{"id":1889,"date":"2024-06-18T18:03:43","date_gmt":"2024-06-18T18:03:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1889"},"modified":"2024-11-21T18:10:52","modified_gmt":"2024-11-21T18:10:52","slug":"introduction-to-power-and-polynomial-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-power-and-polynomial-functions-learn-it-1\/","title":{"raw":"Introduction to Power and Polynomial Functions: Learn It 1","rendered":"Introduction to Power and Polynomial Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Identify power functions and its end behavior.<\/li>\r\n \t<li>Identify polynomial functions and its degree and leading coefficient.<\/li>\r\n \t<li><span data-sheets-root=\"1\">Identifying local behavior of polynomial functions.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Power Functions<\/h2>\r\nHave you ever noticed how a small increase in the dimensions of an object can lead to a much larger increase in its volume? For example, doubling the side length of a cube results in a volume that's eight times larger!\r\n\r\nThis fascinating relationship is due to power functions, a special type of mathematical function that can model a variety of real-world phenomena, from the growth of populations to the way light diminishes over distance. A <strong>power function <\/strong>is a function with a single term that is the product of a real number,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a coefficient.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>power function<\/h3>\r\nA <strong>power function<\/strong> is a function that can be represented in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\r\nwhere [latex]a[\/latex]\u00a0and [latex]n[\/latex]\u00a0are real numbers and [latex]a[\/latex] is known as the <strong>coefficient<\/strong>.\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Unlike a polynomial function, in which all the variable powers must be non-negative integers, a power function only requires the power on the exponent be a real number.<\/section><section class=\"textbox example\">Which of the following functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill &amp; \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill &amp; \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill &amp; \\text{Quadratic}\\text{ }\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill &amp; \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill &amp; \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill &amp; \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt{x}\\hfill &amp; \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill &amp; \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"82786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"82786\"]\r\n\r\nAll of the listed functions are power functions.\r\n\r\nThe constant and identity functions are power functions because they can be written as [latex]f\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.\r\n\r\nThe quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].\r\n\r\nThe <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].\r\n\r\nThe square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?\r\n<\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]24598[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify power functions and its end behavior.<\/li>\n<li>Identify polynomial functions and its degree and leading coefficient.<\/li>\n<li><span data-sheets-root=\"1\">Identifying local behavior of polynomial functions.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Power Functions<\/h2>\n<p>Have you ever noticed how a small increase in the dimensions of an object can lead to a much larger increase in its volume? For example, doubling the side length of a cube results in a volume that&#8217;s eight times larger!<\/p>\n<p>This fascinating relationship is due to power functions, a special type of mathematical function that can model a variety of real-world phenomena, from the growth of populations to the way light diminishes over distance. A <strong>power function <\/strong>is a function with a single term that is the product of a real number,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a coefficient.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>power function<\/h3>\n<p>A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\n<p>where [latex]a[\/latex]\u00a0and [latex]n[\/latex]\u00a0are real numbers and [latex]a[\/latex] is known as the <strong>coefficient<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Unlike a polynomial function, in which all the variable powers must be non-negative integers, a power function only requires the power on the exponent be a real number.<\/section>\n<section class=\"textbox example\">Which of the following functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill & \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill & \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill & \\text{Quadratic}\\text{ }\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill & \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill & \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill & \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt{x}\\hfill & \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill & \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q82786\">Show Solution<\/button><\/p>\n<div id=\"q82786\" class=\"hidden-answer\" style=\"display: none\">\n<p>All of the listed functions are power functions.<\/p>\n<p>The constant and identity functions are power functions because they can be written as [latex]f\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.<\/p>\n<p>The quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].<\/p>\n<p>The <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].<\/p>\n<p>The square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<br \/>\n<\/strong><\/p>\n<hr \/>\n<p>No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm24598\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24598&theme=lumen&iframe_resize_id=ohm24598&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":206,"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\/1889"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1889\/revisions"}],"predecessor-version":[{"id":6212,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1889\/revisions\/6212"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/206"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1889\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1889"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1889"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1889"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1889"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}