{"id":309222,"date":"2026-11-09T17:41:06","date_gmt":"2026-11-09T16:41:06","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=309222"},"modified":"2026-11-09T17:45:20","modified_gmt":"2026-11-09T16:45:20","slug":"polynome-second-degre","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/polynome-second-degre\/","title":{"rendered":"Fonction polyn\u00f4me du second degr\u00e9 : d\u00e9finition, parabole et racines"},"content":{"rendered":"

\ud83e\udde0 \u00c0 retenir :<\/p><\/div>

Une fonction polyn\u00f4me du second degr\u00e9 s\u2019\u00e9crit f(x) = ax\u00b2 + bx + c, avec a \u2260 0. Sa courbe est une parabole tourn\u00e9e vers le haut si a > 0, vers le bas sinon, de sommet d\u2019abscisse \u03b1 = \u2212b\/(2a). Son discriminant \u0394 = b\u00b2 \u2212 4ac donne 2, 1 ou 0 racines.<\/p><\/div><\/section>\n\n

Les fonctions polyn\u00f4mes du second degr\u00e9<\/strong> jouent un r\u00f4le central en math\u00e9matiques<\/a>, leur usage s\u2019\u00e9tendant de la simple r\u00e9solution d\u2019\u00e9quations du second degr\u00e9 \u00e0 des applications plus complexes comme l\u2019analyse des trajectoires paraboliques. Ces fonctions, au programme de sp\u00e9cialit\u00e9 math\u00e9matiques en premi\u00e8re, sont pr\u00e9sent\u00e9es ici \u00e0 travers leur d\u00e9finition, leurs propri\u00e9t\u00e9s graphiques, leurs variations et leurs sym\u00e9tries.<\/p>\n\n\n

D\u00e9finition des fonctions polyn\u00f4mes du second degr\u00e9<\/h2>\n\n\n

Une fonction polyn\u00f4me du second degr\u00e9 est d\u00e9finie sur \u211d par une expression de la forme f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi><\/math>, o\u00f9 a<\/strong>, b<\/strong> et c<\/strong> sont des constantes r\u00e9elles et a<\/mi>\u2260<\/mo>0<\/mn><\/math><\/strong>. Ce type de fonction est \u00e9galement connu sous le nom de fonction quadratique<\/strong>. L\u2019expression ax\u00b2 + bx + c, appel\u00e9e trin\u00f4me, peut s\u2019\u00e9crire sous forme d\u00e9velopp\u00e9e ou canonique, et sous forme factoris\u00e9e quand son discriminant est positif ou nul, ce qui facilite son \u00e9tude.<\/p>\n\n
\n
\n \n
\n
\n
\n \"Logo\n <\/div>\n
\n
\n
\n
\n \n <\/div>\n

Louise<\/p>

Mines ParisTech<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

24\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Nicolas<\/p>

CentraleSup\u00e9lec<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

27\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Fabien<\/p>

T\u00e9l\u00e9com Paris<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

20\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Cl\u00e9mence<\/p>

HEC Paris<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

21\u20ac\/h\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Bastien<\/p>

Polytechnique<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

26\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Pierre<\/p>

ESSEC<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

26\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Simon<\/p>

4e ann\u00e9e de m\u00e9decine<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

26\u20ac\/h<\/p> <\/div>\n <\/div>\n

\n
\n \n <\/div>\n

Jade<\/p>

Sciences Po Paris<\/p>

\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n \n \n <\/svg>\n <\/div>\n

21\u20ac\/h<\/p> <\/div>\n <\/div>\n <\/div>\n <\/div>\n<\/div>\n

\n
\n \"Logo\n <\/div>\n

Besoin d’un prof particulier<\/span> de maths ? \u2728<\/span><\/p>\n<\/div>\n

Nos Sherpas sont l\u00e0 pour t’aider \u00e0 progresser et prendre confiance en toi !<\/p>\n<\/div>\n

\n \n JE PRENDS UN COURS OFFERT !\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n <\/section>\n\n\n

Un polyn\u00f4me du second degr\u00e9 poss\u00e8de trois termes distincts :<\/p>\n\n

    \n
  • Le terme quadratique (a<\/mi>x<\/mi>2<\/mn><\/msup><\/math>)<\/li>\n
  • Le terme lin\u00e9aire (b<\/mi>x<\/mi><\/math>)<\/li>\n
  • Le terme constant (c<\/mi><\/math>)<\/li>\n<\/ul>\n\n

    Ces termes influencent la forme et la position de la courbe repr\u00e9sent\u00e9e par la fonction sur un graphique.<\/p>\n\n\n

    Forme canonique et forme factoris\u00e9e<\/h3>\n\n\n\n

    Toute fonction polyn\u00f4me du second degr\u00e9 a une forme canonique a(x \u2212 \u03b1)\u00b2 + \u03b2, o\u00f9 S(\u03b1 ; \u03b2) est son sommet, et, si \u0394 > 0, une forme factoris\u00e9e a(x \u2212 x\u2081)(x \u2212 x\u2082). Exemple : 2x\u00b2 \u2212 4x \u2212 6 = 2(x \u2212 1)\u00b2 \u2212 8 = 2(x + 1)(x \u2212 3).<\/p>\n\n\n