{"id":62,"date":"2025-02-13T22:43:19","date_gmt":"2025-02-13T22:43:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions\/"},"modified":"2026-04-01T08:54:38","modified_gmt":"2026-04-01T08:54:38","slug":"linear-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions\/","title":{"raw":"Graphs of Linear Functions: Learn It 1","rendered":"Graphs of Linear Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Make a table of values and use it to graph a linear function<\/li>\r\n \t<li>Find and understand the slope and intercepts of a linear graph<\/li>\r\n \t<li>Look at a graph and write the equation of the line shown<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Linear Functions<\/h2>\r\nImagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly [latex]1.5[\/latex] inches every hour [footnote]\"Fastest growing plant,\" Guinness World Records, accessed: April 1, 2026, http:\/\/www.guinnessworldrecords.com\/world-records\/fastest-growing-plant\/.[\/footnote]. In a twenty-four hour period, this bamboo plant grows about [latex]36[\/latex] inches, or an incredible [latex]3[\/latex] feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.\r\n\r\nJust as with the growth of a bamboo plant, there are many situations that involve constant change over time. For example, consider the first commercial Maglev train in the world, the Shanghai Maglev Train. It carries passengers comfortably for a [latex]30[\/latex]-kilometer trip from the airport to the subway station in only [latex]8[\/latex] minutes.[footnote]\"Shanghai Maglev Train \u2014 The Fastest Train in the World,\" China Highlights, accessed: April 1, 2026, http:\/\/www.chinahighlights.com\/shanghai\/transportation\/maglev-train.htm[\/footnote]\r\n\r\n<center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223047\/CNX_Precalc_Figure_02_01_0112.jpg\" alt=\"The Shanghai Maglev train.\" width=\"396\" height=\"263\" \/><\/center><center><strong><span style=\"font-size: 10pt;\">A view of the Shanghai Maglev Train. (credit: Rolf Wilhelm Pfennig)<\/span><\/strong><\/center>Suppose that a Maglev train were to travel a long distance, and the train maintains a constant speed of [latex]83[\/latex] meters per second for a period of time once it is [latex]250[\/latex] meters from the station. How can we analyze the train\u2019s distance from the station as a function of time? In this section, we will investigate a type of function that is useful for this purpose and use it to investigate real-world situations such as the train\u2019s distance from the station at a given point in time.\r\n\r\nThe function describing the train\u2019s motion is a <strong>linear function<\/strong>, which is defined as a function with a constant rate of change, that is, a polynomial of degree [latex]1[\/latex]. There are several ways to represent a linear function including word form, function notation, tabular form and graphical form. We will describe the train\u2019s motion as a function using each method.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>linear function<\/h3>\r\nA <strong>linear function<\/strong> is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex] or as the graph of a straight line.\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137759903\">\r\n<h3>Representing a Linear Function in Word Form<\/h3>\r\n<p id=\"fs-id1165137588695\">Let\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.<\/p>\r\n\r\n<ul id=\"fs-id1165135526954\">\r\n \t<li><em>The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.<\/em><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135188466\">The speed is the rate of change.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">A rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable.<\/section>The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.\r\n<h3>Representing a Linear Function in Tabular Form<\/h3>\r\nAnother method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in the table below. From the table, we can see that the distance changes by [latex]83[\/latex] meters for every [latex]1[\/latex] second increase in time.\r\n\r\n[caption id=\"attachment_4929\" align=\"aligncenter\" width=\"539\"]<img class=\"wp-image-4929 \" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-300x94.png\" alt=\"Table with the first row, labeled t, containing the seconds from 0 to 3, and with the second row, labeled D(t), containing the meters 250 to 499. The first row goes up by 1 second, and the second row goes up by 83 meters.\" width=\"539\" height=\"169\" \/> Tabular representation of the function D showing selected input and output values.[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>representing a linear function in tabular form<\/h3>\r\nIn a table representing a linear function, each input-output pair forms a consistent pattern, exhibiting a constant rate of change between [latex]y[\/latex]-values. To identify the function as linear, ensure that the difference between consecutive [latex]y[\/latex]-values is the same when the [latex]x[\/latex]-values increase by a consistent amount.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318704[\/ohm_question]<\/section><\/section><section id=\"fs-id1165137837055\"><section id=\"fs-id1165137804818\"><section id=\"fs-id1165137894282\"><section id=\"fs-id1165135696154\" class=\"key-concepts\">\r\n<dl id=\"fs-id1165135195656\" class=\"definition\">\r\n \t<dd id=\"fs-id1165137635107\"><\/dd>\r\n<\/dl>\r\n<\/section><\/section><\/section><\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Make a table of values and use it to graph a linear function<\/li>\n<li>Find and understand the slope and intercepts of a linear graph<\/li>\n<li>Look at a graph and write the equation of the line shown<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Linear Functions<\/h2>\n<p>Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly [latex]1.5[\/latex] inches every hour <a class=\"footnote\" title=\"&quot;Fastest growing plant,&quot; Guinness World Records, accessed: April 1, 2026, http:\/\/www.guinnessworldrecords.com\/world-records\/fastest-growing-plant\/.\" id=\"return-footnote-62-1\" href=\"#footnote-62-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. In a twenty-four hour period, this bamboo plant grows about [latex]36[\/latex] inches, or an incredible [latex]3[\/latex] feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.<\/p>\n<p>Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. For example, consider the first commercial Maglev train in the world, the Shanghai Maglev Train. It carries passengers comfortably for a [latex]30[\/latex]-kilometer trip from the airport to the subway station in only [latex]8[\/latex] minutes.<a class=\"footnote\" title=\"&quot;Shanghai Maglev Train \u2014 The Fastest Train in the World,&quot; China Highlights, accessed: April 1, 2026, http:\/\/www.chinahighlights.com\/shanghai\/transportation\/maglev-train.htm\" id=\"return-footnote-62-2\" href=\"#footnote-62-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223047\/CNX_Precalc_Figure_02_01_0112.jpg\" alt=\"The Shanghai Maglev train.\" width=\"396\" height=\"263\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">A view of the Shanghai Maglev Train. (credit: Rolf Wilhelm Pfennig)<\/span><\/strong><\/div>\n<p>Suppose that a Maglev train were to travel a long distance, and the train maintains a constant speed of [latex]83[\/latex] meters per second for a period of time once it is [latex]250[\/latex] meters from the station. How can we analyze the train\u2019s distance from the station as a function of time? In this section, we will investigate a type of function that is useful for this purpose and use it to investigate real-world situations such as the train\u2019s distance from the station at a given point in time.<\/p>\n<p>The function describing the train\u2019s motion is a <strong>linear function<\/strong>, which is defined as a function with a constant rate of change, that is, a polynomial of degree [latex]1[\/latex]. There are several ways to represent a linear function including word form, function notation, tabular form and graphical form. We will describe the train\u2019s motion as a function using each method.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>linear function<\/h3>\n<p>A <strong>linear function<\/strong> is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[\/latex] or as the graph of a straight line.<\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137759903\">\n<h3>Representing a Linear Function in Word Form<\/h3>\n<p id=\"fs-id1165137588695\">Let\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.<\/p>\n<ul id=\"fs-id1165135526954\">\n<li><em>The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.<\/em><\/li>\n<\/ul>\n<p id=\"fs-id1165135188466\">The speed is the rate of change.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">A rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable.<\/section>\n<p>The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.<\/p>\n<h3>Representing a Linear Function in Tabular Form<\/h3>\n<p>Another method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in the table below. From the table, we can see that the distance changes by [latex]83[\/latex] meters for every [latex]1[\/latex] second increase in time.<\/p>\n<figure id=\"attachment_4929\" aria-describedby=\"caption-attachment-4929\" style=\"width: 539px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4929\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-300x94.png\" alt=\"Table with the first row, labeled t, containing the seconds from 0 to 3, and with the second row, labeled D(t), containing the meters 250 to 499. The first row goes up by 1 second, and the second row goes up by 83 meters.\" width=\"539\" height=\"169\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-300x94.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-65x20.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-225x70.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1-350x109.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02213804\/3.1.6.L.1.png 677w\" sizes=\"(max-width: 539px) 100vw, 539px\" \/><figcaption id=\"caption-attachment-4929\" class=\"wp-caption-text\">Tabular representation of the function D showing selected input and output values.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>representing a linear function in tabular form<\/h3>\n<p>In a table representing a linear function, each input-output pair forms a consistent pattern, exhibiting a constant rate of change between [latex]y[\/latex]-values. To identify the function as linear, ensure that the difference between consecutive [latex]y[\/latex]-values is the same when the [latex]x[\/latex]-values increase by a consistent amount.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318704\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318704&theme=lumen&iframe_resize_id=ohm318704&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section id=\"fs-id1165137837055\">\n<section id=\"fs-id1165137804818\">\n<section id=\"fs-id1165137894282\">\n<section id=\"fs-id1165135696154\" class=\"key-concepts\">\n<dl id=\"fs-id1165135195656\" class=\"definition\">\n<dd id=\"fs-id1165137635107\"><\/dd>\n<\/dl>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-62-1\">\"Fastest growing plant,\" Guinness World Records, accessed: April 1, 2026, http:\/\/www.guinnessworldrecords.com\/world-records\/fastest-growing-plant\/. <a href=\"#return-footnote-62-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-62-2\">\"Shanghai Maglev Train \u2014 The Fastest Train in the World,\" China Highlights, accessed: April 1, 2026, http:\/\/www.chinahighlights.com\/shanghai\/transportation\/maglev-train.htm <a href=\"#return-footnote-62-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":6,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","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":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"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\/62"}],"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\/6"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions"}],"predecessor-version":[{"id":6097,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/62\/revisions\/6097"}],"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\/62\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=62"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=62"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}