Content code
m1121
Slug (identifier)
the-properties-of-a-linear-function
Grades
Secondary III
Secondary IV
Topic
Mathematics
Tags
domaine
propriétés
affine
contexte
taux de variation
croissante
propriétés droite
propriété droite
propriété fonction linéaire
propriétées fonction linéaire
propriétés fonction affine
propriété fonction affine
Content
Contenu
Corps

In the following animation, experiment with the values of parameters |a| and |b| of the first-degree polynomial function (linear function) and observe the effect on the function’s properties.

Corps

Properties

Linear function of the form |y=ax+b| 

Domain

 |\mathbb{R}| or depending on the context 

Range (codomain)

|\mathbb{R}| or depending on the context

|x|-intercept

|\displaystyle x = \frac{-b}{a}| or replace |y| with |0|, and then isolate |x| 

Sign of the function

If |a>0| , the function is negative over |(-∞,\frac{-b}{a}]| and positive over |[\frac{-b}{a},+∞).|

If |a<0| , the function is positive over |(-∞,\frac{-b}{a}]| and negative over |[\frac{-b}{a},+∞).|  

|y|-intercept

The value of |b|. 

Extrema

None, or depending on the context.

Increasing intervals

If the slope is positive |(a>0),| the function is increasing over its entire domain.

Decreasing intervals

 If the slope is negative |(a<0),| the function is decreasing over its entire domain.

Content
Corps

Determine the various properties of the function |y = 2x + 1|.

It is very useful to draw a graph to help determine the properties.

Image
Graph
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  • The domain of the function is the set of real numbers, denoted |\mathbb{R}|, since the function has no restrictions on its domain.

  • The range of the function is the set of real numbers, denoted |\mathbb{R}|, since the function outputs every real number.

  • The |x|-intercept of the function is calculated as follows: |\displaystyle x = \frac{-b}{a} = \frac{-1}{2}|. It is possible to replace |y| with |0|, and then isolate |x|.
    ||\begin{align} 0 &= 2x + 1 \\ -1 &= 2x \\ \displaystyle \frac{-1}{2} &= x \end{align}||

  • The sign of the function is negative over |(-∞,\frac{-1}{2}]| and positive over |[\frac{-1}{2},+∞).|

  • The |y|-intercept is |b = 1.| 

  • The function has no extrema.

  • The function is increasing, because |a>0.|

Contenu
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Pour valider ta compréhension des propriétés des fonctions de façon interactive, consulte la MiniRécup suivante :

MiniRécup
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