In the 7th and 8th grade, a schedule of direct proportionality is studied.
How to build a schedule of direct proportionality?
Consider the schedule of direct proportionality.
Graph line proportionality formula
The schedule of direct proportionality represents the function.
In general, direct proportionality has a formula
The angle of inclination of the direct proportionality chart relative to the X axis is depends on the magnitude and sign of the direct proportionality coefficient.
Schedule direct proportionality passes
The schedule of direct proportionality passes through the origin of the coordinates.
The schedule of direct proportionality is straight. Direct is set by two points.
Thus, when constructing a graph of direct proportionality, it suffices to determine the position of two points.
But we always know one of them is the beginning of the coordinates.
It remains to find the second. Let's see an example of building a direct proportion graph.
Build a schedule of direct proportionality Y \u003d 2X
A task .
Build a schedule of direct proportionality specified by the formula
Decision .
There are all numbers.
We take any number from the field of definition of direct proportionality, let it be 1.
Find the value of the function at X is equal to 1
Y \u003d 2x \u003d
2 * 1 = 2
that is, with x \u003d 1, we get y \u003d 2. The point with these coordinates belongs to the graph of the function y \u003d 2x.
We know that the schedule of direct proportionality is straight, and the direct is set by two points.
Linear function
Linear function - this is a function that can be set by the formula y \u003d kx + b,
where X is an independent variable, K and B - some numbers.
The graph of the linear function is straight.
The number K is called angular coefficient direct - Function graphics y \u003d kx + b.
If k\u003e 0, then the angle of the straight line y \u003d kx + b to the axis h. acute; If K.< 0, то этот угол тупой.
If the angular coefficients of direct, the graphs of two linear functions are different, then these direct intersect. And if the angular coefficients are the same, then direct are parallel.
Schedule function y \u003d.kX +.b.where k ≠ 0, there is a straight line, parallel straight y \u003d kx.
Direct proportionality.
Direct proportionality A function that can be specified by the y \u003d kx formula, where X is an independent variable, k is not zero number. The number K is called direct proportionate coefficient.
The direct patterns of direct proportion is a direct, passing through the origin of the coordinates (see Sinun).
Direct proportionality is a special case of a linear function.
Properties functiony \u003d.kX:
Inverse proportionality
Inverse proportionality It is called a function that can be set by the formula:
k.
y \u003d -
x.
where x. - independent variable, and k. - Not equal to zero number.
A graph of inverse proportionality is the curve called hyperboloic(See sinok).
For a curve that is a graph of this function, the axis x. and y. act as an asymptot. Asymptote - It is straight, to which the point of the curve is approaching as they are removed into infinity.
k.
Properties functiony \u003d -:
X.
Example
1.6 / 2 \u003d 0.8; 4/5 \u003d 0.8; 5.6 / 7 \u003d 0.8, etc.Proportionality coefficient
The unchanged relationship of proportional values \u200b\u200bis called coefficient of proportionality. The coefficient of proportionality shows how many units of one value are per unit another.
Direct proportionality
Direct proportionality - Functional dependence in which some value depends on another value in such a way that their relationship remains constant. In other words, these variables change proportionalIn equal shares, that is, if the argument has changed twice in any direction, then the function varies also twice in the same direction.
Mathematically direct proportion is written in the formula:
f.(x.) = a.x.,a. = c.o.n.s.t.
Inverse proportionality
Inverse proportionality - This is a functional dependence at which an increase in the independent value (argument) causes a proportional reduction in the dependent value (function).
Mathematically reverse proportion is written in the formula:
Properties function:
Sources
Wikimedia Foundation. 2010.
Example
1.6 / 2 \u003d 0.8; 4/5 \u003d 0.8; 5.6 / 7 \u003d 0.8, etc.Proportionality coefficient
The unchanged relationship of proportional values \u200b\u200bis called coefficient of proportionality. The coefficient of proportionality shows how many units of one value are per unit another.
Direct proportionality
Direct proportionality - Functional dependence in which some value depends on another value in such a way that their relationship remains constant. In other words, these variables change proportionalIn equal shares, that is, if the argument has changed twice in any direction, then the function varies also twice in the same direction.
Mathematically direct proportion is written in the formula:
f.(x.) = a.x.,a. = c.o.n.s.t.
Inverse proportionality
Inverse proportionality - This is a functional dependence at which an increase in the independent value (argument) causes a proportional reduction in the dependent value (function).
Mathematically reverse proportion is written in the formula:
Properties function:
Sources
Wikimedia Foundation. 2010.
Watch what is "direct proportionality" in other dictionaries:
direct proportionality - - [A.S.Goldberg. English Russian energy dictionary. 2006] Themes Energy as a whole en Direct Ratio ... Technical translator directory
direct proportionality - Tiesioginis Proporcingumas Statusas T Sritis Fizika Atitikmenys: Angl. Direct Proportionality Vok. Direkte proportionalität, f rus. Direct proportionality, F PRANC. Proportionnalité Directe, F ... Fizikos Terminų žodynas
- (from the lat. Proportionalis is proportional, proportional). Proportionality. A dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. The proportionality of delates. Proportionalis proportional. Proportionality. Explanation 25000 ... ... Dictionary of foreign words of the Russian language
Proportionality, proportionality, mn. No, wives (Book.). 1. Distractors. SUD To proportional. The proportionality of the parts. Proportionality of the physique. 2. Such a relationship between values \u200b\u200bwhen they are proportional (see proportional ... Explanatory Dictionary Ushakov
Proportional is two mutually dependent values, if the ratio of their values \u200b\u200bremains unchanged .. Content 1 Example 2 Proportional ratio ... Wikipedia
Proportionality, and, wives. 1. See proportional. 2. In mathematics: such a relationship between values, with a swarm, an increase in one of them changes the change in the other at the same time. Direct n. (With swarm with an increase in one value ... ... Explanatory dictionary of Ozhegov
AND; g. 1. To proportional (1 zn); proportionality. P. Parts. P. Body. P. Representative offices in parliament. 2. Mat. Dependence between proportionally changing values. Proportionality coefficient. Direct p. (At which with ... ... encyclopedic Dictionary
Proportionality is the relationship between two values \u200b\u200bat which the change in one of them entails the change in the other for the same time.
Proportionality is direct and reverse. In this lesson, we will look at each of them.
Design of lessonDirect proportionality
Suppose that the car moves at a speed of 50 km / h. We remember that the speed is the distance traveled per unit of time (1 hour, 1 minute or 1 second). In our example, the car moves at a speed of 50 km / h, that is, in one hour it will drive the distance equal to fifty kilometers.
Pictures in the picture the distance traveled by a car in 1 hour
Let the car drove another hour at the same speed equal to fifty kilometers per hour. Then it turns out that the car will drive 100 km
As can be seen from the example, the increase in time twice led to an increase in the distance traveled at the same time, that is, twice.
Such values \u200b\u200bas time and distance are called directly proportional. And the relationship between such values \u200b\u200bis called direct proportionality.
Direct proportionality is called the relationship between two values, at which an increase in one of them entails an increase in the other at the same time.
and on the contrary, if one value decreases in a certain number of times, the other decreases at the same time.
Suppose it was originally planned to travel by car 100 km in 2 hours, but driving 50 km, the driver decided to relax. Then it turns out that having reduced the distance twice, the time will decrease at the same time. In other words, the decrease in the distance traveled will lead to a decrease in time at the same time.
An interesting feature of direct proportional values \u200b\u200bis that their attitude is always constantly. That is, when changing the values \u200b\u200bof direct proportional values, their ratio remains unchanged.
In the considered example, the distance first was 50 km away, and time to one hour. The ratio of the distance to the time is the number 50.
But we increased the time of movement by 2 times, making it for two hours. As a result, the distance passed increased to the same time, that is, it became 100 km. The attitude of a hundred kilometers to two hours Again there is a number 50
The number 50 is called direct proportionate coefficient. It shows how much distance it comes from the hour of movement. In this case, the coefficient plays the role of the speed of movement, since the speed is the ratio of the distance towards the time.
Of course, proportional values \u200b\u200bcan be proportions. For example, relationships and make up the proportion:
Fifty kilometers so belong to one hour, as one hundred kilometers belong to two hours.
Example 2.. The cost and number of purchased goods are directly proportional to the values. If 1 kg of candy costs 30 rubles, then 2 kg of these same candies will cost 60 rubles, 3 kg in 90 rubles. With an increase in the cost of purchased goods, its number increases at the same time.
Since the cost of goods and its amount are directly proportional to the values, then their relationship is always constantly.
We write down the ratio of thirty rubles to one kilogram
Now we write down the ratio of sixty rubles to two kilograms. This ratio will be equal to thirty again:
Here, the ratio of direct proportionality is the number 30. This coefficient shows how many rubles falls on a kilogram of candy. In this example, the coefficient plays the role of the price of one kilogram of goods, since the price is the ratio of the value of the goods for its number.
Inverse proportionality
Consider the following example. The distance between the two cities is 80 km. The motorcyclist went from the first city, and at a speed of 20 km / h reached the second city in 4 hours.
If the speed of the motorcyclist amounted to 20 km / h, this means that every hour he drove a distance equal to twenty kilometers. I will shown the distance traveled by a motorcyclist, and the time of its movement:
On the way back the speed of the motorcyclist was 40 km / h, and on the same path he spent 2 hours.
It is easy to see that when changing the speed, the time of movement has changed at the same time. And changed in the opposite direction - that is, the speed has increased, and the time is the opposite decreased.
Such values \u200b\u200bas speed and time are called inversely proportional. And the relationship between such values \u200b\u200bis called inverse proportionality.
In reverse proportion is called the relationship between two values, in which an increase in one of them entails a decrease in the other at the same time.
and on the contrary, if one value decreases to a certain number of times, the other increases at the same time.
For example, if on the way back the speed of the motorcyclist would be 10 km / h, then the same 80 km would overcome in 8 hours:
As can be seen from the example, the reduction of the speed led to an increase in the time of movement at the same time.
The peculiarity of inverse proportional values \u200b\u200bis that their work is always constantly. That is, when changing the values \u200b\u200binversely proportional values, their product remains unchanged.
In the considered example, the distance between cities was equal to 80 km. When changing the speed and time of the motorcyclist movement, this distance has always remained unchanged
The motorcyclist could drive this distance at a speed of 20 km / h in 4 hours, and at a speed of 40 km / h in 2 hours, and at a speed of 10 km / h in 8 hours. In all cases, the product of speed and time was equal to 80 km
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