{"id":1533,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1533"},"modified":"2017-04-03T15:03:38","modified_gmt":"2017-04-03T15:03:38","slug":"key-concepts-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-3\/","title":{"raw":"Key Concepts","rendered":"Key Concepts"},"content":{"raw":"<section id=\"fs-id1165137661989\" class=\"key-equations\" data-depth=\"1\"><h2 data-type=\"title\">Key Equations<\/h2>\r\n<table id=\"fs-id2055298\" summary=\"...\"><tbody><tr><td>General Form for the Translation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr><\/tbody><\/table><\/section><section id=\"fs-id1165137447701\" class=\"key-concepts\" data-depth=\"1\"><h2 data-type=\"title\">Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137447708\"><li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em data-effect=\"italics\">y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain [latex]\\left(-\\infty , \\infty \\right)[\/latex], range [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote [latex]y=0[\/latex].<\/li>\r\n\t<li>If [latex]b&gt;1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\r\n\t<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n\t<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a&gt;0[\/latex], represents a vertical stretch if [latex]|a|&gt;1[\/latex] or compression if [latex]0&lt;|a|&lt;1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n\t<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\r\n\t<li>All translations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\r\n\t<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\r\n<\/ul><\/section>","rendered":"<section id=\"fs-id1165137661989\" class=\"key-equations\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Equations<\/h2>\n<table id=\"fs-id2055298\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137447701\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165137447708\">\n<li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em data-effect=\"italics\">y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain [latex]\\left(-\\infty , \\infty \\right)[\/latex], range [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote [latex]y=0[\/latex].<\/li>\n<li>If [latex]b>1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\n<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\n<li>The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a>0[\/latex], represents a vertical stretch if [latex]|a|>1[\/latex] or compression if [latex]0<|a|<1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n<li>All translations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\n<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\n<\/ul>\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-1533\">\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":4,"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-1533","chapter","type-chapter","status-publish","hentry"],"part":1518,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1533","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1533\/revisions"}],"predecessor-version":[{"id":3002,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1533\/revisions\/3002"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1518"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1533\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1533"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1533"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1533"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}