{"id":1438,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1438"},"modified":"2015-11-12T18:35:29","modified_gmt":"2015-11-12T18:35:29","slug":"key-concepts-glossary-40","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/key-concepts-glossary-40\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<section id=\"fs-id1165137659195\" class=\"key-equations\" data-depth=\"1\"><h1 data-type=\"title\">Key Equations<\/h1>\n<table id=\"eip-id1362369\" summary=\"..\"><tbody><tr><td data-valign=\"middle\" data-align=\"left\">Rational Function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{P\\left(x\\right)}{Q\\left(x\\right)}=\\frac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr><\/tbody><\/table><\/section><section id=\"fs-id1165137793507\" class=\"key-concepts\" data-depth=\"1\"><h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1165137603314\"><li>We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\\left(x\\right)=\\frac{1}{x}\\\\[\/latex] and [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}\\\\[\/latex].<\/li>\n\t<li>A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.<\/li>\n\t<li>Application problems involving rates and concentrations often involve rational functions.<\/li>\n\t<li>The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.<\/li>\n\t<li>The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.<\/li>\n\t<li>A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.<\/li>\n\t<li>A rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.<\/li>\n\t<li>Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.<\/li>\n\t<li>If a rational function has <em data-effect=\"italics\">x<\/em>-intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}\\\\[\/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\\dots ,{v}_{m}[\/latex], and no [latex]{x}_{i}=\\text{any }{v}_{j}\\\\[\/latex], then the function can be written in the form\u00a0[latex]f\\left(x\\right)=a\\frac{{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}}{{\\left(x-{v}_{1}\\right)}^{{q}_{1}}{\\left(x-{v}_{2}\\right)}^{{q}_{2}}\\cdots {\\left(x-{v}_{m}\\right)}^{{q}_{n}}}\\\\[\/latex]<\/li>\n<\/ul><div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165137758530\" class=\"definition\"><dt><strong>arrow notation<\/strong><\/dt><dd id=\"fs-id1165135154402\">a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value<\/dd><\/dl><dl id=\"fs-id1165135154407\" class=\"definition\"><dt><strong>horizontal asymptote<\/strong><\/dt><dd id=\"fs-id1165135154413\">a horizontal line <em>y\u00a0<\/em>= <em>b<\/em>\u00a0where the graph approaches the line as the inputs increase or decrease without bound.<\/dd><\/dl><dl id=\"fs-id1165135192626\" class=\"definition\"><dt><strong>rational function<\/strong><\/dt><dd id=\"fs-id1165134401081\">a function that can be written as the ratio of two polynomials<\/dd><\/dl><dl id=\"fs-id1165134401085\" class=\"definition\"><dt><strong>removable discontinuity<\/strong><\/dt><dd id=\"fs-id1165134401090\">a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function<\/dd><\/dl><dl id=\"fs-id1165137426312\" class=\"definition\"><dt><strong>vertical asymptote<\/strong><\/dt><dd id=\"fs-id1165137426317\">a vertical line <em>x\u00a0<\/em>= <em>a<\/em>\u00a0where the graph tends toward positive or negative infinity as the inputs approach\u00a0<em>a<\/em><\/dd><\/dl><\/div>\n<\/section>","rendered":"<section id=\"fs-id1165137659195\" class=\"key-equations\" data-depth=\"1\">\n<h1 data-type=\"title\">Key Equations<\/h1>\n<table id=\"eip-id1362369\" summary=\"..\">\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">Rational Function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{P\\left(x\\right)}{Q\\left(x\\right)}=\\frac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137793507\" class=\"key-concepts\" data-depth=\"1\">\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1165137603314\">\n<li>We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\\left(x\\right)=\\frac{1}{x}\\\\[\/latex] and [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}\\\\[\/latex].<\/li>\n<li>A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.<\/li>\n<li>Application problems involving rates and concentrations often involve rational functions.<\/li>\n<li>The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.<\/li>\n<li>The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.<\/li>\n<li>A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.<\/li>\n<li>A rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.<\/li>\n<li>Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.<\/li>\n<li>If a rational function has <em data-effect=\"italics\">x<\/em>-intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}\\\\[\/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\\dots ,{v}_{m}[\/latex], and no [latex]{x}_{i}=\\text{any }{v}_{j}\\\\[\/latex], then the function can be written in the form\u00a0[latex]f\\left(x\\right)=a\\frac{{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}}{{\\left(x-{v}_{1}\\right)}^{{q}_{1}}{\\left(x-{v}_{2}\\right)}^{{q}_{2}}\\cdots {\\left(x-{v}_{m}\\right)}^{{q}_{n}}}\\\\[\/latex]<\/li>\n<\/ul>\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165137758530\" class=\"definition\">\n<dt><strong>arrow notation<\/strong><\/dt>\n<dd id=\"fs-id1165135154402\">a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135154407\" class=\"definition\">\n<dt><strong>horizontal asymptote<\/strong><\/dt>\n<dd id=\"fs-id1165135154413\">a horizontal line <em>y\u00a0<\/em>= <em>b<\/em>\u00a0where the graph approaches the line as the inputs increase or decrease without bound.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135192626\" class=\"definition\">\n<dt><strong>rational function<\/strong><\/dt>\n<dd id=\"fs-id1165134401081\">a function that can be written as the ratio of two polynomials<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134401085\" class=\"definition\">\n<dt><strong>removable discontinuity<\/strong><\/dt>\n<dd id=\"fs-id1165134401090\">a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137426312\" class=\"definition\">\n<dt><strong>vertical asymptote<\/strong><\/dt>\n<dd id=\"fs-id1165137426317\">a vertical line <em>x\u00a0<\/em>= <em>a<\/em>\u00a0where the graph tends toward positive or negative infinity as the inputs approach\u00a0<em>a<\/em><\/dd>\n<\/dl>\n<\/div>\n<\/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-1438\">\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>","protected":false},"author":276,"menu_order":8,"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-1438","chapter","type-chapter","status-publish","hentry"],"part":1406,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1438","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\/1438\/revisions"}],"predecessor-version":[{"id":2372,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1438\/revisions\/2372"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1406"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1438\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/media?parent=1438"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1438"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1438"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/license?post=1438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}