{"id":3969,"date":"2024-12-05T10:09:56","date_gmt":"2024-12-05T09:09:56","guid":{"rendered":"https:\/\/jhmaths.fr\/?page_id=3969"},"modified":"2025-09-23T17:17:18","modified_gmt":"2025-09-23T15:17:18","slug":"second-degre","status":"publish","type":"page","link":"https:\/\/jhmaths.fr\/index.php\/premiere-generale\/second-degre\/","title":{"rendered":"Second degr\u00e9"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center\">Fonction du second degr\u00e9<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-rich is-provider-prise-en-charge-des-contenus-embarqu-s wp-block-embed-prise-en-charge-des-contenus-embarqu-s wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Polyn\u00f4me du Second Degr\u00e9 2nde et Forme Canonique du Trin\u00f4me - Mathrix\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Slh5PcEnwHU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"propriete\"><strong>D\u00e9finition :<\/strong><br>Une fonction du second degr\u00e9 s&rsquo;\u00e9crit $f(x)=ax^2+bx+c$ o\u00f9 $a,b,c$ sont trois nombres donn\u00e9s.<\/p>\n\n\n\n<p><strong>Exemples :<\/strong><\/p>\n\n\n\n<p>$x^2+4$ ($a$ vaut 1, $b$ vaut 0 et $c$ vaut 4)<br>$3x^2-2x+5$ ($a$ vaut 3, $b$ vaut -2 et $c$ vaut 5)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><br>La parabole &#8211; \u00e9l\u00e9ments caract\u00e9ristiques<\/h2>\n\n\n\n<p class=\"propriete\"><strong>Propri\u00e9t\u00e9 :<br><\/strong>Une fonction du second degr\u00e9 $f : x \\mapsto ax^2+bx+c$ <br>est repr\u00e9sent\u00e9e sous la forme d&rsquo;une <strong>parabole<\/strong>, dont l&rsquo;orientation d\u00e9pend du signe de $a$.<br>Cette parabole poss\u00e8de un <strong>axe de sym\u00e9trie vertical<\/strong>  d&rsquo;\u00e9quation $x=-\\dfrac{b}{2a}$ qui passe par son <strong>sommet<\/strong> $S(\\alpha;\\beta)$.<\/p>\n\n\n\n<p class=\"propriete\"><strong>Remarque :<br><\/strong>On a donc :<br>$\\alpha=-\\dfrac{b}{2a}$<br>$\\beta=f\\left(\\alpha \\right)$<\/p>\n\n\n\n<p><strong>Exemples :<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>avec $a$ n\u00e9gatif (il vaut -3), sommet $S(2;4)$ et axe de sym\u00e9trie d&rsquo;\u00e9quation $x=2$<br><br><img decoding=\"async\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/parabole-avec-elemts-carac-300x155.jpg\" alt=\"\"><br><\/li>\n\n\n\n<li>avec (a) positif (il vaut 4), sommet (S(1;1)) et axe de sym\u00e9trie d&rsquo;\u00e9quation (x=1)<br><br><img decoding=\"async\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/parabole-avec-elemts-carac2-300x170.jpg\" alt=\"\"><\/li>\n<\/ul>\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Les 2 formes \u00e0 conna\u00eetre<\/h2>\n\n\n\n<div class=\"wp-block-group propriete has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\">\n<h3 class=\"wp-block-heading\">Forme canonique $f(x)=a(x-\\alpha)^2+\\beta$<\/h3>\n\n\n\n<p><strong>Utilit\u00e9 : <\/strong><br>Le sommet et l&rsquo;orientation de la courbe permettent de conna\u00eetre les variations de la fonction.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-group propriete has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\">\n<h3 class=\"wp-block-heading\">Forme factoris\u00e9e (lorsqu&rsquo;elle existe) : $f(x)=a(x-x_1)(x-x_2)$<\/h3>\n\n\n\n<p><strong>Utilit\u00e9 : <\/strong><br>Les racines et l&rsquo;orientation de la courbe permettent de conna\u00eetre le signe de la fonction.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"238\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/parabole-factorisee-300x238.jpg\" alt=\"\" class=\"wp-image-3211\" srcset=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/parabole-factorisee-300x238.jpg 300w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/parabole-factorisee.jpg 445w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n\n<div style=\"height:133px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading has-text-align-center\">Quizz<\/h2>\n\n\n\n<p>\\(P_1\\), \\(P_2\\), \\(P_3\\), \\(P_4\\) repr\u00e9sentent quatre fonctions polyn\u00f4mes de degr\u00e9 2.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"280\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/quizz-paraboles-55-300x280.jpg\" alt=\"\" class=\"wp-image-3214\" srcset=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/quizz-paraboles-55-300x280.jpg 300w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/quizz-paraboles-55-558x520.jpg 558w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/10\/quizz-paraboles-55.jpg 645w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n\n\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-55\" class=\"h5p-iframe\" data-content-id=\"55\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"1 Techno 2nd degr\u00e9 - parabole\"><\/iframe><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Fonction du second degr\u00e9 D\u00e9finition :Une fonction du second degr\u00e9 s&rsquo;\u00e9crit $f(x)=ax^2+bx+c$ o\u00f9 $a,b,c$ sont trois nombres donn\u00e9s. Exemples : $x^2+4$ ($a$ vaut 1, $b$ vaut 0 et $c$ vaut 4)$3x^2-2x+5$ ($a$ vaut 3, $b$ vaut -2 et $c$ vaut 5) La parabole &#8211; \u00e9l\u00e9ments caract\u00e9ristiques Propri\u00e9t\u00e9 :Une fonction du second degr\u00e9 $f : x [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3972,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"folder":[47],"class_list":["post-3969","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3969","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/comments?post=3969"}],"version-history":[{"count":2,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3969\/revisions"}],"predecessor-version":[{"id":4101,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3969\/revisions\/4101"}],"up":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3972"}],"wp:attachment":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/media?parent=3969"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/folder?post=3969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}