In grades 7 and 8, the graph of direct proportionality is studied.
How to construct a direct proportionality graph?
Let's look at the direct proportionality graph using examples.
Direct proportionality graph formula
A direct proportionality graph represents a function.
In general, direct proportionality has the formula
The inclination angle of the direct proportionality graph relative to the x-axis depends on the magnitude and sign of the coefficient of direct proportionality.
Direct proportionality graph goes through
A direct proportionality graph passes through the origin.
A direct proportionality graph is a straight line. A straight line is defined by two points.
Thus, when constructing a graph of direct proportionality, it is enough to determine the position of two points.
But we always know one of them - this is the origin of coordinates.
All that remains is to find the second one. Let's look at an example of constructing a graph of direct proportionality.
Graph direct proportionality y = 2x
Task .
Plot a graph of direct proportionality given by the formula
Solution .
All the numbers are there.
Take any number from the domain of direct proportionality, let it be 1.
Find the value of the function when x is equal to 1
Y=2x=
2 * 1 = 2
that is, for x = 1 we get y = 2. The point with these coordinates belongs to the graph of the function y = 2x.
We know that the graph of direct proportionality is a straight line, and a straight line is defined by two points.
Linear function
Linear function is a function that can be specified by the formula y = kx + b,
where x is the independent variable, k and b are some numbers.
The graph of a linear function is a straight line.
The number k is called slope of a straight line– graph of the function y = kx + b.
If k > 0, then the angle of inclination of the straight line y = kx + b to the axis X spicy; if k< 0, то этот угол тупой.
If the slopes of the lines that are graphs of two linear functions are different, then these lines intersect. And if the angular coefficients are the same, then the lines are parallel.
Graph of a function y =kx +b, where k ≠ 0, is a line parallel to the line y = kx.
Direct proportionality.
Direct proportionality is a function that can be specified by the formula y = kx, where x is an independent variable, k is a non-zero number. The number k is called coefficient of direct proportionality.
The graph of direct proportionality is a straight line passing through the origin of coordinates (see figure).
Direct proportionality is a special case of a linear function.
Function Propertiesy =kx:
Inverse proportionality
Inverse proportionality is called a function that can be specified by the formula:
k
y = -
x
Where x is the independent variable, and k– a non-zero number.
The graph of inverse proportionality is a curve called hyperbole(see picture).
For a curve that is the graph of this function, the axis x And y act as asymptotes. Asymptote- this is the straight line to which the points of the curve approach as they move away to infinity.
k
Function Propertiesy = -:
x
The two quantities are called directly proportional, if when one of them increases several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.
The relationship between such quantities is a direct proportional relationship. Examples of direct proportional dependence:
1) at a constant speed, the distance traveled is directly proportional to time;
2) the perimeter of a square and its side are directly proportional quantities;
3) the cost of a product purchased at one price is directly proportional to its quantity.
To distinguish a direct proportional relationship from an inverse one, you can use the proverb: “The further into the forest, the more firewood.”
It is convenient to solve problems involving directly proportional quantities using proportions.
1) To make 10 parts you need 3.5 kg of metal. How much metal will go into making 12 of these parts?
(We reason like this:
1. In the filled column, place an arrow in the direction from the largest number to the smallest.
2. The more parts, the more metal needed to make them. This means that this is a directly proportional relationship.
Let x kg of metal be needed to make 12 parts. We make up the proportion (in the direction from the beginning of the arrow to its end):
12:10=x:3.5
To find , you need to divide the product of the extreme terms by the known middle term:
This means that 4.2 kg of metal will be required.
Answer: 4.2 kg.
2) For 15 meters of fabric they paid 1680 rubles. How much does 12 meters of such fabric cost?
(1. In the filled column, place an arrow in the direction from the largest number to the smallest.
2. The less fabric you buy, the less you have to pay for it. This means that this is a directly proportional relationship.
3. Therefore, the second arrow is in the same direction as the first).
Let x rubles cost 12 meters of fabric. We make a proportion (from the beginning of the arrow to its end):
15:12=1680:x
To find the unknown extreme term of the proportion, divide the product of the middle terms by the known extreme term of the proportion:
This means that 12 meters cost 1344 rubles.
Answer: 1344 rubles.
Dependency Types
Let's look at charging the battery. As the first quantity, let's take the time it takes to charge. The second value is the time it will work after charging. The longer you charge the battery, the longer it will last. The process will continue until the battery is fully charged.
Dependence of battery operating time on the time it is charged
Note 1
This dependence is called straight:
As one value increases, so does the second. As one value decreases, the second value also decreases.
Let's look at another example.
The more books a student reads, the fewer mistakes he will make in the dictation. Or the higher you rise in the mountains, the lower the atmospheric pressure will be.
Note 2
This dependence is called reverse:
As one value increases, the second decreases. As one value decreases, the second value increases.
Thus, in case direct dependence both quantities change equally (both either increase or decrease), and in the case inverse relationship– opposite (one increases and the other decreases, or vice versa).
Determining dependencies between quantities
Example 1
The time it takes to visit a friend is $20$ minutes. If the speed (first value) increases by $2$ times, we will find how the time (second value) that will be spent on the path to a friend changes.
Obviously, the time will decrease by $2$ times.
Note 3
This dependence is called proportional:
The number of times one quantity changes, the number of times the second quantity changes.
Example 2
For $2$ loaves of bread in the store you need to pay 80 rubles. If you need to buy $4$ loaves of bread (the quantity of bread increases by $2$ times), how many times more will you have to pay?
Obviously, the cost will also increase $2$ times. We have an example of proportional dependence.
In both examples, proportional dependencies were considered. But in the example with loaves of bread, the quantities change in one direction, therefore, the dependence is straight. And in the example of going to a friend’s house, the relationship between speed and time is reverse. Thus there is directly proportional relationship And inversely proportional relationship.
Direct proportionality
Let's consider $2$ proportional quantities: the number of loaves of bread and their cost. Let $2$ loaves of bread cost $80$ rubles. If the number of buns increases by $4$ times ($8$ buns), their total cost will be $320$ rubles.
The ratio of the number of buns: $\frac(8)(2)=4$.
Bun cost ratio: $\frac(320)(80)=$4.
As you can see, these relations are equal to each other:
$\frac(8)(2)=\frac(320)(80)$.
Definition 1
The equality of two ratios is called proportion.
With a directly proportional dependence, a relationship is obtained when the change in the first and second quantities coincides:
$\frac(A_2)(A_1)=\frac(B_2)(B_1)$.
Definition 2
The two quantities are called directly proportional, if when one of them changes (increases or decreases), the other value also changes (increases or decreases, respectively) by the same amount.
Example 3
The car traveled $180$ km in $2$ hours. Find the time during which he will cover $2$ times the distance at the same speed.
Solution.
Time is directly proportional to distance:
$t=\frac(S)(v)$.
How many times will the distance increase, at a constant speed, by the same amount will the time increase:
$\frac(2S)(v)=2t$;
$\frac(3S)(v)=3t$.
The car traveled $180$ km in $2$ hours
The car will travel $180 \cdot 2=360$ km - in $x$ hours
The further the car travels, the longer it will take. Consequently, the relationship between the quantities is directly proportional.
Let's make a proportion:
$\frac(180)(360)=\frac(2)(x)$;
$x=\frac(360 \cdot 2)(180)$;
Answer: The car will need $4$ hours.
Inverse proportionality
Definition 3
Solution.
Time is inversely proportional to speed:
$t=\frac(S)(v)$.
By how many times does the speed increase, with the same path, the time decreases by the same amount:
$\frac(S)(2v)=\frac(t)(2)$;
$\frac(S)(3v)=\frac(t)(3)$.
Let's write the problem condition in the form of a table:
The car traveled $60$ km - in $6$ hours
The car will travel $120$ km – in $x$ hours
The faster the car speeds, the less time it will take. Consequently, the relationship between the quantities is inversely proportional.
Let's make a proportion.
Because the proportionality is inverse, the second relation in the proportion is reversed:
$\frac(60)(120)=\frac(x)(6)$;
$x=\frac(60 \cdot 6)(120)$;
Answer: The car will need $3$ hours.
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:
- Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
- The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
- Does not have maximum or minimum values.
- It is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not intersect the coordinate axes.
- Has no zeros.
- 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.
- 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.
website, when copying material in full or in part, a link to the source is required.