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

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


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)


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.


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  \,axThe 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 :

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

Curve of a linear function


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

II. Linear functions and percentages

1. percentages of increase and decrease

  • 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}.

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\,=500i.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


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.

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

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