: The lines are perpendicular.<\/li>\r\n<\/ul>\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-cdacdecc-Uq8pSFaXPmQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Uq8pSFaXPmQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdacdecc-Uq8pSFaXPmQ\" 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=12780695&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cdacdecc-Uq8pSFaXPmQ&vembed=0&video_id=Uq8pSFaXPmQ&video_target=tpm-plugin-cdacdecc-Uq8pSFaXPmQ'><\/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+Tell+If+Two+Lines+Are+Parallel%2C+Perpendicular%2C+or+Neither%3F_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Tell If Two Lines Are Parallel, Perpendicular, or Neither?\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Equations of Parallel and Perpendicular Lines<\/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\">Parallel lines have the same slope but different [latex]y[\/latex]-intercepts.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perpendicular lines have slopes that are negative reciprocals of each other.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Point-slope form and slope-intercept form are key tools for writing these equations.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Given a line and a point, we can write equations of parallel or perpendicular lines passing through that point.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The process involves manipulating slopes and using known points to determine y-intercepts.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Writing Equations of Parallel Lines<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Given a line [latex]f(x) = mx + b[\/latex] and a point [latex](x_0, y_0)[\/latex], to write the equation of a parallel line through the point:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the slope [latex]m[\/latex] of the original line.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use point-slope form with the new point: [latex]y - y_0 = m(x - x_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to slope-intercept form: [latex]y = mx + (y_0 - mx_0)[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Writing Equations of Perpendicular Lines<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Given a line [latex]f(x) = mx + b[\/latex] and a point [latex](x_0, y_0)[\/latex], to write the equation of a perpendicular line through the point:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find the negative reciprocal of the slope: [latex]m_{\\perp} = -\\frac{1}{m}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use point-slope form with the new point: [latex]y - y_0 = m_{\\perp}(x - x_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to slope-intercept form: [latex]y = m_{\\perp}x + (y_0 - m_{\\perp}x_0)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(x) = 3x + 1[\/latex] and point [latex](2, 7)[\/latex], find a parallel line.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"570974\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"570974\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Solution:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope [latex]m = 3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y - 7 = 3(x - 2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = 3x + 1[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(x) = 2x + 4[\/latex] and point [latex](3, 1)[\/latex], find a perpendicular line.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"576372\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"576372\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Solution:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]m_{\\perp} = -\\frac{1}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y - 1 = -\\frac{1}{2}(x - 3)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = -\\frac{1}{2}x + \\frac{5}{2}[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A line passes through the points, [latex](\u20132, \u201315)[\/latex] and [latex](2, \u20133)[\/latex]. Find the equation of a perpendicular line that passes through the point, [latex](6, 4)[\/latex].[reveal-answer q=\"678591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"678591\"][latex]y=-\\frac{1}{3}x+6[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Create and interpret equations of linear functions<\/li>\n<li>Identify and graph lines that are vertical or horizontal<\/li>\n<li>Graph straight lines by plotting points, using slope and y-intercept, and make changes like shifts to graphs<\/li>\n<li>Write equations for lines that run parallel or at a right angle to another line<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Forms of Linear Equations<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope-intercept form: [latex]y = mx + b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Essential Components<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope (m): Rate of change<\/li>\n<li class=\"whitespace-normal break-words\">y-intercept (b): Point where the line crosses the y-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Determining the Equation<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">From a graph<\/li>\n<li class=\"whitespace-normal break-words\">From two points<\/li>\n<li class=\"whitespace-normal break-words\">From a point and slope<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Graphical Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify two points on the line<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the slope: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Find the y-intercept visually or by calculation<\/li>\n<li class=\"whitespace-normal break-words\">Formulate the equation: [latex]y = mx + b[\/latex]<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Algebraic Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Given two points: [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate slope: [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use point-slope form: [latex]y - y_1 = m(x - x_1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rearrange to slope-intercept form if needed<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Consider the following graph of a linear function given below. Find the equation of this line in slope-intercept form.<\/p>\n<figure id=\"attachment_4008\" aria-describedby=\"caption-attachment-4008\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4008\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409.png\" alt=\"Graph of a red line crossing the y-axis at (0, 1) with a positive slope. The line also passes through the point (4, 4)\" width=\"497\" height=\"504\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409.png 633w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409-296x300.png 296w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409-65x66.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409-225x228.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16171430\/Screenshot-2024-09-16-131409-350x355.png 350w\" sizes=\"(max-width: 497px) 100vw, 497px\" \/><figcaption id=\"caption-attachment-4008\" class=\"wp-caption-text\">Graph of a linear function<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q142381\">Show Answer<\/button><\/p>\n<div id=\"q142381\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify two points on the line: [latex](0, 1)[\/latex] and [latex](4, 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the slope:\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} m &=& \\frac{y_2 - y_1}{x_2 - x_1} \\\\ &=& \\frac{4 - 1}{4 - 0} \\\\ &=& \\frac{3}{4} \\end{array}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identify the y-intercept: The line passes through [latex](0, 1)[\/latex], so [latex]b = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the equation in slope-intercept form: [latex]y = \\frac{3}{4}x + 1[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation of the line is [latex]y = \\frac{3}{4}x + 1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A car rental company charges a base fee of [latex]$30[\/latex] plus [latex]$0.25[\/latex] per mile driven. Write a linear equation to represent the total cost [latex]C[\/latex] of renting a car as a function of the number of miles [latex]m[\/latex] driven.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q142317\">Show Answer<\/button><\/p>\n<div id=\"q142317\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the components of the linear equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept ([latex]b[\/latex]): This is the base fee, [latex]$30[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): This is the per-mile charge, [latex]$0.25[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Use the slope-intercept form [latex]y = mx + b[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]C[\/latex] represents the total cost (y)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]m[\/latex] represents the number of miles driven ([latex]x[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute the values: [latex]C = 0.25m + 30[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation representing the total cost [latex]C[\/latex] as a function of miles driven [latex]m[\/latex] is:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]C = 0.25m + 30[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Interpretation: For every mile driven, the cost increases by [latex]$0.25[\/latex], starting from a base cost of [latex]$30[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Describing Horizontal and Vertical Lines<\/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\"><strong>Horizontal Lines<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation: [latex]y = b[\/latex] or [latex]f(x) = b[\/latex], where [latex]b[\/latex] is constant<\/li>\n<li class=\"whitespace-normal break-words\">Slope: [latex]m = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]Y[\/latex]-intercept = [latex](0, b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept (unless [latex]b = 0[\/latex])<\/li>\n<li>Represent constant output ([latex]y[\/latex]-value)<\/li>\n<li>Parallel to [latex]x[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Vertical Lines<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Equation: [latex]x = a[\/latex], where [latex]a[\/latex] is constant<\/li>\n<li class=\"whitespace-normal break-words\">Slope: Undefined<\/li>\n<li class=\"whitespace-normal break-words\">No [latex]y[\/latex]-intercept (unless [latex]a = 0[\/latex])<\/li>\n<li>[latex]X[\/latex]-intercept = [latex](a, 0)[\/latex]<\/li>\n<li>Represent constant input ([latex]x[\/latex]-value)<\/li>\n<li class=\"whitespace-normal break-words\">Not a function<\/li>\n<li>Parallel to [latex]y[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<h2 data-type=\"title\">Graphing Linear Functions<\/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\">Three primary methods for graphing linear functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Plotting points<\/li>\n<li class=\"whitespace-normal break-words\">Using y-intercept and slope<\/li>\n<li class=\"whitespace-normal break-words\">Applying transformations to [latex]f(x) = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Linear function equation: [latex]f(x) = mx + b[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]m[\/latex]: slope (rate of change)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: [latex]y[\/latex]-intercept (where the line crosses the [latex]y[\/latex]-axis)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Slope-intercept form provides direct access to key graph characteristics<\/li>\n<li class=\"whitespace-normal break-words\">Transformations of the identity function offer insights into function behavior<\/li>\n<li class=\"whitespace-normal break-words\">Each method provides unique perspectives on the function&#8217;s properties<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Method 1: Plotting Points<\/strong><\/p>\n<p class=\"font-600 text-lg font-bold\">Steps:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose at least two input values ([latex]x[\/latex]-coordinates)<\/li>\n<li class=\"whitespace-normal break-words\">Calculate corresponding output values ([latex]y[\/latex]-coordinates)<\/li>\n<li class=\"whitespace-normal break-words\">Plot the points on a coordinate plane<\/li>\n<li class=\"whitespace-normal break-words\">Draw a line through the points<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Method 2: Using [latex]y[\/latex]-intercept and Slope<\/strong><\/p>\n<p class=\"font-600 text-lg font-bold\">Key Concepts:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">y-intercept ([latex]b[\/latex]): Point where the line crosses the y-axis [latex](0, b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Slope ([latex]m[\/latex]): Rate of change, [latex]m = \\frac{\\text{rise}}{\\text{run}} = \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-600 text-lg font-bold\"><strong>Steps:<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the y-intercept ([latex]b[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Identify the slope ([latex]m[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Plot the [latex]y[\/latex]-intercept<\/li>\n<li class=\"whitespace-normal break-words\">Use the slope to plot additional points<\/li>\n<li class=\"whitespace-normal break-words\">Draw the line through the points<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Method 3: Transformations of [latex]f(x) = x[\/latex]<\/strong><\/p>\n<p class=\"font-600 text-lg font-bold\">Key Transformations:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical stretch\/compression: [latex]f(x) = mx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical shift: [latex]f(x) = x + b[\/latex]<\/li>\n<\/ol>\n<h3 class=\"font-600 text-lg font-bold\">Steps:<\/h3>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with [latex]f(x) = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply vertical stretch\/compression by factor [latex]m[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply vertical shift by [latex]b[\/latex] units<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex] by plotting points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q156351\">Show Solution<\/button><\/p>\n<div id=\"q156351\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_2803\" aria-describedby=\"caption-attachment-2803\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2803 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15184550\/CNX_Precalc_Figure_02_02_0022.jpg\" alt=\"cnx_precalc_figure_02_02_0022\" width=\"487\" height=\"316\" \/><\/a><figcaption id=\"caption-attachment-2803\" class=\"wp-caption-text\">Graph of a linear function with three points labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f(x) = -\\frac{3}{2}x + 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q784642\">Show Answer<\/button><\/p>\n<div id=\"q784642\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">y-intercept: [latex](0, 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Slope: [latex]-\\frac{3}{2}[\/latex]<\/li>\n<\/ul>\n<figure id=\"attachment_4025\" aria-describedby=\"caption-attachment-4025\" style=\"width: 511px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4025 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005.png\" alt=\"The image shows a graph of the linear function f(x) = -3\/2x + 4. The graph is a straight line with a negative slope, crossing the y-axis at y = 4. The line descends from left to right due to the negative slope of -3\/2.\" width=\"511\" height=\"528\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005.png 511w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005-290x300.png 290w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005-65x67.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005-225x232.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16180037\/Screenshot-2024-09-16-140005-350x362.png 350w\" sizes=\"(max-width: 511px) 100vw, 511px\" \/><figcaption id=\"caption-attachment-4025\" class=\"wp-caption-text\">Graph of a linear function<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q350962\">Show Solution<\/button><\/p>\n<div id=\"q350962\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_2804\" aria-describedby=\"caption-attachment-2804\" style=\"width: 510px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2804 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/15185737\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"cnx_precalc_figure_02_02_0092\" width=\"510\" height=\"520\" \/><\/a><figcaption id=\"caption-attachment-2804\" class=\"wp-caption-text\">Graph of three linear functions<\/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-fgfcchbc-h9zn_ODlgbM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fgfcchbc-h9zn_ODlgbM\" 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=12844470&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fgfcchbc-h9zn_ODlgbM&#38;vembed=0&#38;video_id=h9zn_ODlgbM&#38;video_target=tpm-plugin-fgfcchbc-h9zn_ODlgbM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Linear+Function+as+a+Transformation+of+f(x)%3Dx_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Linear Function as a Transformation of f(x)=x\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Determining Whether Lines are Parallel or Perpendicular<\/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\">Parallel lines never intersect and have the same slope.\n<ul>\n<li>\n<p class=\"whitespace-pre-wrap break-words\">Key characteristics:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Same slope ([latex]m[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Different y-intercept ([latex]b[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Equation form: [latex]y = mx + b_1[\/latex] and [latex]y = mx + b_2[\/latex], where [latex]b_1 \\neq b_2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Perpendicular lines intersect at a 90-degree angle and have slopes that are negative reciprocals of each other.\n<ul>\n<li>\n<p class=\"whitespace-pre-wrap break-words\">Key characteristics:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slopes are negative reciprocals: [latex]m_1 = -\\frac{1}{m_2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product of slopes equals [latex]-1[\/latex]: [latex]m_1 \\cdot m_2 = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Equation form: [latex]y = m_1x + b_1[\/latex] and [latex]y = -\\frac{1}{m_1}x + b_2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The slope-intercept form of a line ([latex]y = mx + b[\/latex]) is key to identifying parallel and perpendicular lines.<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = 3x + 2[\/latex] and [latex]g(x) = -\\frac{1}{3}x + 5[\/latex], are these lines perpendicular?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q882910\">Show Answer<\/button><\/p>\n<div id=\"q882910\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Slopes: [latex]m_1 = 3[\/latex], [latex]m_2 = -\\frac{1}{3}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product of slopes: [latex]3 \\cdot (-\\frac{1}{3}) = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Conclusion: The lines are perpendicular.<\/li>\n<\/ul>\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-cdacdecc-Uq8pSFaXPmQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Uq8pSFaXPmQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdacdecc-Uq8pSFaXPmQ\" 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=12780695&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cdacdecc-Uq8pSFaXPmQ&#38;vembed=0&#38;video_id=Uq8pSFaXPmQ&#38;video_target=tpm-plugin-cdacdecc-Uq8pSFaXPmQ\"><\/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+Tell+If+Two+Lines+Are+Parallel%2C+Perpendicular%2C+or+Neither%3F_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Tell If Two Lines Are Parallel, Perpendicular, or Neither?\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Equations of Parallel and Perpendicular Lines<\/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\">Parallel lines have the same slope but different [latex]y[\/latex]-intercepts.<\/li>\n<li class=\"whitespace-normal break-words\">Perpendicular lines have slopes that are negative reciprocals of each other.<\/li>\n<li class=\"whitespace-normal break-words\">Point-slope form and slope-intercept form are key tools for writing these equations.<\/li>\n<li class=\"whitespace-normal break-words\">Given a line and a point, we can write equations of parallel or perpendicular lines passing through that point.<\/li>\n<li class=\"whitespace-normal break-words\">The process involves manipulating slopes and using known points to determine y-intercepts.<\/li>\n<\/ul>\n<p class=\"font-600 text-lg font-bold\"><strong>Writing Equations of Parallel Lines<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Given a line [latex]f(x) = mx + b[\/latex] and a point [latex](x_0, y_0)[\/latex], to write the equation of a parallel line through the point:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the slope [latex]m[\/latex] of the original line.<\/li>\n<li class=\"whitespace-normal break-words\">Use point-slope form with the new point: [latex]y - y_0 = m(x - x_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to slope-intercept form: [latex]y = mx + (y_0 - mx_0)[\/latex]<\/li>\n<\/ol>\n<p class=\"font-600 text-lg font-bold\"><strong>Writing Equations of Perpendicular Lines<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Given a line [latex]f(x) = mx + b[\/latex] and a point [latex](x_0, y_0)[\/latex], to write the equation of a perpendicular line through the point:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find the negative reciprocal of the slope: [latex]m_{\\perp} = -\\frac{1}{m}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use point-slope form with the new point: [latex]y - y_0 = m_{\\perp}(x - x_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to slope-intercept form: [latex]y = m_{\\perp}x + (y_0 - m_{\\perp}x_0)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(x) = 3x + 1[\/latex] and point [latex](2, 7)[\/latex], find a parallel line.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q570974\">Show Answer<\/button><\/p>\n<div id=\"q570974\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Solution:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope [latex]m = 3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y - 7 = 3(x - 2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = 3x + 1[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(x) = 2x + 4[\/latex] and point [latex](3, 1)[\/latex], find a perpendicular line.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q576372\">Show Answer<\/button><\/p>\n<div id=\"q576372\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Solution:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]m_{\\perp} = -\\frac{1}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y - 1 = -\\frac{1}{2}(x - 3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = -\\frac{1}{2}x + \\frac{5}{2}[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A line passes through the points, [latex](\u20132, \u201315)[\/latex] and [latex](2, \u20133)[\/latex]. Find the equation of a perpendicular line that passes through the point, [latex](6, 4)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q678591\">Show Solution<\/button><\/p>\n<div id=\"q678591\" class=\"hidden-answer\" style=\"display: none\">[latex]y=-\\frac{1}{3}x+6[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Graph a Linear Function as a Transformation of f(x)=x \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=h9zn_ODlgbM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How To Tell If Two Lines Are Parallel, Perpendicular, or Neither? \",\"author\":\"\",\"organization\":\"The Organic Chemistry Tutor\",\"url\":\"https:\/\/www.youtube.com\/watch?v=Uq8pSFaXPmQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":164,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Graph a Linear Function as a Transformation of f(x)=x ","author":"","organization":"Mathispower4u","url":"https:\/\/www.youtube.com\/watch?v=h9zn_ODlgbM","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"How To Tell If Two Lines Are Parallel, Perpendicular, or Neither? 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