The course on affine functions with the definition, vocabulary and different properties of these functions is necessary for the student’s progress. Moreover, the latter must be able to study the representative curve and the direction of variation

The student should be able to calculate an image or an antecedent and also know how to draw the curve of an affine function using two distinct points. Also, he/she must develop skills on affine functions by knowing how to determine the value of the directing coefficient a and the y-intercept b.
We will end this lesson with concrete examples from everyday life in the third grade.

## I. Affine functions: definition and vocabulary.

Definition:

Let ” a ” and ” b ” be two fixed numbers, by associating to each number ” x ” a number ” ax + b ” called ” image of x “,

we define an affine function.

This function will be noted as follows: .

The image of x will be noted: g(x).

Example:

Let g be the affine function defined by: .

then :

• the image of 5 is: .
• the image of (-3) is: .
• the image of 0 is: .

Remark:

The function is the linear function associated with g.

A linear function is affine, the reciprocal is false.

If b=0, we obtain the associated linear function .

## II. Graphical representation of an affine function

Ownership:

Let g be the affine function defined by: .The set of points M with coordinates is called the graphical representation of the affine function.

In a reference frame, this representation is the line :

• parallel to the line representing the associated linear function.
• passing through the point with coordinates .

We say that this line has the equation: .

• “a” is the directing coefficient.
• “b” is theintercept. It indicates the “height” at which the line intersects the y-axis.

Remarks:

– If a = 0, the line of equation is parallel to the x-axis.

– Any line that is not parallel to the y-axis has an equation of the form y = ax + b, and is therefore an affine function.

## III. Direction of variation of an affine function

Ownership:

Let a and b be two relative numbers.

Let g be the affine function defined by .

• If a>0 then g is increasing.
• If a<0 then g is decreasing.

Example:

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