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 concept of direct proportionality
Imagine that you are planning to buy your favorite candies (or anything that you really like). Sweets in the store have their own price. Let's say 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 kilograms, pay 900 rubles. This seems to be all clear, right?
If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the relationship of two quantities dependent 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 with the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our example about candy, the price is a constant value, a constant. It does not increase or decrease, no matter how many candies you decide to buy. The independent variable (argument)x is how many kilograms of candy 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 into the formula and get: 600 rubles. = 300 rub. * 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 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 coefficient, and it is always a non-zero number. It is easy to calculate k - 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 = 60*t, and this formula is similar to the function of direct proportionality y = k *x. Let's draw a parallel further: if k = 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 = 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 subsets);
function is odd;
the change in variables is directly proportional along the entire length of the number line.
Direct proportionality and its graph
The graph of a direct proportionality function is a straight line that intersects the origin. To build it, it is enough to mark only one more point. And connect it and the origin of coordinates with a straight 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.
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 located parallel to the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.
Sample problems
Now let's solve a couple direct proportionality problems
Let's start with something simple.
Problem 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: Let's denote the unknown by kx. And we will reason as follows: how many times more chickens have there become? Divide 20 by 5 and find out that it is 4 times. 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 = 80 eggs will be laid by 20 hens in 20 days.
Now the example is a little more complicated, let’s paraphrase the problem from Newton’s “General Arithmetic”. Problem 2: A writer can compose 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 volume 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 conditions of the task, more time is given for the work, the number of assistants increases not by 30 times, but in this way: x = 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 in our examples.
Problem 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 took 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 distance, we can equate the two expressions: 70*2 = V*7. How 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 functions of direct proportionality. Sometimes problems require finding the coefficient k.
Task 4: Given the functions y = - x/16 and y = 5x/2, determine their proportionality coefficients.
Solution: As you remember, k = y/x. This means that for the first function the coefficient is equal to -1/16, and for the second k = 5/2.
You may also encounter a task like Task 5: Write down direct proportionality with a formula. Its graph and the graph of the function y = -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 formula familiar to us: y = k *x. Coefficient k = -5, direct proportionality: y = -5*x.
Conclusion
Now you have learned (or remembered, if you have already covered this topic before) what is called direct proportionality, and looked at it examples. We also talked about the direct proportionality function and its graph, and solved several example problems.
If this article was useful and helped you understand the topic, tell us about it in the comments. So that we know if we could benefit you.
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Trikhleb Daniil, 7th grade student
acquaintance with direct proportionality and the coefficient of direct proportionality (introduction of the concept of angular coefficient”);
constructing a direct proportionality graph;
consideration of the relative position of graphs of direct proportionality and linear functions with identical angular coefficients.
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Slide captions:
Direct proportionality and its graph
What is the argument and value of a function? Which variable is called independent or dependent? What is a function? REVIEW What is the domain of a function?
Methods for specifying a function. Analytical (using a formula) Graphical (using a graph) Tabular (using a table)
The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are the corresponding values of the function. FUNCTION SCHEDULE
1) 2) 3) 4) 5) 6) 7) 8) 9)
COMPLETE THE TASK Construct a graph of the function y = 2 x +1, where 0 ≤ x ≤ 4. Make a table. Using the graph, find the value of the function at x=2.5. At what value of the argument does the function value equal 8?
Definition Direct proportionality is a function that can be specified by a formula of the form y = k x, where x is an independent variable, k is a non-zero number. (k-coefficient of direct proportionality) Direct proportionality
8 Graph of direct proportionality - a straight line passing through the origin of coordinates (point O(0,0)) To construct a graph of the function y= kx, two points are enough, one of which is O (0,0) For k > 0, the graph is located at I and III coordinate quarters. At k
Graphs of functions of direct proportionality y x k>0 k>0 k
Task Determine which of the graphs shows the function of direct proportionality.
Task Determine which function graph is shown in the figure. Choose a formula from the three offered.
Oral work. Can the graph of a function given by the formula y = k x, where k
Determine which of the points A(6,-2), B(-2,-10), C(1,-1), E(0,0) belong to the graph of direct proportionality given by the formula y = 5x 1) A( 6;-2) -2 = 5 6 - 2 = 30 - incorrect. Point A does not belong to the graph of the function y=5x. 2) B(-2;-10) -10 = 5 (-2) -10 = -10 - correct. Point B belongs to the graph of the function y=5x. 3) C(1;-1) -1 = 5 1 -1 = 5 - incorrect Point C does not belong to the graph of the function y=5x. 4) E (0;0) 0 = 5 0 0 = 0 - true. Point E belongs to the graph of the function y=5x
TEST 1 option 2 option No. 1. Which of the functions given by the formula are directly proportional? A. y = 5x B. y = x 2 /8 C. y = 7x(x-1) D . y = x+1 A. y = 3x 2 +5 B. y = 8/x C. y = 7(x + 9) D. y = 10x
No. 2. Write down the numbers of lines y = kx, where k > 0 1 option k
No. 3. Determine which of the points belong to the graph of direct proportionality, given by the formula Y = -1 /3 X A (6 -2), B (-2 -10) 1 option C (1, -1), E (0.0 ) Option 2
y =5x y =10x III A VI and IV E 1 2 3 1 2 3 No. Correct answer Correct answer No.
Complete the task: Show schematically how the graph of the function given by the formula is located: y =1.7 x y =-3,1 x y=0.9 x y=-2.3 x
TASK From the following graphs, select only direct proportionality graphs.
1) 2) 3) 4) 5) 6) 7) 8) 9)
Functions y = 2x + 3 2. y = 6/ x 3. y = 2x 4. y = - 1.5x 5. y = - 5/ x 6. y = 5x 7. y = 2x – 5 8. y = - 0.3x 9. y = 3/ x 10. y = - x /3 + 1 Select functions of the form y = k x (direct proportionality) and write them down
Functions of direct proportionality Y = 2x Y = -1.5x Y = 5x Y = -0.3x y x
y Linear functions that are not functions of direct proportionality 1) y = 2x + 3 2) y = 2x – 5 x -6 -4 -2 0 2 4 6 6 3 -3 -6 y = 2x + 3 y = 2x - 5
Homework: paragraph 15 pp. 65-67, No. 307; No. 308.
Let's repeat it again. What new things have you learned? What have you learned? What did you find particularly difficult?
I liked the lesson and the topic is understood: I liked the lesson, but I still don’t understand everything: I didn’t like the lesson and the topic is not clear.
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.
Proportionality is a relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.
Proportionality can be direct or inverse. In this lesson we will look at each of them.
Lesson contentDirect proportionality
Let's assume that the car is moving at a speed of 50 km/h. We remember that speed is the distance traveled per unit of time (1 hour, 1 minute or 1 second). In our example, the car is moving at a speed of 50 km/h, that is, in one hour it will cover a distance of fifty kilometers.
Let us depict in the figure the distance traveled by the car in 1 hour.
Let the car drive for another hour at the same speed of fifty kilometers per hour. Then it turns out that the car will travel 100 km
As can be seen from the example, doubling the time led to an increase in the distance traveled by the same amount, that is, twice.
Quantities such as time and distance are called directly proportional. And the relationship between such quantities is called direct proportionality.
Direct proportionality is the relationship between two quantities in which an increase in one of them entails an increase in the other by the same amount.
and vice versa, if one quantity decreases by a certain number of times, then the other decreases by the same number of times.
Let's assume that the original plan was to drive a car 100 km in 2 hours, but after driving 50 km, the driver decided to rest. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, reducing the distance traveled will lead to a decrease in time by the same amount.
An interesting feature of directly proportional quantities is that their ratio is always constant. That is, when the values of directly proportional quantities change, their ratio remains unchanged.
In the example considered, the distance was initially 50 km and the time was one hour. The ratio of distance to time is the number 50.
But we increased the travel time by 2 times, making it equal to two hours. As a result, the distance traveled increased by the same amount, that is, it became equal to 100 km. The ratio of one hundred kilometers to two hours is again the number 50
The number 50 is called coefficient of direct proportionality. It shows how much distance there is per hour of movement. In this case, the coefficient plays the role of movement speed, since speed is the ratio of the distance traveled to the time.
Proportions can be made from directly proportional quantities. For example, the ratios make up the proportion:
Fifty kilometers is to one hour as one hundred kilometers is to two hours.
Example 2. The cost and quantity of goods purchased are directly proportional. If 1 kg of sweets costs 30 rubles, then 2 kg of the same sweets will cost 60 rubles, 3 kg 90 rubles. As the cost of a purchased product increases, its quantity increases by the same amount.
Since the cost of a product and its quantity are directly proportional quantities, their ratio is always constant.
Let's write down what is the ratio of thirty rubles to one kilogram
Now let’s write down what the ratio of sixty rubles to two kilograms is. This ratio will again be equal to thirty:
Here the coefficient of direct proportionality is the number 30. This coefficient shows how many rubles are per kilogram of sweets. In this example, the coefficient plays the role of the price of one kilogram of goods, since price is the ratio of the cost of the goods to its quantity.
Inverse proportionality
Consider the following example. The distance between the two cities is 80 km. The motorcyclist left the first city and, at a speed of 20 km/h, reached the second city in 4 hours.
If a motorcyclist's speed was 20 km/h, this means that every hour he covered a distance of twenty kilometers. Let us depict in the figure the distance traveled by the motorcyclist and the time of his movement:
On the way back, the motorcyclist's speed was 40 km/h, and he spent 2 hours on the same journey.
It is easy to notice that when the speed changes, the time of movement changes by the same amount. Moreover, it changed in the opposite direction - that is, the speed increased, but the time, on the contrary, decreased.
Quantities such as speed and time are called inversely proportional. And the relationship between such quantities is called inverse proportionality.
Inverse proportionality is the relationship between two quantities in which an increase in one of them entails a decrease in the other by the same amount.
and vice versa, if one quantity decreases by a certain number of times, then the other increases by the same number of times.
For example, if on the way back the motorcyclist’s speed was 10 km/h, then he would cover the same 80 km in 8 hours:
As can be seen from the example, a decrease in speed led to an increase in movement time by the same amount.
The peculiarity of inversely proportional quantities is that their product is always constant. That is, when the values of inversely proportional quantities change, their product remains unchanged.
In the example considered, the distance between cities was 80 km. When the speed and time of movement of the motorcyclist changed, this distance always remained unchanged
A motorcyclist could travel 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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