Today we will look at what quantities are called inversely proportional, what an inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside of school.

Such different proportions

Proportionality name two quantities that are mutually dependent on each other.

The dependence can be direct and inverse. Consequently, the relationships between quantities are described by direct and inverse proportionality.

Direct proportionality– this is such a relationship between two quantities in which an increase or decrease in one of them leads to an increase or decrease in the other. Those. their attitude does not change.

For example, the more effort you put into studying for exams, the higher your grades. Or the more things you take with you on a hike, the heavier your backpack will be to carry. Those. The amount of effort spent preparing for exams is directly proportional to the grades obtained. And the number of things packed in a backpack is directly proportional to its weight.

Inverse proportionality– this is a functional dependence in which a decrease or increase by several times in an independent value (it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in a dependent value (it is called a function).

Let's illustrate with a simple example. You want to buy apples at the market. The apples on the counter and the amount of money in your wallet are in inverse proportion. Those. The more apples you buy, the less money you will have left.

Function and its graph

The inverse proportionality function can be described as y = k/x. In which x≠ 0 and k≠ 0.

This function has the following properties:

1. Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
2. The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
3. Does not have maximum or minimum values.
4. It is odd and its graph is symmetrical about the origin.
5. Non-periodic.
6. Its graph does not intersect the coordinate axes.
7. Has no zeros.
8. If k> 0 (i.e. the argument increases), the function decreases proportionally on each of its intervals. If k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
9. As the argument increases ( k> 0) negative values ​​of the function are in the interval (-∞; 0), and positive values ​​are in the interval (0; +∞). When the argument decreases ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).

The graph of an inverse proportionality function is called a hyperbola. Shown as follows:

Inverse proportionality problems

To make it clearer, let's look at several tasks. They are not too complicated, and solving them will help you visualize what inverse proportionality is and how this knowledge can be useful in your everyday life.

Task No. 1. A car is moving at a speed of 60 km/h. It took him 6 hours to get to his destination. How long will it take him to cover the same distance if he moves at twice the speed?

We can start by writing down a formula that describes the relationship between time, distance and speed: t = S/V. Agree, it reminds us very much of the inverse proportionality function. And it indicates that the time a car spends on the road and the speed at which it moves are in inverse proportion.

To verify this, let's find V 2, which, according to the condition, is 2 times higher: V 2 = 60 * 2 = 120 km/h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it’s not difficult to find out the time t 2 that is required from us according to the conditions of the problem: t 2 = 360/120 = 3 hours.

As you can see, travel time and speed are indeed inversely proportional: at a speed 2 times higher than the original speed, the car will spend 2 times less time on the road.

The solution to this problem can also be written as a proportion. So let's first create this diagram:

↓ 60 km/h – 6 h

↓120 km/h – x h

Arrows indicate an inversely proportional relationship. They also suggest that when drawing up a proportion, the right side of the record must be turned over: 60/120 = x/6. Where do we get x = 60 * 6/120 = 3 hours.

Task No. 2. The workshop employs 6 workers who can complete a given amount of work in 4 hours. If the number of workers is halved, how long will it take the remaining workers to complete the same amount of work?

Let us write down the conditions of the problem in the form of a visual diagram:

↓ 6 workers – 4 hours

↓ 3 workers – x h

Let's write this as a proportion: 6/3 = x/4. And we get x = 6 * 4/3 = 8 hours. If there are 2 times fewer workers, the remaining ones will spend 2 times more time doing all the work.

Task No. 3. There are two pipes leading into the pool. Through one pipe, water flows at a speed of 2 l/s and fills the pool in 45 minutes. Through another pipe, the pool will fill in 75 minutes. At what speed does water enter the pool through this pipe?

To begin with, let us reduce all the quantities given to us according to the conditions of the problem to the same units of measurement. To do this, we express the speed of filling the pool in liters per minute: 2 l/s = 2 * 60 = 120 l/min.

Since the condition implies that the pool fills more slowly through the second pipe, this means that the rate of water flow is lower. The proportionality is inverse. Let us express the unknown speed through x and draw up the following diagram:

↓ 120 l/min – 45 min

↓ x l/min – 75 min

And then we make up the proportion: 120/x = 75/45, from where x = 120 * 45/75 = 72 l/min.

In the problem, the filling rate of the pool is expressed in liters per second; let’s reduce the answer we received to the same form: 72/60 = 1.2 l/s.

Task No. 4. A small private printing house prints business cards. A printing house employee works at a speed of 42 business cards per hour and works a full day - 8 hours. If he worked faster and printed 48 business cards in an hour, how much earlier could he go home?

We follow the proven path and draw up a diagram according to the conditions of the problem, designating the desired value as x:

↓ 42 business cards/hour – 8 hours

↓ 48 business cards/h – x h

We have an inversely proportional relationship: the number of times more business cards an employee of a printing house prints per hour, the same number of times less time he will need to complete the same work. Knowing this, let's create a proportion:

42/48 = x/8, x = 42 * 8/48 = 7 hours.

Thus, having completed the work in 7 hours, the printing house employee could go home an hour earlier.

Conclusion

It seems to us that these inverse proportionality problems are really simple. We hope that now you also think of them that way. And the main thing is that knowledge about the inversely proportional dependence of quantities can really be useful to you more than once.

Not only in math lessons and exams. But even then, when you get ready to go on a trip, go shopping, decide to earn a little extra money during the holidays, etc.

Tell us in the comments what examples of inverse and direct proportional relationships you notice around you. Let it be such a game. You'll see how exciting it is. Don't forget to share this article on social networks so that your friends and classmates can also play.

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First level

Inverse relationship. First level.

Now we will talk about inverse dependence, or in other words - inverse proportionality, as a function. Do you remember that a function is a certain kind of dependency? If you haven't read the topic yet, I strongly recommend that you drop everything and read it, because you can't study any specific function without understanding what it is - a function.

It is also very useful to master two simpler functions before starting this topic: and . There you will reinforce the concept of a function and learn to work with coefficients and graphs.

So, do you remember what a function is?
Let us repeat: a function is a rule according to which each element of one set (argument) is associated with a certain ( the only one!) element of another set (set of function values). That is, if you have a function, this means that for every valid value of a variable (called an “argument”) there is a corresponding value of a variable (called a “function”). What does "acceptable" mean? If you can’t answer this question, return to the “” topic again! It's all in the concept "domain": For some functions, not all arguments are equally useful and can be substituted into dependencies. For example, for a function, negative argument values ​​are not allowed.

Function describing inverse relationship

This is a function of the form where.

In another way, it is called inverse proportionality: an increase in the argument causes a proportional decrease in the function.
Let's define the domain of definition. What can it be equal to? Or, in other words, what cannot it be equal to?

The only number that cannot be divided by is therefore:

or, what is the same,

(such a notation means that it can be any number, except: the “ ” sign denotes the set of real numbers, that is, all possible numbers; the “ ” sign denotes the exclusion of something from this set (analogous to the “minus” sign), and a number in curly brackets means just a number; it turns out that from all possible numbers we exclude).

The set of function values, it turns out, is exactly the same: after all, if, then no matter what we divide it by, it will not work:

Some variations of the formula are also possible. For example, this is also a function that describes an inverse relationship.
Determine the domain of definition and range of values ​​of this function yourself. It should look like this:

Let's look at this function: . Is it inversely related?

At first glance, it is difficult to say: after all, with an increase, both the denominator of the fraction and the numerator increase, so it is not clear whether the function will decrease, and if so, will it decrease proportionally? To understand this, we need to transform the expression so that there is no variable in the numerator:

Indeed, we received an inverse relationship, but with a caveat: .

Here's another example: .

It’s more complicated here: after all, the numerator and denominator now certainly do not cancel. But we can still try:

Do you understand what I did? In the numerator, I added and subtracted the same number (), so I didn't seem to change anything, but now there is a part in the numerator that is equal to the denominator. Now I will divide term by term, that is, I will split this fraction into the sum of two fractions:

(indeed, if we reduce what I got to a common denominator, we will get our initial fraction):

Wow! It works again inverse relationship, only now a number is added to it.
This method will be very useful to us later when constructing graphs.

Now transform the expressions yourself into an inverse relationship:

Answers:

2. Here you need to remember how a square trinomial is factorized (this is described in detail in the topic “”). Let me remind you that for this you need to find the roots of the corresponding quadratic equation: . I will find them verbally using Vieta's theorem: , . How it's done? You can learn this by reading the topic.
So, we get: , therefore:

3. Have you already tried to solve it yourself? What's the catch? Surely the fact is that we have in the numerator and in the denominator - it’s simple. It's no problem. We will need to reduce by, so in the numerator we should put it out of brackets (so that in brackets we get it without the coefficient):

Inverse relationship graph

As always, let's start with the simplest case: .
Let's make a table:

Let's draw points on the coordinate plane:

Now they need to be smoothly connected, but how? It can be seen that the points on the right and left sides form seemingly unconnected curved lines. The way it is. The graph will look like this:

This graph is called "hyperbola"(there's something like a "parabola" in that name, right?). Like a parabola, a hyperbola has two branches, only they are not connected to each other. Each of them strives with its ends to get closer to the axes and, but never reaches them. If you look at the same hyperbole from afar, you get the following picture:

This is understandable: since the graph cannot cross the axis. But also, so the graph will never touch the axis.

Well, now let's see what the coefficients influence. Let's consider these functions:
:

Wow, what a beauty!
All graphs are plotted in different colors to make it easier to distinguish them from each other.

So, what should we pay attention to first? For example, if a function has a minus before the fraction, then the graph is flipped, that is, displayed symmetrically relative to the axis.

Second: the larger the number in the denominator, the further the graph “runs away” from the origin.

What if the function looks more complex, for example, ?

In this case, the hyperbole will be exactly the same as the usual one, only it will shift a little. Let's think, where?

What can't it be equal to now? Right, . This means that the graph will never reach a straight line. What can't it be equal to? Now. This means that now the graph will tend to the straight line, but will never cross it. So, now the straight lines play the same role as the coordinate axes for the function. Such lines are called asymptotes(lines that the graph tends to but does not reach):

We will learn more about how such graphs are constructed in the topic.

Now try to solve a few examples to consolidate:

1. The figure shows a graph of a function. Define.

2. The figure shows the graph of the function. Define

3. The figure shows the graph of the function. Define.

4. The figure shows the graph of the function. Define.

5. The figure shows graphs of functions and.

Choose the correct ratio:

Answers:

Inverse dependence in life

Where do we find such a function in practice? There are many examples. The most common is movement: the greater the speed at which we move, the less time it will take us to cover the same distance. Indeed, let us remember the formula for speed: , where is speed, is travel time, is distance (path).

From here we can express time:

Example:

A person goes to work at an average speed of km/h and gets there in an hour. How many minutes will he spend on the same road if he drives at a speed of km/h?

Solution:

In general, you already solved such problems in 5th and 6th grade. You made up the proportion:

That is, the concept of inverse proportionality is already familiar to you. So we remembered. And now the same thing, only in an adult way: through a function.

Function (that is, dependence) of time in minutes on speed:

It is known that, then:

Need to find:

Now come up with a few examples from life in which inverse proportionality is present.
Invented? Well done if you do. Good luck!

REVERSE DEPENDENCE. BRIEFLY ABOUT THE MAIN THINGS

1. Definition

Function describing inverse dependence is a function of the form where.

In another way, this function is called inverse proportionality, since an increase in the argument causes a proportional decrease in the function.

or, what is the same,

The inverse graph is a hyperbola.

2. Coefficients, and.

Responsible for “flatness” and direction of the graph: the larger this coefficient, the further the hyperbola is located from the origin, and, therefore, it “turns” less steeply (see figure). The sign of the coefficient affects which quarters the graph is located in:

• if, then the branches of the hyperbola are located in and quarters;
• if, then in and.

x=a is vertical asymptote, that is, the vertical to which the graph tends.

The number is responsible for shifting the function graph upward by an amount if , and shifting it down if .

Therefore, this is horizontal asymptote.

1 lesson on the topic

Performed:

Telegina L.B.

The purpose of the lesson:

1. repeat all the material studied on functions.
2. introduce the definition of inverse proportionality and teach how to build its graph.
3. develop logical thinking.
4. cultivate attention, accuracy, precision.

Lesson plan:

1. Repetition.
2. Explanation of new material.
3. Physical education minute.
4. Consolidation.

Equipment: posters.

During the classes:

1. The lesson begins with repetition. Students are asked to solve a crossword puzzle (which is prepared in advance on a large sheet of paper).

Crossword questions:

1. Dependence between variables, in which each value of the independent variable corresponds to a single value of the dependent variable. [Function].

2. Independent variable. [Argument].

3. The set of points of the abscissa coordinate plane, which are equal to the values ​​of the argument, and the ordinates are equal to the values ​​of the function. [Schedule].

4. Function given by the formula y=kx+b. [Linear].

5. What coefficient is a number called? k in the formula y=kx+b? [Corner].

6. What is the graph of a linear function? [Straight].

7. If k≠0, then the graph y=kx+b intersects this axis, and if k=0, then it is parallel to it. What letter is this axis designated by? [X].

8. The word in the name of the function y=kx? [Proportionality].

9. Function given by the formula y=x 2. [Quadratic].

10. Name of the graph of a quadratic function. [Parabola].

11. A letter of the Latin alphabet, which often denotes a function. [Igrek].

12. One of the ways to specify a function. [Formula].

Teacher : What are the main ways of specifying a function that we know?

(One student receives a task at the board: fill out a table of values ​​of the function 12/x using the given values ​​of its argument, and then plot the corresponding points on the coordinate plane).

The rest answer the teacher’s questions: (which are written in advance on the board)

1. What are the names of the following functions given by formulas: y=kx, y=kx+b, y=x 2 , y=x 3 ?

2. Specify the domain of definition of the following functions: y=x 2 +8, y=1/x-7, y= 4x-1/5, y=2x, y=7-5x, y=2/x, y=x 3 , y=-10/x.

Then students work according to the table, answering the questions posed by the teacher:

1. Which figure from the table shows the graphs:

a) linear function;

b) direct proportionality;

c) quadratic function;

d) functions of the form y=kx 3 ?

2. What sign does the coefficient k have in formulas of the form y=kx+b, which correspond to the graphs in Figures 1, 2, 4, 5 of the table?

3. Find in the table graphs of linear functions whose slopes are:

a) equal;

b) equal in magnitude and opposite in sign.

(Then the whole class checks whether the student called to the board correctly filled out the table and placed the points on the coordinate plane).

2. Explanation begins with motivation.

Teacher: As you know, every function describes some processes occurring in the world around us.

Consider, for example, a rectangle with sides x and y and area 12 cm 2 . It is known that x*y=12, but what happens if you start changing one of the sides of the rectangle, let’s say a side with length x?

Side length y can be found from the formula y=12/x. If x increase by 2 times, it will have y=12/2x, i.e. side y will decrease by 2 times. If the value x increase by 3, 4, 5... times, then the value y will decrease by the same amount. On the contrary, if x decrease several times, then y will increase by the same amount. (Work according to the table).

Therefore, a function of the form y=12/x is called inverse proportionality. In general, it is written as y=k/x, where k is a constant, and k≠0.

This is the topic of today's lesson, we wrote it down in our notebooks. I give a strict definition. For the function y=12/x, which is a special type of inverse proportionality, we have already written down a number of values ​​of the argument and function in the table and will depict the corresponding points on the coordinate plane. What does the graph of this function look like? It is difficult to judge the entire graph based on the constructed points, because the points can be connected in any way. Let's try together to draw conclusions about the graph of a function arising from consideration of the table and formula.

Questions for the class:

1. What is the domain of definition of the function y=12/x?
2. Are y values ​​positive or negative if

a) x

b) x>0?

3. How the value of a variable changes y with changing value x?

So,

1. point (0,0) does not belong to the graph, i.e. it does not intersect either the OX or OY axis;
2. the graph is in Ι and ΙΙΙ coordinate quarters;
3. smoothly approaches the coordinate axes both in the Ι coordinate quarter and in the ΙΙΙ, and it approaches the axes as close as desired.

Having this information, we can already connect the dots in the figure (the teacher does this himself on the board) and see the entire graph of the function y=12/x. The resulting curve is called a hyperbola, which in Greek means “passing through something.” This curve was discovered by mathematicians of the ancient Greek school around the 4th century BC. The term, hyperbole, was introduced by Apollonius from the city of Pergamum (Asia Minor), who lived in the 6th-8th centuries. BC.

Now, next to the graph of the function y=12/x, we will construct a graph of the function y=-12/x. (Students complete this task in notebooks, and one student at the blackboard).

Comparing both graphs, students notice that the second occupies 2 and 4 coordinate quarters. In addition, if the graph of the function y=12/x is displayed symmetrically relative to the op-amp axis, then the graph of the function y=-12/x will be obtained.

Question: How does the location of the graph of the hyperbola y=k/x depend on the sign and the value of the coefficient k?

Students are convinced that if k>0, then the graph is located in Ι And ΙΙΙ coordinate quarters, and if k

1. The physical education lesson is conducted by the teacher.

1. Consolidation of what is being studied takes place when completing No. 180, 185 from the textbook.

1. The lesson is summarized, grades, homework: p. 8 No. 179, 184.

Lesson 2 on the topic

“The inverse proportionality function and its graph.”

Performed:

Telegina L.B.

The purpose of the lesson:

1. consolidate the skill of constructing a graph of an inverse proportionality function;
2. develop interest in the subject, logical thinking;
3. cultivate independence and attention.

Lesson plan:

1. Checking homework completion.
2. Oral work.
3. Problem solving.
4. Physical education minute.
5. Multi-level independent work.
6. Summing up, assessments, homework.

Equipment: cards.

During the classes:

1. The teacher announces the topic of the lesson, objectives and lesson plan.

Then two students complete the assigned house numbers 179, 184 on the board.

1. The rest of the students work frontally, answering the teacher's questions.

Questions:

• Define the inverse proportionality function.
• What is the graph of an inverse proportionality function.
• How does the location of the graph of the hyperbola y=k/x depend on the value of the coefficient k?

Tasks:

1. Among the functions specified by the formulas are the functions of inverse proportionality:

a) y=x 2 +5, b) y=1/x, c) y= 4x-1, d) y=2x, e) y=7-5x, f) y=-11/x, g) y=x 3, h) y=15/x-2.

2. For functions of inverse proportionality, name the coefficient and indicate in which quarters the graph lies.

3. Find the domain of definition for functions of inverse proportionality.

(Then students check each other’s homework with a pencil based on the teacher-checked solutions to the numbers on the board and give a grade).

Frontal work according to textbook No. 190, 191, 192, 193 (oral).

1. Execution in notebooks and on the board from textbook No. 186(b), 187(b), 182.

4. A physical education lesson is conducted by the teacher.

5. Independent work is given in three options of varying complexity (distributed on cards).

Ι c. (lightweight).

Plot a graph of the inverse proportionality function y=-6/x using the table:

Using the graph, find:

a) the value of y if x = - 1.5; 2;

b) the value of x at which y = - 1; 4.

ΙΙ century (medium difficulty)

Plot a graph of the inverse proportionality function y=16/x, having first filled in the table.

Using the graph, find at what values x y >0.

ΙΙΙ century (increased difficulty)

Plot a graph of the inverse proportionality function y=10/x-2, having first filled in the table.

Find the domain of definition of this function.

(Students hand in sheets with plotted graphs for testing).

6. Summarizes the lesson, assessments, homework: No. 186 (a), 187 (a).

Let's repeat the theory about functions. A function is a rule according to which each element of one set (argument) is associated with a certain ( the only one!) element of another set (set of function values). That is, if there is a function \(y = f(x)\), this means that for each valid value of the variable \(x\)(which is called an “argument”) corresponds to one value of the variable \(y\)(called a "function").

Function describing inverse dependence

This is a function of the form \(y = \frac(k)(x)\), where \(k\ne 0.\)

In another way, it is called inverse proportionality: an increase in the argument causes a proportional decrease in the function.
Let's define the domain of definition. What can \(x\) be equal to? Or, in other words, what cannot it be equal to?

The only number that cannot be divided by is 0, so \(x\ne 0.\):

\(D(y) = (- \infty ;0) \cup (0; + \infty)\)

or, which is the same:

\(D(y) = R\backslash \( 0\).\)

This notation means that \(x\) can be any number except 0: the sign “R” denotes the set of real numbers, that is, all possible numbers; the sign “\” indicates the exclusion of something from this set (analogous to the “minus” sign), and the number 0 in curly brackets simply means the number 0; It turns out that from all possible numbers we exclude 0.

The set of function values, it turns out, is exactly the same: after all, if \(k \ne 0.\) , then no matter what we divide it by, 0 will not work:

\(E(y) = (- \infty ;0) \cup (0; + \infty)\)

or \(E(y) = R\backslash \( 0\).\)

Some variations of the formula are also possible \(y = \frac(k)(x)\)​​. For example, \(y = \frac(k)((x + a))\)​​is also a function that describes an inverse relationship. The scope and range of values ​​of this function are as follows:

\(D(y) = (- \infty ; - a) \cup (- a; + \infty)\)

\(E(y) = (- \infty ;0) \cup (0; + \infty).\)

Let's consider example, let us reduce the expression to the form of an inverse relationship:

\(y = \frac((x + 2))((x - 3)).\)

\(y = \frac((x + 2))((x - 3)) = \frac((x - 3 + 3 + 2))((x - 3)) = \frac(((x - 3 ) + 5))((x - 3)).\)

We artificially introduced the value 3 into the numerator, and now we divide the numerator by the denominator term by term, we get:

\(y = \frac(((x - 3) + 5))((x - 3)) = \frac((x - 3))((x - 3)) + \frac(5)((x - 3)) = 1 + \frac(5)((x - 3)).\)

We got the inverse relationship plus the number 1.

Inverse relationship graph

Let's start with a simple case \(y = \frac(1)(x).\)

Let's create a table of values:

Let's draw points on the coordinate plane:

Connect the dots, the graph will look like this:

This graph is called "hyperbola". Like a parabola, a hyperbola has two branches, only they are not connected to each other. Each of them tends to move its ends closer to the axes Ox And Oy, but never reaches them.

Let's note some features of the function:

1. If a function has a minus before the fraction, then the graph is flipped, that is, it is displayed symmetrically relative to the axis Ox.
2. The larger the number in the denominator, the further the graph “runs away” from the origin.

Inverse dependence in life

Where do we find such a function in practice? There are many examples. The most common is movement: the greater the speed at which we move, the less time it will take us to cover the same distance. Let's remember the speed formula:

\(v = \frac(S)(t),\)

where v is speed, t is travel time, S is distance (path).

From here we can express time: \(t = \frac(S)(v).\)