Linear functions: 3rd grade math course to download.

sommaire

1 I. Linear functions :
2 II. Linear functions and percentages
- 2.1 1. percentages of increase and decrease
- 2.2 2. application of percentages to linear functions

A course on linear functions with the definition, vocabulary and properties as well as the study of percentages is always necessary for your progress. The student will need to have a good grasp of the concept of proportionality which leads to a linear function. Then, he/she must develop the skills to calculate an image or an antecedent, or to draw the curve of a linear function or to determine the value of the directing coefficient in third grade.

I. Linear functions :

1. definition and vocabulary

Definition:

Let “a” be a fixed number. By associating to each number ” x ” a number ” ax ” called ” image of x “, we define a linear function of coefficient a.

This function will be noted as follows: $f:x\,\mapsto \,ax$

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

x is called the antecedent of f(x)

Example:

Let f be the linear function of coefficient 2.

It is noted : $f:x\,\mapsto \,2x$

then :

The image of 5 is: $f(5)\,=\,2\times \,5\,=\,10$ .
The image of (-3) is: $f(-3)\,=\,2\,\times \,(-3)\,=\,-6$ .
The image of 1 is: $f(1)\,=\,2\,\times \,1\,=\,2$ .

Remark:

These results can be grouped in a table:

x	5	-3	1
f(x)	10	-6	2

This is a proportionality table. And the proportionality coefficient that allows to express f(x) as a function of x is 2 ! Hence the equality: $f(x)\,=\,2\,x$ .

2.graphical representation :

Property and vocabulary:

Let f be the linear function defined by : $f\,:\,x\,\mapsto \,ax$ The set of points with coordinates $(x\,;\,ax)$ is called the graphical representation of the linear function.

In a reference frame, this representation is the line passing through :

The origin of the benchmark.
The point with coordinates $(1\,;\,a)$ .

We say that this line has the equation: $y\,=\,ax$ .

“a” is the directing coefficient of the line. It indicates the “inclination” of the right.

3.direction of variation of a linear function :

Ownership:

If a>0 then the linear function is increasing;
If a<0 then the linear function is decreasing.

Remark:

If a = 0, the representation of the line merges with the x-axis.

II. Linear functions and percentages

1. percentages of increase and decrease

Ownership:

Increasing a number by t% is equivalent to multiplying the number by $k=1+\frac{t}{100}$ .
Decreasing a number by t% is equivalent to multiplying the number by $k=1-\frac{t}{100}$ .

Examples:
If a 400 g can is sold with 25% more product, its new mass (in g) is :

$m\,=\,400\,\times \,\,(\,1+\frac{4}{100}\,\,)\,=\,400\times \,1,25\,=500$ i.e. m = 500 g.

In France, a decrease of 4% was recorded on an annual number of 750 000 births.
The new workforce is :

$N\,=\,750\,000\,\times \,\,(\,1-\frac{4}{100}\,\,)\,=\,750\,000\times \,0,96=720\,000$ i.e. N = 720 000.

2. application of percentages to linear functions

	Take 5% of x.	Increase x by 5%.	Decrease x by 5%.
Calculation to be made	Multiply by 0.05	Multiply by 1.05	Multiply by 0.95
Linear function	$f\,:\,x\,\mapsto \,0,05\,x$	$g\,:\,x\,\mapsto \,1,05\,x$	$h\,:\,x\,\mapsto \,0,95\,x$
Example:	Take 5% of 20 : $f(20)\,=\,0,05\,\times \,20\,=\,1$	Increase 20 by 5%: $g(20)\,=\,1,05\,\times \,20\,=\,21$	Decrease 20 by 5%: $h(20)\,=\,0,95\,\times \,20\,=\,19$

Ownership:

In a general way, we can associate a linear function to any variation of k %.Let us note the function f which to the starting value x associates the value f(x) after variation.

For an increase of k%, we have $f(x)=\,(\,1+\frac{k}{100}\,\,)x$ .
For a reduction of k%, we have $f(x)=\,(\,1-\frac{k}{100}\,\,)x$ .

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Linear functions: 3rd grade math course to download in PDF.

I. Linear functions :

1. definition and vocabulary

2.graphical representation :

3.direction of variation of a linear function :

II. Linear functions and percentages

1. percentages of increase and decrease

2. application of percentages to linear functions

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