Usual functions: 2nd grade math course to download in PDF.

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1 I. Linear functions :
2 II. Affine functions:
3 III. The square function:
- 3.1 1.Definition:
- 3.2 2. graphic representation
4 IV. The cube function:
- 4.1 1.Definition:
- 4.2 2. graphic representation
5 V. The inverse function:
- 5.1 1.Definition:
- 5.2 2. graphic representation

A math course on usual functions in second grade is always beneficial and allows you to progress well throughout the year.

This math course on usual functions involves the following concepts:

definition of a numerical function;
image and antecedent;
set or domain of definition of a function;
affine and linear function;
square function;
square root function;
inverse function;
power function.

This math course on usual functions is available for free download in PDF format.

I. Linear functions :

1.Definition:

Definition:

A linear function is any function defined by :

$f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,ax$ where a is a given real.

2. graphic representation

Ownership:

In an orthonormal frame of reference, the graphical representation of a linear function defined on $\mathbb{R}$ by $f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,ax$ is the line D of equation y=ax passing through the origin of the frame of reference (a is a given real number).

Properties:

If a = 0, the linear function is the null function on $\mathbb{R}$ , we have for all x, f(x)=0.
If a>0 , the linear function is strictly increasing on $\mathbb{R}$ .
If a<0 , the linear function is strictly decreasing on $\mathbb{R}$ .

3.Characteristic property of linear functions

Ownership:

If f is a linear function, then whatever the real numbers m and p, the rate of change between m and p is constant.

More precisely, if , then, whatever the real m and p are: $a=\frac{f(m)-f(p)}{m-p}$ .

This constant number a is the directing coefficient of the line D representative of the function f.

II. Affine functions:

1.Definition:

Definition:

We call affine function, any function defined by :

$f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,mx+p$ where m and p are given real numbers.

2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of an affine function defined on $\mathbb{R}$ by $f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,mx+p$ is the line D of equation y=mx+p where m and p are given real numbers.

Ownership:

If m = 0, the affine function is a constant function on $\mathbb{R}$ , we have for all x, f(x)=p.
If m>0 , the affine function is strictly increasing on $\mathbb{R}$ .
If m<0 , the affine function is strictly decreasing on $\mathbb{R}$ .

3.Characteristic property of affine functions

Ownership:

If f is an affine function, then whatever the real numbers a and b are, the rate of change between a and b is constant.

More precisely, if , then, whatever the real numbers a and b are: $m=\frac{f(b)-f(a)}{b-a}$ .

This constant number m is the directing coefficient of the line D representative of the function f.

The number p is called theintercept. We have p=f(0).

4. particular affine functions:

Properties:

If p=0 then the affine function is linear.

In this case f(x) is proportional x (m is the proportionality coefficient).

The graphs of linear functions are straight lines that pass through the origin of the reference frame. Their equation is: y=mx.

If m=0 then the affine function is constant. We have for all x, f(x)=p.

The graphs of constant functions are lines parallel to the x-axis. Their equation is: y=p.

III. The square function:

1.Definition:

Definition:

We call square function, any function defined by :

$f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,x^2$ .

2. graphic representation

Properties:

In an orthonormal plane, the graphical representation of the square function defined on $\mathbb{R}$ by $f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,x^2$ is the straight parabola of equation y=x^2 .

Ownership:

The square function is strictly increasing on $%5B0;+\infty%5B$ .
The square function is strictly decreasing on $%5D-\infty;0%5D$ .

IV. The cube function:

1.Definition:

Definition:

We call a cube function, any function defined by :

$f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,x^3$ .

2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of the cube function defined on $\mathbb{R}$ by $f:\mathbb{R}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,x^3$ is the curve of equation y=x^3 .

Ownership:

The cube function is strictly increasing on $%5B0;+\infty%5B$ .
The cube function is strictly increasing on $%5D-\infty;0%5D$ .

V. The inverse function:

1.Definition:

Definition:

We call inverse function, any function defined by :

$f:\mathbb{R^*}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,\frac{1}{x}$ .

2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of the inverse function defined on $\mathbb{R^*}$ by $f:\mathbb{R^*}\to\,\mathbb{R}\\\,\,\,\,\,\,\,x\,\mapsto \,\frac{1}{x}$ is the curve of equation $y=\,\frac{1}{x}$ .

Ownership:

The inverse function is strictly decreasing on $%5B0;+\infty%5B$ .
The cube function is strictly decreasing on $%5D-\infty;0%5D$ .

Cette publication est également disponible en : Français (French) العربية (Arabic)

Usual functions : maths course in 2nd grade to download in PDF.

I. Linear functions :

1.Definition:

2. graphic representation

3.Characteristic property of linear functions

II. Affine functions:

1.Definition:

2. graphic representation

3.Characteristic property of affine functions

4. particular affine functions:

III. The square function:

1.Definition:

2. graphic representation

IV. The cube function:

1.Definition:

2. graphic representation

V. The inverse function:

1.Definition:

2. graphic representation

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