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What is direct proportionality? Linear function. Direct proportionality

Example

1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.

Proportionality factor

The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which some quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionately, in equal shares, that is, if the argument has changed twice in any direction, then the function also changes twice in the same direction.

Mathematically, direct proportionality is written as a formula:

f(x) = ax,a = const

Inverse proportionality

Inverse proportion- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

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I. Directly proportional values.

Let the value y depends on the size X. If with an increase X several times the size at increases by the same factor, then such values X And at are called directly proportional.

Examples.

1 . The quantity of the purchased goods and the cost of the purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, so many times more and paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer the path, how many times more time we will spend on it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than the other, then its mass will be 2 times larger)

II. The property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrary values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.

Task 1. For raspberry jam 12 kg raspberries and 8 kg Sahara. How much sugar will be required if taken 9 kg raspberries?

Solution.

We argue like this: let it be necessary x kg sugar on 9 kg raspberries. The mass of raspberries and the mass of sugar are directly proportional: how many times less raspberries, the same amount of sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:

12: 9=8: X;

x=9 · 8: 12;

x=6. Answer: on 9 kg raspberries to take 6 kg Sahara.

The solution of the problem could have been done like this:

Let on 9 kg raspberries to take x kg Sahara.

(The arrows in the figure are directed in one direction, and it does not matter up or down. Meaning: how many times the number 12 more number 9 , the same number 8 more number X, i.e., there is a direct dependence here).

Answer: on 9 kg raspberries to take 6 kg Sahara.

Task 2. car for 3 hours traveled distance 264 km. How long will it take him 440 km if it travels at the same speed?

Solution.

Let for x hours the car will cover the distance 440 km.

Answer: the car will pass 440 km in 5 hours.

The concept of direct proportionality

Imagine that you are thinking of buying your favorite candy (or whatever you really like). The sweets in the store have their own price. Suppose 300 rubles per kilogram. The more candies you buy, the more money pay. That is, if you want 2 kilograms - pay 600 rubles, and if you want 3 kilos - give 900 rubles. Everything seems to be clear with this, right?

If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the ratio of two quantities that depend on each other. And the ratio of these quantities remains unchanged and constant: by how many parts one of them increases or decreases, by the same number of parts the second increases or decreases proportionally.

Direct proportionality can be described by the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our candy example, the price is a constant, a constant. It does not increase or decrease, no matter how many sweets you decide to buy. The independent variable (argument) x is how many kilograms of sweets you are going to buy. And the dependent variable f(x) (function) is how much money you end up paying for your purchase. So we can substitute the numbers in the formula and get: 600 r. = 300 r. * 2 kg.

The intermediate conclusion is this: if the argument increases, the function also increases, if the argument decreases, the function also decreases

Function and its properties

Direct proportional function is special case linear function. If the linear function is y = k*x + b, then for direct proportionality it looks like this: y = k*x, where k is called the proportionality factor, and this is always a non-zero number. Calculating k is easy - it is found as a quotient of a function and an argument: k = y/x.

To make it clearer, let's take another example. Imagine that a car is moving from point A to point B. Its speed is 60 km/h. If we assume that the speed of movement remains constant, then it can be taken as a constant. And then we write the conditions in the form: S \u003d 60 * t, and this formula is similar to the direct proportionality function y \u003d k * x. Let's draw a parallel further: if k \u003d y / x, then the speed of the car can be calculated, knowing the distance between A and B and the time spent on the road: V \u003d S / t.

And now, from the applied application of knowledge about direct proportionality, let's return back to its function. The properties of which include:

its domain of definition is the set of all real numbers (as well as its subset);

the function is odd;

the change in variables is directly proportional to the entire length of the number line.

Direct proportionality and its graph

A graph of a direct proportional function is a straight line that intersects the origin point. To build it, it is enough to mark only one more point. And connect it and the origin of the line.

In the case of a graph, this is slope. If the slope is less than zero (k< 0), то угол между графиком функции прямой пропорциональности и осью абсцисс тупой, а функция убывающая. Если угловой коэффициент больше нуля (k >0), the graph and the x-axis form sharp corner, and the function is increasing.

And one more property of the graph of the direct proportionality function is directly related to the slope k. Suppose we have two non-identical functions and, accordingly, two graphs. So, if the coefficients k of these functions are equal, their graphs are parallel on the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.

Task examples

Let's decide a couple direct proportionality problems

Let's start simple.

Task 1: Imagine that 5 hens laid 5 eggs in 5 days. And if there are 20 hens, how many eggs will they lay in 20 days?

Solution: Denote the unknown as x. And we will argue as follows: how many times have there been more chickens? Divide 20 by 5 and find out that 4 times. And how many times more eggs will 20 hens lay in the same 5 days? Also 4 times more. So, we find ours like this: 5 * 4 * 4 \u003d 80 eggs will be laid by 20 hens in 20 days.

Now the example is a little more complicated, let's rephrase the problem from Newton's "General Arithmetic". Task 2: A writer can write 14 pages of a new book in 8 days. If he had assistants, how many people would it take to write 420 pages in 12 days?

Solution: We reason that the number of people (writer + assistants) increases with the increase in the amount of work if it had to be done in the same amount of time. But how many times? Dividing 420 by 14, we find out that it increases by 30 times. But since, according to the condition of the task, more time is given for work, the number of assistants does not increase by 30 times, but in this way: x \u003d 1 (writer) * 30 (times): 12/8 (days). Let's transform and find out that x = 20 people will write 420 pages in 12 days.

Let's solve another problem similar to those that we had in the examples.

Task 3: Two cars set off on the same journey. One was moving at a speed of 70 km/h and covered the same distance in 2 hours as the other in 7 hours. Find the speed of the second car.

Solution: As you remember, the path is determined through speed and time - S = V *t. Since both cars traveled the same way, we can equate the two expressions: 70*2 = V*7. Where do we find that the speed of the second car is V = 70*2/7 = 20 km/h.

And a couple more examples of tasks with direct proportionality functions. Sometimes in problems it is required to find the coefficient k.

Task 4: Given the functions y \u003d - x / 16 and y \u003d 5x / 2, determine their proportionality coefficients.

Solution: As you remember, k = y/x. Hence, for the first function, the coefficient is -1/16, and for the second, k = 5/2.

And you may also come across a task like Task 5: Write down the direct proportionality formula. Its graph and the graph of the function y \u003d -5x + 3 are located in parallel.

Solution: The function that is given to us in the condition is linear. We know that direct proportionality is a special case of a linear function. And we also know that if the coefficients of k functions are equal, their graphs are parallel. This means that all that is required is to calculate the coefficient of a known function and set direct proportionality using the familiar formula: y \u003d k * x. Coefficient k \u003d -5, direct proportionality: y \u003d -5 * x.

Conclusion

Now you have learned (or remembered, if you have already covered this topic before), what is called direct proportionality, and considered it examples. We also talked about the direct proportionality function and its graph, solved a few problems for example.

If this article was useful and helped to understand the topic, tell us about it in the comments. So that we know if we could benefit you.

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Direct and inverse proportionality

If t is the time the pedestrian is moving (in hours), s is the distance traveled (in kilometers), and he moves uniformly at a speed of 4 km/h, then the relationship between these quantities can be expressed by the formula s = 4t. Since each value of t corresponds to a unique value of s, we can say that a function is given using the formula s = 4t. It is called direct proportionality and is defined as follows.

Definition. Direct proportionality is a function that can be specified using the formula y \u003d kx, where k is a non-zero real number.

The name of the function y \u003d k x is due to the fact that in the formula y \u003d kx there are variables x and y, which can be values of quantities. And if the ratio of two values \u200b\u200bis equal to some number other than zero, they are called directly proportional . In our case = k (k≠0). This number is called proportionality factor.

The function y \u003d k x is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described above. Another example: if there are 2 kg of flour in one package, and x such packages are bought, then the entire mass of the purchased flour (we denote it by y) can be represented as a formula y \u003d 2x, i.e. the relationship between the number of packages and the total mass of purchased flour is directly proportional with the coefficient k=2.

Recall some properties of direct proportionality, which are studied in the school course of mathematics.

1. The domain of the function y \u003d k x and the domain of its values is the set of real numbers.

2. The graph of direct proportionality is a straight line passing through the origin. Therefore, to construct a graph of direct proportionality, it is enough to find only one point that belongs to it and does not coincide with the origin, and then draw a straight line through this point and the origin.

For example, to plot the function y = 2x, it is enough to have a point with coordinates (1, 2), and then draw a straight line through it and the origin (Fig. 7).

3. For k > 0, the function y = kx increases over the entire domain of definition; for k< 0 - убывает на всей области определения.

4. If the function f is a direct proportionality and (x 1, y 1), (x 2, y 2) - pairs of corresponding values of the variables x and y, and x 2 ≠ 0 then.

Indeed, if the function f is a direct proportionality, then it can be given by the formula y \u003d kx, and then y 1 \u003d kx 1, y 2 \u003d kx 2. Since at x 2 ≠0 and k≠0, then y 2 ≠0. That's why and means .

If the values of the variables x and y are positive real numbers, then the proved property of direct proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y increases (decreases) by the same amount.

This property is inherent only in direct proportionality, and it can be used in solving word problems in which directly proportional quantities are considered.

Task 1. In 8 hours, the turner made 16 parts. How many hours will it take a turner to make 48 parts if he works at the same productivity?

Solution. The problem considers the quantities - the time of the turner, the number of parts made by him and productivity (i.e. the number of parts manufactured by the turner in 1 hour), the latter value being constant, and the other two taking various meanings. In addition, the number of parts made and the time of work are directly proportional, since their ratio is equal to a certain number that is not equal to zero, namely, the number of parts made by a turner in 1 hour. If the number of parts made is denoted by the letter y, the work time is x, and performance - k, then we get that = k or y = kx, i.e. the mathematical model of the situation presented in the problem is direct proportionality.

The problem can be solved in two arithmetic ways:

1 way: 2 way:

1) 16:8 = 2 (children) 1) 48:16 = 3 (times)

2) 48:2 = 24(h) 2) 8-3 = 24(h)

Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 2, and then, knowing that y \u003d 2x, we found the value of x, provided that y \u003d 48.

When solving the problem in the second way, we used the property of direct proportionality: how many times the number of parts made by a turner increases, the amount of time for their manufacture increases by the same amount.

Let us now turn to the consideration of a function called inverse proportionality.

If t is the time of the pedestrian's movement (in hours), v is his speed (in km/h) and he walked 12 km, then the relationship between these values can be expressed by the formula v∙t = 20 or v = .

Since each value of t (t ≠ 0) corresponds to a single value of velocity v, we can say that a function is given using the formula v = . It is called inverse proportionality and is defined as follows.

Definition. Inverse proportionality is a function that can be specified using the formula y \u003d, where k is a non-zero real number.

The name of this function comes from the fact that y= there are variables x and y, which can be values of quantities. And if the product of two quantities is equal to some number other than zero, then they are called inversely proportional. In our case, xy = k(k ≠ 0). This number k is called the coefficient of proportionality.

Function y= is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described before the definition of inverse proportionality. Another example: if you bought 12 kg of flour and put it in l: cans of y kg each, then the relationship between these quantities can be represented as x-y= 12, i.e. it is inversely proportional with the coefficient k=12.

Recall some properties of inverse proportionality, known from the school course of mathematics.

1. Function scope y= and its range x is the set of non-zero real numbers.

2. The inverse proportionality graph is a hyperbola.

3. For k > 0, the branches of the hyperbola are located in the 1st and 3rd quadrants and the function y= is decreasing on the entire domain of x (Fig. 8).

Rice. 8 Fig.9

When k< 0 ветви гиперболы расположены во 2-й и 4-й четвертях и функция y= is increasing over the entire domain of x (Fig. 9).

4. If the function f is inversely proportional and (x 1, y 1), (x 2, y 2) are pairs of corresponding values of the variables x and y, then.

Indeed, if the function f is inversely proportional, then it can be given by the formula y= ,and then . Since x 1 ≠0, x 2 ≠0, x 3 ≠0, then

If the values of the variables x and y are positive real numbers, then this property of inverse proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y decreases (increases) by the same amount.

This property is inherent only in inverse proportionality, and it can be used in solving word problems in which inversely proportional quantities are considered.

Problem 2. A cyclist, moving at a speed of 10 km/h, covered the distance from A to B in 6 hours.

Solution. The problem considers the following quantities: the speed of the cyclist, the time of movement and the distance from A to B, the latter value being constant, and the other two taking different values. In addition, the speed and time of movement are inversely proportional, since their product is equal to a certain number, namely the distance traveled. If the time of the cyclist's movement is denoted by the letter y, the speed is x, and the distance AB is k, then we get that xy \u003d k or y \u003d, i.e. the mathematical model of the situation presented in the problem is inverse proportionality.

You can solve the problem in two ways:

1 way: 2 way:

1) 10-6 = 60 (km) 1) 20:10 = 2 (times)

2) 60:20 = 3(4) 2) 6:2 = 3(h)

Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 60, and then, knowing that y \u003d, we found the value of y, provided that x \u003d 20.

When solving the problem in the second way, we used the inverse proportionality property: how many times the speed of movement increases, the time to travel the same distance decreases by the same amount.

Note that when solving specific problems with inversely proportional or directly proportional quantities, some restrictions are imposed on x and y, in particular, they can be considered not on the entire set of real numbers, but on its subsets.

Problem 3. Lena bought x pencils, and Katya bought 2 times more. Denote the number of pencils Katya bought as y, express y in terms of x, and plot the established correspondence graph, provided that x ≤ 5. Is this match a function? What is its domain of definition and range of values?

Solution. Katya bought u = 2 pencils. When plotting the function y=2x, it must be taken into account that the variable x denotes the number of pencils and x≤5, which means that it can only take on the values 0, 1, 2, 3, 4, 5. This will be the domain of this function. To get the range of this function, you need to multiply each value x from the domain of definition by 2, i.e. it will be a set (0, 2, 4, 6, 8, 10). Therefore, the graph of the function y \u003d 2x with the domain of definition (0, 1, 2, 3, 4, 5) will be the set of points shown in Figure 10. All these points belong to the line y \u003d 2x.

The concept of direct proportionality

Imagine that you are thinking of buying your favorite candy (or whatever you really like). The sweets in the store have their own price. Suppose 300 rubles per kilogram. The more candies you buy, the more money you pay. That is, if you want 2 kilograms - pay 600 rubles, and if you want 3 kilos - give 900 rubles. Everything seems to be clear with this, right?

The intermediate conclusion is this: if the argument increases, the function also increases, if the argument decreases, the function also decreases

Function and its properties

Direct proportional function is a special case of a linear function. If the linear function is y = k*x + b, then for direct proportionality it looks like this: y = k*x, where k is called the proportionality factor, and this is always a non-zero number. Calculating k is easy - it is found as a quotient of a function and an argument: k = y/x.

And now, from the applied application of knowledge about direct proportionality, let's return back to its function. The properties of which include:

its domain of definition is the set of all real numbers (as well as its subset);

the function is odd;

the change in variables is directly proportional to the entire length of the number line.

Direct proportionality and its graph

A graph of a direct proportional function is a straight line that intersects the origin point. To build it, it is enough to mark only one more point. And connect it and the origin of the line.

In the case of a graph, k is the slope. If the slope is less than zero (k< 0), то угол между графиком функции прямой пропорциональности и осью абсцисс тупой, а функция убывающая. Если угловой коэффициент больше нуля (k >0), the graph and the x-axis form an acute angle, and the function is increasing.

Task examples

Let's decide a couple direct proportionality problems

Let's start simple.

Task 1: Imagine that 5 hens laid 5 eggs in 5 days. And if there are 20 hens, how many eggs will they lay in 20 days?

Let's solve another problem similar to those that we had in the examples.

Task 3: Two cars set off on the same journey. One was moving at a speed of 70 km/h and covered the same distance in 2 hours as the other in 7 hours. Find the speed of the second car.

And a couple more examples of tasks with direct proportionality functions. Sometimes in problems it is required to find the coefficient k.

Task 4: Given the functions y \u003d - x / 16 and y \u003d 5x / 2, determine their proportionality coefficients.

Solution: As you remember, k = y/x. Hence, for the first function, the coefficient is -1/16, and for the second, k = 5/2.

And you may also come across a task like Task 5: Write down the direct proportionality formula. Its graph and the graph of the function y \u003d -5x + 3 are located in parallel.