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Maths » Collège » Calcul numérique
Comment déterminer une fonction affine quand on connait deux antécédents et leurs images.

Comment déterminer une fonction affine connaissant deux images et leurs antécédents ?

Le problème.

On sait que la fonction recherchée est une fonction affine (sinon le problème n'a pas de sens) et on connaît deux antécédents et leurs images par cette fonction. Il s'agit de déterminer la fonction affine.

Les propriétés.

On sait qu'une fonction affine est, pour a et b données, le procédé qui à tout nombre x associe le nombre ax + b. Ce qu'on écrit
Une fonction affine est entièrement déterminée par la donnée de a (coefficient de proportionnalité) et b (image de 0) dans .
Le coefficient de proportionnalité ou taux d'accroissement d'une fonction affine est donné par la formule :

où et sont deux antécédents distincts.

La méthode.

Elle est similaire au calcul d'une équation de droite connaissant deux points.

On calcul le coefficient de proportionnalité qui est égal au taux d'accroissement.

Pour trouver l'image de 0 : b, on écrit une équation en se servant de la propriété 2 dans laquelle a est le coefficient calculé, et x un des deux antécédents donnés (n'importe lequel) et f(x) son image.

L'exemple.

Soit une fonction affine f. On donne et .

On calcule le coefficient a.

La fonction est telle que

On cherche maintenant à déterminer b.

Prenons et écrivons l'équation or f(2) = 1 ; on a donc :

La fonction affine f est donc

Last update: 2006-02-09 18:56
Author: Henry Serge

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Maths » Collège » Calcul numérique Comment déterminer une fonction affine quand on connait deux antécédents et leurs images.

Le problème.

Les propriétés.

La méthode.

L'exemple.

Maths » Collège » Calcul numérique
Comment déterminer une fonction affine quand on connait deux antécédents et leurs images.