{"id":1057,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1057"},"modified":"2015-11-12T18:35:32","modified_gmt":"2015-11-12T18:35:32","slug":"determine-whether-a-linear-function-is-increasing-decreasing-or-constant","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/determine-whether-a-linear-function-is-increasing-decreasing-or-constant\/","title":{"raw":"Determine whether a linear function is increasing, decreasing, or constant","rendered":"Determine whether a linear function is increasing, decreasing, or constant"},"content":{"raw":"<section id=\"fs-id1165137749252\" data-depth=\"1\">\n\n[caption id=\"\" align=\"alignnone\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201019\/CNX_Precalc_Figure_02_01_004abc2.jpg\" alt=\"Three graphs depicting an increasing function, a decreasing function, and a constant function.\" width=\"975\" height=\"375\" data-media-type=\"image\/jpg\"\/><b>Figure 4<\/b>[\/caption]\n<p id=\"fs-id1165135482019\">The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant.<\/p>\n\u00a0\n\nFor an <strong>increasing function<\/strong>, as with the train example,\n<p style=\"text-align: center;\"><strong><em>the output values increase as the input values increase. <\/em><\/strong><\/p>\n<p style=\"text-align: center;\"\/>\nThe graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in <strong>(a)<\/strong>.\n\n\u00a0\n\nFor a <strong>decreasing function<\/strong>, the slope is negative.\n<p style=\"text-align: center;\"\/>\n<p style=\"text-align: center;\"><strong><em>The output values decrease as the input values increase. <\/em><\/strong><\/p>\n\u00a0\n\nA line with a negative slope slants downward from left to right as in <strong>(b)<\/strong>. If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in <strong>(c)<\/strong>.<span id=\"fs-id1165137453957\" data-type=\"media\" data-alt=\"Three graphs depicting an increasing function, a decreasing function, and a constant function.\">\n<\/span>\n<div id=\"fs-id1165137446154\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Increasing and Decreasing Functions<\/h3>\n<p id=\"fs-id1165134085973\">The slope determines if the function is an <strong>increasing linear function<\/strong>, a <strong>decreasing linear function<\/strong>, or a constant function.<\/p>\n\n<ul id=\"eip-643\" data-bullet-style=\"none\"><li>[latex]f\\left(x\\right)=mx+b\\text{ is an increasing function if }m&gt;0[\/latex].<\/li>\n\t<li>[latex]f\\left(x\\right)=mx+b\\text{ is an decreasing function if }m&lt;0[\/latex].<\/li>\n\t<li>[latex]f\\left(x\\right)=mx+b\\text{ is a constant function if }m=0[\/latex].<\/li>\n<\/ul><\/div>\n<div id=\"Example_02_01_02\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137405281\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135571684\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 2: Deciding whether a Function Is Increasing, Decreasing, or Constant<\/h3>\n<p id=\"fs-id1165137400625\">Some recent studies suggest that a teenager sends an average of 60 texts per day.[footnote]<a href=\"http:\/\/www.cbsnews.com\/8301-501465_162-57400228-501465\/teens-are-sending-60-texts-a-day-study-says\/\" target=\"_blank\">http:\/\/www.cbsnews.com\/8301-501465_162-57400228-501465\/teens-are-sending-60-texts-a-day-study-says\/<\/a>[\/footnote]\u00a0For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.<\/p>\n\n<ol id=\"fs-id1165137807449\" data-number-style=\"lower-alpha\"><li>The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.<\/li>\n\t<li>A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.<\/li>\n\t<li>A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.<\/li>\n<\/ol><\/div>\n<div id=\"fs-id1165137435336\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137508045\">Analyze each function.<\/p>\n\n<ol id=\"fs-id1165137778959\" data-number-style=\"lower-alpha\"><li>The function can be represented as [latex]f\\left(x\\right)=60x[\/latex] where [latex]x[\/latex] is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.<\/li>\n\t<li>The function can be represented as [latex]f\\left(x\\right)=500 - 60x[\/latex] where [latex]x[\/latex] is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after [latex]x[\/latex] days.<\/li>\n\t<li>The cost function can be represented as [latex]f\\left(x\\right)=50[\/latex] because the number of days does not affect the total cost. The slope is 0 so the function is constant.<\/li>\n<\/ol><\/div>\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id1165137837055\" data-depth=\"1\"><div id=\"fs-id1165137410528\" class=\"note precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"\/>\n<\/section><section id=\"fs-id1165135208792\" data-depth=\"1\"\/>","rendered":"<section id=\"fs-id1165137749252\" data-depth=\"1\">\n<div style=\"width: 985px\" 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\/11\/25201019\/CNX_Precalc_Figure_02_01_004abc2.jpg\" alt=\"Three graphs depicting an increasing function, a decreasing function, and a constant function.\" width=\"975\" height=\"375\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135482019\">The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant.<\/p>\n<p>\u00a0<\/p>\n<p>For an <strong>increasing function<\/strong>, as with the train example,<\/p>\n<p style=\"text-align: center;\"><strong><em>the output values increase as the input values increase. <\/em><\/strong><\/p>\n<p style=\"text-align: center;\">\nThe graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in <strong>(a)<\/strong>.<\/p>\n<p>\u00a0<\/p>\n<p>For a <strong>decreasing function<\/strong>, the slope is negative.\n<\/p>\n<p style=\"text-align: center;\">\n<p style=\"text-align: center;\"><strong><em>The output values decrease as the input values increase. <\/em><\/strong><\/p>\n<p>\u00a0<\/p>\n<p>A line with a negative slope slants downward from left to right as in <strong>(b)<\/strong>. If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in <strong>(c)<\/strong>.<span id=\"fs-id1165137453957\" data-type=\"media\" data-alt=\"Three graphs depicting an increasing function, a decreasing function, and a constant function.\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1165137446154\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Increasing and Decreasing Functions<\/h3>\n<p id=\"fs-id1165134085973\">The slope determines if the function is an <strong>increasing linear function<\/strong>, a <strong>decreasing linear function<\/strong>, or a constant function.<\/p>\n<ul id=\"eip-643\" data-bullet-style=\"none\">\n<li>[latex]f\\left(x\\right)=mx+b\\text{ is an increasing function if }m>0[\/latex].<\/li>\n<li>[latex]f\\left(x\\right)=mx+b\\text{ is an decreasing function if }m<0[\/latex].<\/li>\n<li>[latex]f\\left(x\\right)=mx+b\\text{ is a constant function if }m=0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_02_01_02\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137405281\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135571684\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 2: Deciding whether a Function Is Increasing, Decreasing, or Constant<\/h3>\n<p id=\"fs-id1165137400625\">Some recent studies suggest that a teenager sends an average of 60 texts per day.<a class=\"footnote\" title=\"http:\/\/www.cbsnews.com\/8301-501465_162-57400228-501465\/teens-are-sending-60-texts-a-day-study-says\/\" id=\"return-footnote-1057-1\" href=\"#footnote-1057-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.<\/p>\n<ol id=\"fs-id1165137807449\" data-number-style=\"lower-alpha\">\n<li>The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.<\/li>\n<li>A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.<\/li>\n<li>A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137435336\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137508045\">Analyze each function.<\/p>\n<ol id=\"fs-id1165137778959\" data-number-style=\"lower-alpha\">\n<li>The function can be represented as [latex]f\\left(x\\right)=60x[\/latex] where [latex]x[\/latex] is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.<\/li>\n<li>The function can be represented as [latex]f\\left(x\\right)=500 - 60x[\/latex] where [latex]x[\/latex] is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after [latex]x[\/latex] days.<\/li>\n<li>The cost function can be represented as [latex]f\\left(x\\right)=50[\/latex] because the number of days does not affect the total cost. The slope is 0 so the function is constant.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137837055\" data-depth=\"1\">\n<div id=\"fs-id1165137410528\" class=\"note precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<\/div>\n<\/section>\n<section id=\"fs-id1165135208792\" data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1057\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1057-1\"><a href=\"http:\/\/www.cbsnews.com\/8301-501465_162-57400228-501465\/teens-are-sending-60-texts-a-day-study-says\/\" target=\"_blank\">http:\/\/www.cbsnews.com\/8301-501465_162-57400228-501465\/teens-are-sending-60-texts-a-day-study-says\/<\/a> <a href=\"#return-footnote-1057-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1057","chapter","type-chapter","status-publish","hentry"],"part":1048,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1057\/revisions"}],"predecessor-version":[{"id":2458,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1057\/revisions\/2458"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1048"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1057\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/media?parent=1057"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1057"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1057"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/license?post=1057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}