Recall on the punctual
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Function y = ax |
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Created it, 05/10/15
Update it, 05/12/29
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CONCEPT OF FUNCTION “1st PART”
This fourth lesson of mathematics is devoted to the examination of some functions.
We will first of all see what it is necessary to understand by the term “function”. That is the subject of the first paragraph.
This acquired concept, we will make a recall on the punctual coordinates. Indeed, the functions being represented graphically, it is necessary to trace them, to know to locate in a plan some their particular points. After having re-examined what into arithmetic one directly calls of the sizes (or conversely) proportional, we will approach in the other paragraphs the examination of the functions themselves.
As we already said, our lessons of mathematics do not constitute a course of mathematics strictly speaking, but contain knowledge necessary to the good comprehension of the other theoretical lessons. This is why the studied functions are those which are most often related to certain electric laws, electrotechnical or electronic.
We endeavoured to give an example of application, either in the theory itself, or of the complementary examples. As always, we invite you to remake each example.
Often, the law of dependence between two sizes can be specified. Thus, when a pedestrian walks at the constant speed of 6 km/h, the distance covered depends on the time during which it went. If one represents speed by the symbol “v”, the distance covered by “d”, the time of the course by “t”, one can write :
Knowing that v = 6 and knowing time “t”, one can calculate the distance “d”. It is said that the distance is a function of time.
We know that at the boundaries of a resistance R given, the tension U existing on its terminals depends on intensity I of the current crossing this resistance. We know the relation which binds these three sizes :
We will say that the tension U is, for a resistance R given, according to intensity I of the current. By definition, one will say :
A size is a known function of another size when, knowing a value of the second, one can calculate the corresponding value of the first.
Numerical example applied to the relation :
Let us generalize our formula of the law of Ohm in a mathematical form.
As we had agreed that the value of resistance was constant, we replace the letter “R” by “a”. As long as with the two other sizes U and I, respectively replace them by y and x. The relation U = RI can then be written :
Thus, to each value of x corresponds a value of y there which one can calculate. It is still said that the number y there according to number x.
When a number is related to another one can, knowing one, to calculate the other.
Number x, to which one gives the values that one wants, is called the variable. The number y, which one calculates, is called the function.
Mathematically, to express that a size (y) is related to a variable x, one writes :
who is read : y equal f of x or is related to x there.
2. - RECALL ON
THE PUNCTUAL COORDINATES
Let us trace two directed perpendicular axes (one says also rectangular) x'Ox and y'Oy (figure 1).
2. 1. - RECALL OF DEFINITIONS
The axis x'x is called the x-axis or x axis ; the axis y'y called there the y-axis or axis of the y.
The x-axis and that of the ordinates form the axes of co-ordinates. Item 0 is the origin of the co-ordinates.
2. 2. - POSITIONING OF A POINT COMPARED TO ITS CO-ORDINATES
With a couple of two unspecified algebraic numbers, example x = - 4 and y = 5, corresponds a point A of the plan which one obtains in the following way :
It is said that the numbers - 4 and + 5 are the co-ordinates of point A ; - 4 is its X-coordinate, + 5 is its ordinate.
2. 3. - DEFINITION OF the PUNCTUAL COORDINATES LOCATES IN the PLAN
That is to say a point B (figure 1) located in the plan.
By this point, one even parallels with the axes of the co-ordinates. The intersection of these parallels with the various axes determines the co-ordinates of the point.
Thus, figure 1, the point B has as co-ordinates x = 6 and y = - 3, which is written : B (6, - 3).
2. 4. - SCALE OF THE GRADUATIONS
To simplify, we carried divisions having even length on the two axes. However, one can relate to one of the axes of divisions length different from that related to the other, i.e. to adopt different scales for the two graduations.
For example, let us take again the representation of the law of OHM which we saw in the preceding lesson.
That is to say the circuit of the figure 2-a. A generator outputs in a fixed resistance of 100 ohms. If the power provided by the generator takes successively the values of 100, 200, 300… volts, the current corresponding will have as a value 1, 2, 3… amps (application of the relation I = U / R).
The amp and the volt are two corresponding units. If one adopts for scale 1 cm = 1 amp, it would be unreasonable to take 1 cm = 1 volt because that would impose a graph of several meters (100 V = 100 cm = 1 m, 200 V = 200 cm = 2 m, etc). Consequently, to represent this relation graphically, one will adopt like scale 1 cm = 100 volts and 1 cm = 1 amp. Moreover, on the same axis, one can find two scales different.
Let us take the example where we must measure the intensity of the current crossing a receiver R according to the value and of the polarities of the tension applied on its terminals (the current is the function, the tension the variable). That is to say the assembly of the figure 3-a.
Let us suppose that measurements of the tension U give values of a few volts and that one records the corresponding values of the I1 current about the milliampere.
We inform the graph appears 3-b by deferring the tensions on the positive x axis (1 cm = 1 V) and the intensities on the positive y axis (1 cm = 1 mA).
Let us modify the assembly as indicates it the figure 3-c by the inversion of the polarities of the generator and the important increase in the value of the resistance which one will call R'.
As previously, tensions of a few volts are always measured but the polarities are reversed. If one agrees for example to take the point B like potential of reference, point A became negative compared to this point B. We thus have negative tensions and their values are to be deferred on the negative x axis with the same scale, 1 cm = 1 volt.
One measures the corresponding intensities of the I2 current which are this time of a few microamperes and a contrary direction to that of I1. We thus defer these values on the negative y axis and one adopts like scale 1 cm = 1 µA, i.e. a scale 1 000 times larger than previously. We have as follows :
3. - CHART
OF SIZES DIRECTLY PROPORTIONAL
Definition : Two sizes are directly proportional when the various values of the one are proportional to the corresponding values of the other.
We saw for example that the distance covered by a pedestrian going at a constant speed depended on the time during which it went: the distance covered is proportional to the up time. In the same way, the intensity of the current crossing a receiver of determined resistance, depends on the tension applied at the boundaries of this receiver. Let us make a numerical application of this last example.
That is to say a receiver of resistance of 10 ohms at the boundaries of which one applies a tension which will take successively the values of 10 V, 20 V, 30 V, etc… the intensity the beam will take the values of 1 respectively A, 2 A, 3 A, etc… since I = U / R.
The tension and the intensity are thus two sizes directly proportional because, when the values of the tension are multiplied by 2, 3, 4, etc… the values of the intensity are also multiplied by 2, 3, 4, etc…
It is noticed that the quotient of two corresponding values of the sizes considered is constant. One has indeed :
This quotient (10 in our example) is called proportionality factor. More generally, “y” and “x” being corresponding measurements of two sizes proportional, “a” the proportionality factor, these measurements are bound by the relation :
In other words, two sizes directly proportional when the measurement of “y” is obtained by multiplying the corresponding measurement of the other “x” by a constant number, are called proportionality factor.
4. - FUNCTION y = ax
4. 1. - CHART
Let us take again our receiver of 10 ohms resistance and make traverse it by a current to which we will give arbitrary values. Let us calculate the various corresponding values of the tension which one will represent graphically.
The terminal voltage of the receiver is given by the relation :
We can thus write following equivalences :
Let us give to x (Thus I) various values and calculate the corresponding values of y (thus U).
|
Arbitrary values of x (I) |
1 |
2 |
3 |
4 |
5 |
|
Computed values of y (U = RI) |
10 |
20 |
30 |
40 |
50 |
Let us trace two axes of co-ordinates, 0x and 0y (figure 4).
As it should be let us graduate 0x in amps and 0y in volts while adopting like scale 1 A = 1 cm and 10 V = 1 cm. Let us place points A (1,10) ; B (2,20) ; C (3,30) ; D (4,40) ; E (5,50).
It is noted that these points are on the same
line 
(delta)
which passes by origin 0 and which one calls
right-hand side representative of the function y = 10 x
from where the rule :
The curve representative of the function y = ax is a line passing by the origin.
We point out that in mathematics, the “curved” word is synonymous with “line”. Be thus not shocked if it is said to you that a “curve” is a “line”.
4. 2. - LAYOUT PRACTICES LINE REPRESENTING THE FUNCTION y = ax
As we have just seen it, the line y = ax passes by origin 0.
Like positioning a line, it is enough to know two of its points, it remains us to determine a second point, the first being known (origin 0). One then takes an arbitrary value of x which one multiplies by the proportionality factor. The product obtained gives us the ordinate of the second point, the X-coordinate being obviously the arbitrary value of x selected.
By preserving the preceding data, one takes for example x = 4 from where y = 10 x 4 = 40 and one obtains the point P1 (4,40) (figure 5).
One joint then item 0 at the P1 point and one
obtains the line 1
which is the line of the function y = 10 x.
Note : For the precision of the drawing, it is advised to take a value of x largest possible, compatible with the size of the graph.
If we had chosen for x the arbitrary value
:
0,5 we would have obtained the point P2 (0,5 ; 7). It is enough
to a light error of positioning of this point so that the information given by
the reading of the curve is false. Thus, if we not locate P2 (figure 5)
with its exact co-ordinates (0,5 ; 7) but with a light error (0,5 ;
approximately 6,5), for the higher values of x, the absolute error
becomes important. While having plotted straight line 2
we see that to the value of x = 4 does not correspond any more y = 40
but 35. The error which corresponds well off 12,5 % is not
negligible.
4. 3. - GENERALIZATION
Let us represent on the same graph (figure 6) the functions y = ax following :
|
y = - 4x |
y = - x |
y = - 0,5x |
y = 0,5x |
y = x |
y = 3x |
To plot the corresponding straight lines, we must determine two points: one is already formed by the intersection of the axes x'x and y'y, it is item 0.
To obtain the second, let us give to x a numerical value :
|
y = - 4x |
for x = 2 |
y = - 4 (2) = - 8 = P1 |
|
y = - x |
for x = 5 |
y = 1 (- 5) = - 5 = P2 |
|
y = - 0,5x |
for x = 6 |
y = - 0,5 (6) = - 3 = P3 |
|
y = 0,5x |
for x = 6 |
y = 0,5 (6) = 3 = P4 |
|
y = x |
for x = 6 |
y = 1 (6) = 6 = P5 |
|
y = 3x |
for x = 2 |
y = 3 (2) = 6 = P6 |
These points being defined now perfectly by their co-ordinates, P1 (2 ; - 8) ; P2 (5 ; - 5) ; P3 (6 ; - 3) etc… we can place them in the plan and plot the straight lines representing each function by making them pass by origin 0 and the points P correspondents (figure 6).
Let us observe figure 6. We notice that in the function y = ax :
1 - When coefficient “a” is positive (a > 0 ) ;
- the line is in the first and the third quadrant,
- when x grows, y also grows : x and vary y in the same direction.
It is said whereas the function is increasing.
2 - When coefficient “a” is negative (a < 0 ) :
- the line is in the second and the fourth quadrant,
- when x grows, y grows (x and y vary in opposite direction).
It is said whereas the function is decreasing.
In short, the function y = ax is increasing for a > 0 and decreasing for a < 0.
4. 4. - SLOPE OF THE RIGHT-HAND SIDE y = ax
Continuously to observe figure 6, we make the following observation: when coefficient “a” increases in absolute value, the angle formed by the x-axis (x'x) and the corresponding line also increases. Thus, angle x 0 P6 (function y = 3 x) is higher than angle x 0 P5 (function y = x).
The coefficient “a” “x” in the relation y = ax is called slope or angular coefficient, of the right-hand side y = ax.
In short, the slope of the right-hand side y = ax increases like the absolute value of the coefficient a of x.
Note : One shows in trigonometry that “a” is the tangent of the angle formed by the x-axis with the line of the function considered.
If 
indicates this angle, one with the relation :
a = tg
(tg = tangent).
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