{"id":1206,"date":"2015-11-12T18:35:31","date_gmt":"2015-11-12T18:35:31","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1206"},"modified":"2017-03-28T05:05:40","modified_gmt":"2017-03-28T05:05:40","slug":"introduction-to-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/introduction-to-quadratic-functions\/","title":{"raw":"Introduction to Quadratic Functions","rendered":"Introduction to Quadratic Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>LEARNING OBJECTIVES<\/h3>\r\nBy the end of this lesson, you will be able to:\r\n<ul>\r\n \t<li>Recognize characteristics of parabolas.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\r\n \t<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\r\n \t<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"Figure_03_02_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201246\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> <b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\r\nWe will see that quadratic functions can be written in the form\r\n<div id=\"fs-id1165137422466\" class=\"equation\" style=\"text-align: center\">$$f\\left(x\\right)=a{x}^{2}+bx+c$$<\/div>\r\n<p id=\"fs-id1165137544673\">where <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0.[\/latex] This is called the\u00a0<strong>general form of a quadratic equation.\u00a0<\/strong><\/p>\r\n<p id=\"fs-id1165134081264\">In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion.<\/p>\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this lesson, you will be able to:<\/p>\n<ul>\n<li>Recognize characteristics of parabolas.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\n<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"Figure_03_02_001\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><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\/25201246\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\n<p>We will see that quadratic functions can be written in the form<\/p>\n<div id=\"fs-id1165137422466\" class=\"equation\" style=\"text-align: center\">$$f\\left(x\\right)=a{x}^{2}+bx+c$$<\/div>\n<p id=\"fs-id1165137544673\">where <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0.[\/latex] This is called the\u00a0<strong>general form of a quadratic equation.\u00a0<\/strong><\/p>\n<p id=\"fs-id1165134081264\">In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion.<\/p>\n<p>&nbsp;<\/p>\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-1206\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Definition. <strong>Authored by<\/strong>: Ernesto D. Calleros. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><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":1,"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.\"},{\"type\":\"original\",\"description\":\"Definition\",\"author\":\"Ernesto D. Calleros\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1206","chapter","type-chapter","status-publish","hentry"],"part":1204,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1206\/revisions"}],"predecessor-version":[{"id":2752,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1206\/revisions\/2752"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1204"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1206\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=1206"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1206"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1206"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=1206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}