{"id":1517,"date":"2024-05-24T20:26:23","date_gmt":"2024-05-24T20:26:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1517"},"modified":"2025-08-13T16:28:23","modified_gmt":"2025-08-13T16:28:23","slug":"piecewise-functions-and-rates-of-change-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/piecewise-functions-and-rates-of-change-learn-it-3\/","title":{"raw":"Rates of Change and Behavior of Graphs: Learn It 3","rendered":"Rates of Change and Behavior of Graphs: Learn It 3"},"content":{"raw":"<h2>Behaviors of Functions Cont.<\/h2>\r\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\r\nWe will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Increasing\/Decreasing<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Constant Function[latex]f\\left(x\\right)={c}[\/latex]<\/td>\r\n<td>Neither increasing nor decreasing<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12510\" align=\"alignnone\" width=\"143\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img class=\"wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194806\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a> Constant function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identity Function[latex]f\\left(x\\right)={x}[\/latex]<\/td>\r\n<td>\u00a0Increasing<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12511\" align=\"alignnone\" width=\"143\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img class=\"wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194807\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a> Identity Function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quadratic Function[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nMinimum at [latex]x=0[\/latex]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12512\" align=\"alignnone\" width=\"138\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img class=\"wp-image-12512 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194808\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"Quadratic function\" width=\"138\" height=\"143\" \/><\/a> Quadratic function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cubic Function[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td>Increasing<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12513\" align=\"alignnone\" width=\"137\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img class=\"wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194809\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a> Cubic function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Reciprocal[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12514\" align=\"alignnone\" width=\"138\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img class=\"wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194810\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a> Reciprocal function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal Squared[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]Decreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12515\" align=\"alignnone\" width=\"134\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img class=\"wp-image-12515 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194812\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a> Reciprocal squared function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cube Root[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n<td>Increasing<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12516\" align=\"alignnone\" width=\"140\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img class=\"wp-image-12516 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194813\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a> Cube root function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Square Root[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12517\" align=\"alignnone\" width=\"138\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img class=\"wp-image-12517 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a> Square root function[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Absolute Value[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nMinimum at [latex]x=0[\/latex]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12518\" align=\"alignnone\" width=\"135\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img class=\"wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a> Absolute value function[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"<h2>Behaviors of Functions Cont.<\/h2>\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\n<p>We will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Increasing\/Decreasing<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant Function[latex]f\\left(x\\right)={c}[\/latex]<\/td>\n<td>Neither increasing nor decreasing<\/td>\n<td>\n<figure id=\"attachment_12510\" aria-describedby=\"caption-attachment-12510\" style=\"width: 143px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194806\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a><figcaption id=\"caption-attachment-12510\" class=\"wp-caption-text\">Constant function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Identity Function[latex]f\\left(x\\right)={x}[\/latex]<\/td>\n<td>\u00a0Increasing<\/td>\n<td>\n<figure id=\"attachment_12511\" aria-describedby=\"caption-attachment-12511\" style=\"width: 143px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194807\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a><figcaption id=\"caption-attachment-12511\" class=\"wp-caption-text\">Identity Function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Quadratic Function[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Minimum at [latex]x=0[\/latex]<\/td>\n<td>\n<figure id=\"attachment_12512\" aria-describedby=\"caption-attachment-12512\" style=\"width: 138px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12512 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194808\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"Quadratic function\" width=\"138\" height=\"143\" \/><\/a><figcaption id=\"caption-attachment-12512\" class=\"wp-caption-text\">Quadratic function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Cubic Function[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td>Increasing<\/td>\n<td>\n<figure id=\"attachment_12513\" aria-describedby=\"caption-attachment-12513\" style=\"width: 137px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194809\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a><figcaption id=\"caption-attachment-12513\" class=\"wp-caption-text\">Cubic function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>\u00a0Reciprocal[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\n<td>\n<figure id=\"attachment_12514\" aria-describedby=\"caption-attachment-12514\" style=\"width: 138px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194810\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a><figcaption id=\"caption-attachment-12514\" class=\"wp-caption-text\">Reciprocal function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal Squared[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]Decreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td>\n<figure id=\"attachment_12515\" aria-describedby=\"caption-attachment-12515\" style=\"width: 134px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12515 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194812\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a><figcaption id=\"caption-attachment-12515\" class=\"wp-caption-text\">Reciprocal squared function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Cube Root[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td>Increasing<\/td>\n<td>\n<figure id=\"attachment_12516\" aria-describedby=\"caption-attachment-12516\" style=\"width: 140px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12516 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194813\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a><figcaption id=\"caption-attachment-12516\" class=\"wp-caption-text\">Cube root function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Square Root[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td>\n<figure id=\"attachment_12517\" aria-describedby=\"caption-attachment-12517\" style=\"width: 138px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12517 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a><figcaption id=\"caption-attachment-12517\" class=\"wp-caption-text\">Square root function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Absolute Value[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Minimum at [latex]x=0[\/latex]<\/td>\n<td>\n<figure id=\"attachment_12518\" aria-describedby=\"caption-attachment-12518\" style=\"width: 135px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a><figcaption id=\"caption-attachment-12518\" class=\"wp-caption-text\">Absolute value function<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":12,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":116,"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1517"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1517\/revisions"}],"predecessor-version":[{"id":7653,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1517\/revisions\/7653"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/116"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1517\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1517"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1517"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1517"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1517"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}