{"id":737,"date":"2025-07-15T14:56:55","date_gmt":"2025-07-15T14:56:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=737"},"modified":"2026-01-09T21:08:27","modified_gmt":"2026-01-09T21:08:27","slug":"linear-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-learn-it-1\/","title":{"raw":"Linear Functions: Learn It 1","rendered":"Linear Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write the equation of a linear function given a point and a slope, two points, or a table of values.<\/li>\r\n \t<li>Graph linear functions given any form of its equation.<\/li>\r\n \t<li>Graph and write the equations of horizontal and vertical lines.<\/li>\r\n \t<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\r\n<\/ul>\r\n<\/section>If we want to write the equation of a linear function there are three forms we can choose from:\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>different forms of a linear function<\/h3>\r\n<ul>\r\n \t<li>Standard form: [latex]Ax+By=C[\/latex]<\/li>\r\n \t<li>Slope-intercept form: [latex]y=mx+b[\/latex] where\r\n<ul>\r\n \t<li>[latex]m[\/latex] is the slope, and<\/li>\r\n \t<li>[latex]b[\/latex] is the [latex]y[\/latex]-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Point-slope form: [latex]y-y_1=m(x-x_1)[\/latex] where\r\n<ul>\r\n \t<li>[latex]m[\/latex] is the slope, and<\/li>\r\n \t<li>[latex](x_1,y_1)[\/latex] is any point on the line.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Slope-Intercept Form<\/h3>\r\nPerhaps the most familiar form of a linear equation is the slope-intercept form, written as [latex]y=mx+b[\/latex], where [latex]m=\\text{ slope }[\/latex] and [latex]b=y-\\text{intercept}[\/latex].\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">To calculate slope given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], [latex]m=\\frac{\\text{rise}}{\\text{run}} = \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex].<\/section><section class=\"textbox example\">Find the equation of the line for the graph below.<img class=\"aligncenter\" 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\" \/><strong>\r\nStep 1: Calculate the Slope ([latex]m[\/latex])<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The slope of a line is calculated using the formula [latex]m=\\frac{\\text{rise}}{\\text{run}} = \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex].<\/span>\r\n<ul>\r\n \t<li>Selecting two points from the graph: [latex](-2, 1)[\/latex] and [latex](2, 3)[\/latex].<\/li>\r\n \t<li>Using these points, we can calculate the slope:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]m = \\dfrac{y_2 - y_1}{x_2 - x_1} = \\dfrac{3 - 1}{2 - (-2)} = \\dfrac{2}{4} = \\dfrac{1}{2}[\/latex]<\/p>\r\n<strong>Step 2: Find the [latex]y[\/latex]-intercept ([latex]b[\/latex])<\/strong>\r\n\r\nThe [latex]y[\/latex]-intercept is the value of [latex]y[\/latex] when [latex]x=0[\/latex]. From the graph, it is apparent that when [latex]x=0, y=2[\/latex]. Therefore, [latex]b=2[\/latex].\r\n\r\n<strong>Step 3: Write the Equation<\/strong>\r\n\r\nNow that we have the slope and [latex]y[\/latex]-intercept, we can write the equation of the line:\r\n<p style=\"text-align: center;\">[latex]y = \\dfrac{1}{2}x+2[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318709[\/ohm_question]<\/section><section class=\"textbox example\">Identify the slope and <em>y-<\/em>intercept given the equation [latex]y=-\\frac{3}{4}x - 4[\/latex].\r\n[reveal-answer q=\"757424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757424\"]As the line is in [latex]y=mx+b[\/latex] form, the given line has a slope of [latex]m=-\\frac{3}{4}[\/latex]. The <em>y-<\/em>intercept is [latex]b=-4[\/latex], or, [latex](0, -4)[\/latex] in ordered pair format.<strong>Analysis of the Solution<\/strong>The <em>y<\/em>-intercept is the point at which the line crosses the <em>y-<\/em>axis. On the <em>y-<\/em>axis, [latex]x=0[\/latex]. We can always identify the <em>y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em>b.<\/em> Or, just substitute [latex]x=0[\/latex] and solve for <em>y.<\/em>[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318710[\/ohm_question]<\/section><section id=\"fs-id1165137543411\"><\/section><section id=\"fs-id1165137543411\"><section id=\"fs-id1165135667863\"><section id=\"fs-id1165137761836\">\r\n<div id=\"Example_02_02_08\" class=\"example\">\r\n<div id=\"fs-id1165137596422\" class=\"exercise\">\r\n<div id=\"fs-id1165135187508\" class=\"commentary\"><section id=\"fs-id1165134093077\">\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>slope-intercept form<\/h3>\r\nThe slope-intercept form of a line is written as:\r\n<p style=\"text-align: center;\">[latex]y = mx+b[\/latex]<\/p>\r\nwhere:\r\n<ul>\r\n \t<li>[latex]m[\/latex] is the slope of the line, representing the rate of change or steepness of the line.<\/li>\r\n \t<li>[latex]b[\/latex] is the y-intercept, which is the point where the line crosses the y-axis. This value indicates where the line will pass through the y-axis when [latex]x=0[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write the equation of a linear function given a point and a slope, two points, or a table of values.<\/li>\n<li>Graph linear functions given any form of its equation.<\/li>\n<li>Graph and write the equations of horizontal and vertical lines.<\/li>\n<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\n<\/ul>\n<\/section>\n<p>If we want to write the equation of a linear function there are three forms we can choose from:<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>different forms of a linear function<\/h3>\n<ul>\n<li>Standard form: [latex]Ax+By=C[\/latex]<\/li>\n<li>Slope-intercept form: [latex]y=mx+b[\/latex] where\n<ul>\n<li>[latex]m[\/latex] is the slope, and<\/li>\n<li>[latex]b[\/latex] is the [latex]y[\/latex]-intercept<\/li>\n<\/ul>\n<\/li>\n<li>Point-slope form: [latex]y-y_1=m(x-x_1)[\/latex] where\n<ul>\n<li>[latex]m[\/latex] is the slope, and<\/li>\n<li>[latex](x_1,y_1)[\/latex] is any point on the line.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<h3>Slope-Intercept Form<\/h3>\n<p>Perhaps the most familiar form of a linear equation is the slope-intercept form, written as [latex]y=mx+b[\/latex], where [latex]m=\\text{ slope }[\/latex] and [latex]b=y-\\text{intercept}[\/latex].<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">To calculate slope given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], [latex]m=\\frac{\\text{rise}}{\\text{run}} = \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex].<\/section>\n<section class=\"textbox example\">Find the equation of the line for the graph below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><strong><br \/>\nStep 1: Calculate the Slope ([latex]m[\/latex])<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The slope of a line is calculated using the formula [latex]m=\\frac{\\text{rise}}{\\text{run}} = \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex].<\/span><\/p>\n<ul>\n<li>Selecting two points from the graph: [latex](-2, 1)[\/latex] and [latex](2, 3)[\/latex].<\/li>\n<li>Using these points, we can calculate the slope:<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]m = \\dfrac{y_2 - y_1}{x_2 - x_1} = \\dfrac{3 - 1}{2 - (-2)} = \\dfrac{2}{4} = \\dfrac{1}{2}[\/latex]<\/p>\n<p><strong>Step 2: Find the [latex]y[\/latex]-intercept ([latex]b[\/latex])<\/strong><\/p>\n<p>The [latex]y[\/latex]-intercept is the value of [latex]y[\/latex] when [latex]x=0[\/latex]. From the graph, it is apparent that when [latex]x=0, y=2[\/latex]. Therefore, [latex]b=2[\/latex].<\/p>\n<p><strong>Step 3: Write the Equation<\/strong><\/p>\n<p>Now that we have the slope and [latex]y[\/latex]-intercept, we can write the equation of the line:<\/p>\n<p style=\"text-align: center;\">[latex]y = \\dfrac{1}{2}x+2[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318709\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318709&theme=lumen&iframe_resize_id=ohm318709&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Identify the slope and <em>y-<\/em>intercept given the equation [latex]y=-\\frac{3}{4}x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q757424\">Show Solution<\/button><\/p>\n<div id=\"q757424\" class=\"hidden-answer\" style=\"display: none\">As the line is in [latex]y=mx+b[\/latex] form, the given line has a slope of [latex]m=-\\frac{3}{4}[\/latex]. The <em>y-<\/em>intercept is [latex]b=-4[\/latex], or, [latex](0, -4)[\/latex] in ordered pair format.<strong>Analysis of the Solution<\/strong>The <em>y<\/em>-intercept is the point at which the line crosses the <em>y-<\/em>axis. On the <em>y-<\/em>axis, [latex]x=0[\/latex]. We can always identify the <em>y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em>b.<\/em> Or, just substitute [latex]x=0[\/latex] and solve for <em>y.<\/em><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318710\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318710&theme=lumen&iframe_resize_id=ohm318710&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section id=\"fs-id1165137543411\"><\/section>\n<section>\n<section id=\"fs-id1165135667863\">\n<section id=\"fs-id1165137761836\">\n<div id=\"Example_02_02_08\" class=\"example\">\n<div id=\"fs-id1165137596422\" class=\"exercise\">\n<div id=\"fs-id1165135187508\" class=\"commentary\">\n<section id=\"fs-id1165134093077\">\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>slope-intercept form<\/h3>\n<p>The slope-intercept form of a line is written as:<\/p>\n<p style=\"text-align: center;\">[latex]y = mx+b[\/latex]<\/p>\n<p>where:<\/p>\n<ul>\n<li>[latex]m[\/latex] is the slope of the line, representing the rate of change or steepness of the line.<\/li>\n<li>[latex]b[\/latex] is the y-intercept, which is the point where the line crosses the y-axis. This value indicates where the line will pass through the y-axis when [latex]x=0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"author":13,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/737"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions"}],"predecessor-version":[{"id":5270,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions\/5270"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/737\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=737"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=737"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}