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:
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 :
- The image of 5 is: .
- The image of (-3) is: .
- The image of 1 is: .
These results can be grouped in a table:
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: .
2.graphical representation :
Let f be the linear function defined by : The set of points with coordinates 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 .
We say that this line has the equation: .
“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.
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 .
- Decreasing a number by t% is equivalent to multiplying the number by .
If a 400 g can is sold with 25% more product, its new mass (in g) is :
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 :
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|
|Example:||Take 5% of 20 :
||Increase 20 by 5%:
||Decrease 20 by 5%:
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 .
- For a reduction of k%, we have .
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