Size proportional to the
square of another size |
Function
y = ax2 |
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Created it, 05/10/15
Update it, 05/12/29
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CONCEPT OF FUNCTION “2nd PART”
5. 1. - CONCRETE STUDY
Let us suppose that a cyclist rests at a given time of his excursion in a point P located at 20 km of a city 0. He takes again his road while moving away from this city 0 at the speed of 15 km / h.
a) With which distance from city 0 will be it at the end of 1 hour ? 2 hours ? 3 hours ? etc…
b) Which is the graph which can represent the distance covered according to time?
Solution :
a) We know that the distance covered by a person or an object who moves, one says a “mobile”, is a function the speed of this mobile and its time of displacement.
The
distance covered by the cyclist in one hour is 15 x 1 =
15 km. It is thus found far away from city 0
from 15 + 20 = 35 km (not P1
figure 7).
In
two hours, it traverses: 15 x 2 = 30 km. It
is found then 30 + 20 = 50 km of 0
(not P2).
Let us generalize : in x hours, the cyclist traverses 15 . x km. It is found then to 15 x + 20 km of 0.
Let us indicate by y the distance cyclist - city 0. The relation which connects this distance (y) to the time of course (x) can thus be written:
If numbers 15 are replaced and 20 by the letters has and B, called parameter, i.e. letters representing of the unspecified but known numbers, the relation becomes:
b) Let us represent the function y graphically = 15 x + 20. Let us take in scale 1 cm = 10 km and 1 cm = 1 hour.
Let us determine the points P1, P2, P3, etc… having for respective co-ordinates the corresponding values of x and y (figure 7).
|
Not P x = 0 |
y = (15 x 0) + 20 = 20 |
|
Not P1 x = 1 |
y = (15 x 1) + 20 = 35 |
|
Not P2 x = 2 |
y = (15 x 2) + 20 = 50 |
|
Not P3 x = 3 |
y = (15 x 3) + 20 = 65 etc…. |
By joining the points the ones to the others, one notes that they are aligned and that consequently the curve representative of the function y = 15x + 20 is a line.
5. 2. - REPRESENTATION OF the FUNCTION y = ax + b
While generalizing what has been just said, one can write :
The curve representative of the function y = ax + b is a straight line parallel with the right-hand side y = ax and cutting the y-axis at the point of ordinate b.
Remarks and definitions :
The
number b is called the ordinate in the
beginning (value of y for x
= 0),
The
line y = ax + b being parallel to the
right-hand side y = ax with the same slope;
parameter “a” is thus the slope (or
angular coefficient) of the right-hand side y = ax + b.
If
“a” is positive, the function y
= ax is increasing. The two lines being parallel, the function y
= ax + b is also increasing.
The
parameter “b” can be positive or
negative. If it is null, the equation is reduced to y =
ax.
For
the same reason, if “a” are negative,
the function y = ax and consequently the
function y = ax + b is decreasing.
In short, in the function y = ax + b :
5. 3. - LAYOUT PRACTICES LINE y = ax + b
To build the line y = ax + b, it is enough, of course, to determine two of its points and to join them. In general, one determines the points which result from the intersection of this line with the axes of co-ordinates :
one makes x = 0 and one calculates y ;
then y = 0 and one calculates x.
First example : That is to say the function y = - 2x + 4
Let us take as scale :
Let us trace the two axes (figure 8)
Let us pose x = 0 ;
the relation y = - 2x + 4 becomes :
Let us pose y = 0 ;
the relation y = - 2x + 4 becomes :
|
0 = - 2x + 4 |
from where |
2 x = 4 |
that is to say |
x = 2 (not P2) |
One joint the two points P1
and P2.
Second example: Let us represent graphically how varies the royalty to pay according to the consumed electric power, knowing that this one is tarifiée 48 centimes the kilowatt-hour and that the meter rental is of 250 frank. It is supposed that consumption varies between 0 and 1 200 kWh.
Resolution :
We should establish the relation existing between
these various data. It is obvious that the royalty, i.e. the paid price, depends,
or is a function, of consumption (a number of kWh)
and of the price of kWh. We can thus write :
Price to be paid = a number of consumed kWh x price of kWh
What is form : y = ax by allotting “y” to the price to be paid, “x” with the number of consumed kWh and “a”, known parameter, at the price of kWh i.e. 48 centimes.
It remains to take account of the meter rental who is fixed and independent of consumption. We have there the factor B and we can write : y = ax + b.
With y = assembling royalty
a = price of kWh (0,48 F)
x = a number of consumed kWh
b = meter rental (250 F)
and, numerically : y = 0,48x + 250
The relation being established, it is necessary for us to transpose graphically.
To trace and graduate the axes of co-ordinates, let us choose the scales :
- X-axis. We have a maximum consumption of 1200 kWh. By taking an axis of 10 cm, we obtain 1200 / 10 = 120 kWh per centimetre.
- Y-axis. To the maximum, the royalty will be of :
(0,48 x 1 200) + 250 = 576 + 250 = 826 frank.
With an axis of 10 cm, we obtain 826 / 10 = 82,6 F per centimetre. For more convenience, we will take 100 F per centimetre.
Let us trace and graduate the two axes (figure 9).
Let us determine the two points necessary to the layout of the right-hand side.
We have y = 0,48 x + 250
For x = 0 y = 250 (P1)
For y = 0 0 = 0,48 x + 250 - 0,48 x = 250
x = - 250 / 0,48 = - 520
We do not have this value on our graph. We could find it by prolonging the x axis towards the negative values. But let us make differently. Let us suppose that our consumption is 1200 kWh what corresponds to a royalty of :
We thus determined the co-ordinates of the second point: P2 (1 200, 826). The two points being found, it any more but does not remain to plot the straight line.
In electrical engineering or electronics, one meets laws whose general form is y = ax + b.
In the study of the transistors, one finds the relation:
IB + ICEO
6. - SIZE
PROPORTIONAL TO THE SQUARE OF ANOTHER SIZE
Definition : A size is proportional squared of another when one having a value, the other takes the square of this value.
Let us consider a resistance in which one runs a current I of various values. A wattmeter connected in the circuit enables us to measure the power P consumed by this resistance:
|
I |
1 |
2 |
3 |
4 |
5 |
|
P |
10 |
40 |
90 |
160 |
250 |
Now let us draw up the relationship between the powers and the square of the corresponding currents.
10 / 12 = 10 ; 40 / 22 = 40 / 4 = 10 ; 90 / 32 = 90 / 9 = 10 ; 160 / 42 = 160 / 16 = 10 ; 250 / 52 = 250 / 25 = 10
We notice that this report/ratio is constant. The value obtained in our example (10) is that of resistance R.
We can thus write :
Generally, if (x) is the measurement of a size and (y) the measurement of another size proportional to the square of the first, the report/ratio of proportionality “a” is expressed by the relation :
The definition of the beginning of the paragraph can thus be expressed as follows: a size (y) is proportional to the square of a size (x) when they are bound by the relation: y = ax2, “a” being a fixed coefficient.
7. - FUNCTION
y = ax2
7. 1. - FUNCTION y = x2
a = 1, the function y = ax2 is reduced to y = x2.
To study the function y graphically = x2, we will give to x positive and negative unspecified values. Each one of these values, high squared, will give the corresponding value of y.
Thus let us write the continuation of the integers and in lower part their squares :
|
x = |
- 5 |
- 4 |
- 3 |
- 2 |
- 1 |
0 |
1 |
2 |
3 |
4 |
5 |
|
y = x2 = |
25 |
16 |
9 |
4 |
1 |
0 |
1 |
4 |
9 |
16 |
25 |
By examining this table we note :
When the negative numbers increase, i.e. when they
approach zero, their squares decrease,
When the positive numbers increase, their squares
increase,
Two opposite numbers have the same square.
The results obtained are graphically represented figure 10 (the process to plot a curve now being supposed known, we will not explain it in detail any more).
We notice that the curve obtained is not any more one line like in the case of two sizes directly proportional. This curve is called a parabola.
In addition, one can observe that :
+),
one can always calculate his square x2 ;
to 0, the function is decreasing ;
when x varies from 0 with +,
it is increasing ;
The study of the variation of the function y = x2 can be summarized by the table according to whether we use still thereafter.
In the first line of this table, one finds
various values of X. We took here the extreme values -
and +
and zero value. In the study of functions a little more complicated,
one takes other particular values as we will see it. The slope of the arrows
gives the direction of variation of the numbers ranging between the two values
located on both sides arrow when one counts value located on the left of this
one to lead to that which is located on the right. Thus, when one leaves -
to arrive at zero, the relative values of the numbers increase as one
approaches 0 ; one summarizes that by an
arrow to which one gives an ascending slope. When one starts from 0 and that one
indefinitely continues to count, the values of the numbers also increase and the
arrow is still ascending.
The second line gives the values of the function
calculated according to those of the variable located in the first line. When
one leaves +
to arrive at zero, it is obvious that the values of the numbers are
increasingly small, from where the downward arrow.
On the contrary and like previously, when one counts of zero until the infinite one, the values increase and the arrow is ascending.
7. 2. - FUNCTION y = 2 x2
This function can be put in the form :
Thus, to calculate y, it is enough to multiply by 2 the values obtained for x2.
Let us make these calculations for some arbitrary values of x, register the results in the following table and symbolize by arrows, as we saw, the direction of the variations.
By analyzing the table, we see that for the values of :
- x varying
from -
to 0 : y
decrease of +
to 0
- x varying
of 0 with +
: y grows of 0
with +
The curve of figure 11 materializes the various results.
Note : One always does not carry in the tables intermediate calculations and values having been used to build the curve. The table is simplified and presented then in the form of this one :
7. 3. - FUNCTION y = (1 / 2) x2
As previously, let us start by drawing the picture of the study of the variations of (y) according to those of X.
The summarized table is as follows :
Let us plot the corresponding curve now (figure 12).
We see that the curve obtained is always a parabola.
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