Class: 7
The function occupies one of the leading places in the school algebra course and has numerous applications in other sciences. At the beginning of the study, for the purpose of motivation and actualization of the question, I inform you that not a single phenomenon, not a single process in nature can be studied, no machine can be constructed and then operate without a complete mathematical description. One tool for this is a function. Its study begins in the 7th grade; as a rule, children do not delve into the definition. Particularly difficult to access concepts are domain of definition and domain of meaning. Using known connections between quantities in problems of movement and value, I translate them into the language of a function, maintaining a connection with its definition. Thus, students develop the concept of function at a conscious level. At the same stage, painstaking work is carried out on new concepts: domain of definition, domain of value, argument, value of a function. I use advanced learning: I introduce the notation D(y), E(y), introduce the concept of zero of a function (analytically and graphically), when solving exercises with areas of constant sign. The earlier and more often students encounter difficult concepts, the better they become aware of them at the level of long-term memory. When studying a linear function, it is advisable to show the connection with the solution of linear equations and systems, and later with the solution of linear inequalities and their systems. At the lecture, students receive a large block (module) of new information, therefore, at the end of the lecture, the material is “wrung out” and a summary is compiled that the students must know. Practical skills are developed in the process of performing exercises using various methods, which are based on individual and independent work.
1. Some information about linear functions.
The linear function is very often encountered in practice. The length of the rod is a linear function of temperature. The length of rails and bridges is also a linear function of temperature. The distance traveled by a pedestrian, train, or car at a constant speed is a linear function of travel time.
A linear function describes a number of physical relationships and laws. Let's look at some of them.
1) l = l о (1+at) – linear expansion of solids.
2) v = v о (1+bt) – volumetric expansion of solids.
3) p=p o (1+at) – dependence of the resistivity of solid conductors on temperature.
4) v = v o + at – speed of uniformly accelerated motion.
5) x= x o + vt – coordinate of uniform motion.
Task 1. Determine the linear function from the tabular data:
| X | 1 | 3 |
| at | -1 | 3 |
Solution. y= kx+b, the problem is reduced to solving a system of equations: 1=k 1+b and 3=k 3 + b
Answer: y = 2x – 3.
Problem 2. Moving uniformly and rectilinearly, the body passed 14 m in the first 8 s, and 12 m in another 4 s. Create an equation of motion based on these data.
Solution. According to the conditions of the problem, we have two equations: 14 = x o +8 v o and 26 = x o +12 v o, solving the system of equations, we obtain v = 3, x o = -10.
Answer: x = -10 + 3t.
Problem 3. A car left the city moving at a speed of 80 km/h. After 1.5 hours, a motorcycle came after him, the speed of which was 100 km/h. How long will it take the motorcycle to catch up with him? At what distance from the city will this happen?
Answer: 7.5 hours, 600 km.
Task 4. The distance between two points at the initial moment is 300m. The points move towards each other at speeds of 1.5 m/s and 3.5 m/s. When will they meet? Where will this happen?
Answer: 60 s, 90 m.
Task 5. A copper ruler at 0 o C has a length of 1 m. Find the increase in its length when its temperature increases by 35 o, by 1000 o C (melting point of copper is 1083 o C)
Answer: 0.6mm.
2. Direct proportionality.
Many laws of physics are expressed through direct proportionality. In most cases, a model is used to write these laws
in some cases -
Let's give a few examples.
1. S = v t (v – const)
2. v = a t (a – const, a – acceleration).
3. F = kx (Hooke’s law: F – force, k – stiffness (const), x – elongation).
4. E= F/q (E is the intensity at a given point of the electric field, E is const, F is the force acting on the charge, q is the magnitude of the charge).
As a mathematical model of direct proportionality, you can use the similarity of triangles or the proportionality of segments (Thales’ theorem).
Problem 1. The train passed the traffic light in 5 s, and passed the platform 150 m long in 15 s. What is the length of the train and its speed?
Solution. Let x be the length of the train, x+150 be the total length of the train and platform. In this problem, the speed is constant, and the time is proportional to the length.
We have the proportion: (x+150) :15 = x: 5.
Where x = 75, v = 15.
Answer. 75 m, 15 m/s.
Problem 2. The boat traveled 90 km downstream in some time. In the same time, he would have traveled 70 km against the current. How far will the raft travel in this time?
Answer. 10 km.
Problem 3. What was the initial temperature of the air if, when heated by 3 degrees, its volume increased by 1% of the original.
Answer. 300 K (Kelvin) or 27 0 C.
Lecture on the topic "Linear function".
Algebra, 7th grade
1. Consider examples of problems using well-known formulas:
S = v t (path formula), (1)
C = ck (value formula). (2)
Problem 1. The car drove 20 km from point A and continued its journey at a speed of 62 km/h. At what distance from point A will the car be after t hours? Make up an expression for the problem, denoting the distance S, find it at t = 1 hour, 2.5 hours, 4 hours.
1) Using formula (1) we find the path traveled by a car at a speed of 62 km/h in time t, S 1 = 62t;
2) Then from point A after t hours the car will be at a distance S = S 1 + 20 or S = 62t + 20, let’s find the value of S:
at t = 1, S = 62*1 + 20, S = 82;
at t = 2.5, S = 62*2.5 + 20, S = 175;
at t = 4, S = 62*4+ 20, S = 268.
We note that when finding S, only the value of t and S changes, i.e. t and S are variables, and S depends on t, each value of t corresponds to a single value of S. Denoting the variable S by Y, and t by x, we obtain a formula for solving this problem:
Y= 62x + 20. (3)
Problem 2. In a store we bought a textbook for 150 rubles and 15 notebooks of n rubles each. How much money did you pay for the purchase? Compose an expression for the problem, denoting the cost C, find it for n = 5,8,16.
1) Using formula (2) we find the cost of notebooks C 1 = 15n;
2) Then the cost of the entire purchase is C = C 1 +150 or C = 15n+150, let’s find the value of C:
with n = 5, C = 15 5 + 150, C = 225;
with n = 8, C = 15 8 + 150, C = 270;
with n = 16, C = 15 16+ 150, C = 390.
Similarly, we note that C and n are variables, for each value of n there corresponds a single value of C. Denoting the variable C as Y, and n as x, we obtain a formula for solving problem 2:
Y= 15x + 150. (4)
Comparing formulas (3) and (4), we are convinced that the variable Y is found through the variable x using the same algorithm. We considered only two different problems that describe the phenomena that surround us every day. In fact, there are many processes that change according to the obtained laws, so such a dependence between variables deserves study.
Solutions to problems show that the values of the variable x are chosen arbitrarily, satisfying the conditions of the problems (positive in problem 1 and natural in problem 2), i.e. x is an independent variable (it is called an argument), and Y is a dependent variable and there is a one-to-one correspondence between them , and by definition such a dependence is a function. Therefore, denoting the coefficient of x by the letter k, and the free term by the letter b, we obtain the formula
Y= kx + b.
Definition: Function of the form y= kx + b, where k, b are some numbers, x is an argument, y is the value of the function, called a linear function.
To study the properties of a linear function, we introduce definitions.
Definition 1. The set of admissible values of an independent variable is called the domain of definition of the function (admissible - this means those numerical values of x for which calculations y are performed) and is denoted D(y).
Definition 2. The set of values of the dependent variable is called the domain of the function (these are the numerical values that y takes) and is denoted E(y).
Definition 3. The graph of a function is the set of points on the coordinate plane whose coordinates turn the formula into a true equality.
Definition 4. The coefficient k of x is called the slope.
Let's consider the properties of a linear function.
1. D(y) – all numbers (multiplication is defined on the set of all numbers).
2. E(y) – all numbers.
3. If y = 0, then x = -b/k, point (-b/k;0) – the point of intersection with the Ox axis, is called the zero of the function.
4. If x = 0, then y = b, point (0; b) is the point of intersection with the Oy axis.
5. Let’s find out which line the linear function on the coordinate plane will line up the points, i.e. which is the graph of the function. To do this, consider the functions
1) y= 2x + 3, 2) y= -3x – 2.
For each function we will create a table of values. Let's set arbitrary values of the x variable and calculate the corresponding values of the Y variable.
| X | -1,5 | -2 | 0 | 1 | 2 |
| Y | 0 | -1 | 3 | 5 | 7 |
Having constructed the resulting pairs (x;y) on the coordinate plane and connecting them for each function separately (we took the x values with a step of 1, if we reduce the step, the points will line up more often, and if the step is close to zero, then the points will merge into a solid line ), we notice that the points line up in a straight line in case 1) and in case 2). Due to the fact that the functions are chosen arbitrarily (construct your own graphs y= 0.5x – 4, y= x + 5), we conclude that that the graph of a linear function is a straight line. Using the property of a straight line: there is only one straight line passing through two points, it is enough to take two points to construct a straight line.
6. From geometry it is known that lines can either intersect or be parallel. Let's study the relative position of graphs of several functions.
1) y= -x + 5, y= -x + 3, y= -x – 4; 2) y= 2x + 2, y= x + 2, y= -0.5x + 2.
Let's build groups of graphs 1) and 2) and draw conclusions.
|
 |
The graphs of functions 1) are located in parallel, examining the formulas, we notice that all functions have the same coefficients for x.
The graphs of functions 2) intersected at one point (0;2). Examining the formulas, we notice that the coefficients are different, and the number b = 2.
In addition, it is easy to notice that straight lines defined by linear functions with k › 0 form an acute angle with the positive direction of the Ox axis, and an obtuse angle with k ‹ 0. Therefore, the coefficient k is called the slope coefficient.
7. Let's consider special cases of a linear function, depending on the coefficients.
1) If b=0, then the function takes the form y= kx, then k = y/x (the ratio shows how many times the difference or what part y is from x).
A function of the form Y= kx is called direct proportionality. This function has all the properties of a linear function, its peculiarity is that for x=0 y=0. The direct proportionality graph passes through the origin point (0;0).
2) If k = 0, then the function takes the form y = b, which means that for any value of x the function takes the same value.
A function of the form y = b is called constant. The graph of the function is a straight line passing through the point (0;b) parallel to the Ox axis; at b=0, the graph of the constant function coincides with the abscissa axis.
Abstract
1. Definition A function of the form Y = kx + b, where k, b are some numbers, x is an argument, Y is the value of the function, is called a linear function.
D(y) – all numbers.
E(y) – all numbers.
The graph of a linear function is a straight line passing through the point (0;b).
2. If b=0, then the function takes the form y= kx, called direct proportionality. A direct proportionality graph passes through the origin.
3. If k = 0, then the function takes the form y= b and is called constant. The graph of a constant function passes through the point (0;b), parallel to the abscissa axis.
4. Mutual arrangement of graphs of linear functions.
The functions y= k 1 x + b 1 and y= k 2 x + b 2 are given.
If k 1 = k 2, then the graphs are parallel;
If k 1 and k 2 are not equal, then the graphs intersect.
5. See above for examples of graphs of linear functions.
Literature.
- Textbook Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov and others. “Algebra, 8.”
- Didactic materials on algebra for grade 8 / V.I. Zhokhov, Yu.N. Makarychev, N.G. Mindyuk. – M.: Education, 2006. – 144 p.
- Supplement to the newspaper September 1 “Mathematics”, 2001, No. 2, No. 4.
Summarize and systematize knowledge on the topic “Linear function”:
- consolidate the ability to read and build graphs of functions given by the formulas y = kx+b, y = kx;
- consolidate the ability to determine the relative position of graphs of linear functions;
- develop skills in working with graphs of linear functions.
Develop ability to analyze, compare, draw conclusions. Development of cognitive interest in mathematics, competent oral mathematical speech, accuracy and precision in construction.
Upbringing attentiveness, independence in work, ability to work in pairs.
Equipment: ruler, pencil, task cards, colored pencils.
Lesson type: lesson on consolidating the material learned.
Lesson plan:
- Organizing time.
- Oral work. Mathematical dictation with self-test and self-assessment. Historical excursion.
- Training exercises.
- Independent work.
- Lesson summary.
- Homework.
During the classes
1. State the purpose of the lesson.
The purpose of the lesson is to summarize and systematize knowledge on the topic “Linear function”.
2. Let's start by testing your theoretical knowledge.
– Define the function. What is an independent variable? Dependent variable?
– Define the graph of a function.
– Formulate the definition of a linear function.
– What is the graph of a linear function?
– How to graph a linear function?
– Formulate the definition of direct proportionality. What is a graph? How to build a graph? How is the graph of the function y = kx located in the coordinate plane for k > 0 and for k< 0?
Mathematical dictation with self-test and self-assessment.
Look at the pictures and answer the questions.
1) Which function's graph is redundant?
2) Which figure shows a graph of direct proportionality?
3) In which figure does the graph of a linear function have a negative slope?
4) Determine the sign of number b. (Write the answer as an inequality)
Checking the work. Rating.
Work in pairs.
Decipher the name of the mathematician who first used the term function. To do this, write in the boxes the letter corresponding to the graph of the given function. Write the letter C in the remaining square. Complete the drawing with a graph of the function corresponding to this letter.
Picture 1
Figure 2
Figure 3
Gottfried Wilhelm Leibniz, 1646-1716, German philosopher, mathematician, physicist and linguist. He and the English scientist I. Newton created (independently of each other) the foundations of an important branch of mathematics - mathematical analysis. Leibniz introduced many concepts and symbols that are still used in mathematics today.
3. 1. Given the functions specified by the formulas: y = x-5; y = 0.5x; y = – 2x; y = 4.
Name the functions. Indicate the graphs of which of these functions will pass through point M (8;4). Show schematically what the drawing will look like if you depict graphs of functions passing through point M on it.
2. The graph of direct proportionality passes through point C (2;1). Create a formula that specifies direct proportionality. At what value of m will the graph pass through point B (-4;m).
3. Graph the function given by y=1/2X. How can you obtain from the graph of a given function a graph of a function given by the formula y=1/2X – 4 and y = 1/2X+3. Analyze the resulting graphs.
4. The functions are given by the formulas:
1) y= 4x+9 and y= 6x-5;
2) y=1/2x-3 and y=0.5x+2;
3) y= x and y= -5x+2.4;
4) y= 3x+6 and y= -2.5x+6.
What is the relative position of the function graphs? Without performing any construction, find the coordinates of the intersection point of the first pair of graphs. (Self-test)
4. Independent work in pairs. (performed on ml paper). Interdisciplinary communication.
It is necessary to construct graphs of functions and select that part of it for the points of which the corresponding inequality holds:
y = x + 6, 4 < X < 6; y = -x + 6, -6 < X < -4; y = – 1/3 x + 10, -6 < X < -3; y = 1/3 x +10, 3 < X < 6; y = -x + 14, 0 < X < 3; y = x + 14, -3 < X < 0; y = 9x – 18, 2 < X < 4; y = – 9x – 18 -4 < X < -2; y = 0, -2 < X < 2.
What kind of drawing did you get? ( Tulip.)
A little about tulips:
About 120 species of tulips are known, distributed mainly in Central, Eastern and Southern Asia and Southern Europe. Botanists believe that the tulip culture originated in Turkey in the 12th century. The plant gained world fame far from its homeland, in Holland, rightly called the Land of Tulips.
Here is the legend about the tulip. Happiness was contained in the golden bud of a yellow tulip. No one could reach this happiness, because there was no such force that could open its bud. But one day a woman with a child was walking through the meadow. The boy escaped from his mother’s arms, ran up to the flower with a ringing laugh, and the golden bud opened. The carefree children's laughter accomplished what no force could do. Since then, it has become a custom to give tulips only to those who feel happiness.
Creative homework. Create a drawing in a rectangular coordinate system, consisting of segments, and create an analytical model of it.
6. Independent work. Differentiated task (in two versions)
Option I:
Sketch the graphs of the functions:
Option II:
Draw schematically the graphs of functions for which the following conditions are met:
7. Lesson summary
Analysis of the work done. Grading.
Maslova Angelina
Research work in mathematics. Angelina compiled a computer model of a linear function, which she used to conduct the research.
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Municipal autonomous educational institution secondary school No. 8 of the urban district of Bor, Nizhny Novgorod region
Research work in computer science and mathematics
Completed by a student of class 7A, Angelina Maslova
Head: computer science teacher, Voronina Anna Alekseevna.
Urban district of Bor - 2015
Introduction
- Exploring Linear Functions in Spreadsheets
Conclusion
Bibliography
Introduction
This year in algebra lessons we were introduced to linear functions. We learned to build a graph of a linear function, determined how the graph of a function should behave depending on its coefficients. A little later, in a computer science lesson, we learned that these actions can be considered mathematical modeling. I decided to see if it was possible to explore a linear function using spreadsheets.
Goal of the work: explore linear function in spreadsheets
Research objectives:
- find and study information about a linear function;
- build a mathematical model of a linear function in a spreadsheet;
- explore a linear function using the constructed model.
Object of study:math modeling.
Subject of study:mathematical model of a linear function.
Modeling as a method of cognition
A person experiences the world almost from his birth. To do this, a person uses models that can be very diverse.
Model is a new object that reflects some essential properties of a real object.
Models of real objects are used in a variety of situations:
- When an object is very large (for example, the Earth is a model: a globe or a map) or, conversely, too small (a biological cell).
- When the object is very complex in its structure (car - model: children's car).
- When an object is dangerous to study (volcano).
- When the object is very far away.
Modeling is the process of creating and studying a model.
We create and use models ourselves, sometimes without even thinking about it. For example, we take photographs of some event in our life, and then show them to our friends.
Based on the type of information, all models can be divided into several groups:
- Verbal models. These models can exist in oral or written form. It could be just a verbal description of an object or a poem, or it could be a newspaper article or an essay - all these are verbal models.
- Graphic models. These are our drawings, photographs, diagrams and graphs.
- Iconic models. These are models written in some symbolic language: notes, mathematical, physical or chemical formulas.
Linear function and its properties
Linear functioncalled a function of the form
The graph of a linear function is a straight line.
1 . To plot a function, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the function equation, and use them to calculate the corresponding y values.
For example, to plot a function, convenient to take and , then the ordinates of these points will be equal And .
We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function:
2 . In the equation of the function y=kx+b, the coefficient k is responsible for the slope of the function graph:
Coefficient b is responsible for shifting the graph along the OY axis:
The figure below shows the graphs of functions; ;
Note that in all these functions the coefficient greater than zero to the right . Moreover, the greater the value, the steeper the straight line goes.
In all functions– and we see that all graphs intersect the OY axis at point (0;3)
Now let's look at the graphs of functions; ;
This time in all functions the coefficient less than zero , and all function graphs are sloped left . The coefficient b is the same, b=3, and the graphs, as in the previous case, intersect the OY axis at point (0;3)
Let's look at the graphs of functions; ;
Now in all function equations the coefficientsare equal. And we got three parallel lines.
But the coefficients b are different, and these graphs intersect the OY axis at different points:
Graph of a function (b=3) intersects the OY axis at point (0;3)
Graph of a function (b=0) intersects the OY axis at the point (0;0) - the origin.
Graph of a function (b=-2) intersects the OY axis at point (0;-2)
So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function looks like.
If k 0, then the graph of the function has the form:
If k>0 and b>0 , then the graph of the function has the form:
If k>0 and b , then the graph of the function has the form:
If k, then the graph of the function has the form:
If k=0 , then the function turns into a functionand its graph looks like:
The ordinates of all points on the graph of the function equal
If b=0 , then the graph of the functionpasses through the origin:
4. Condition for parallelism of two lines:
Graph of a function parallel to the graph of the function, If
5. The condition for the perpendicularity of two straight lines:
Graph of a function perpendicular to the graph of the function, if or
6 . Intersection points of a function graphwith coordinate axes.
With OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero in the equation of the function instead of x. We get y=b. That is, the point of intersection with the OY axis has coordinates (0; b).
With OX axis: The ordinate of any point belonging to the OX axis is equal to zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero in the equation of the function instead of y. We get 0=kx+b. From here. That is, the point of intersection with the OX axis has coordinates (;0):
Exploring Linear Functions in Spreadsheets
To study a linear function in a spreadsheet environment, I compiled the following algorithm:
- Construct a mathematical model of a Linear function in a spreadsheet.
- Fill in the trace table of argument and function values.
- Plot a Linear Function using the Chart Wizard.
- Explore the Linear function depending on the values of the coefficients.
To study the linear function, I used Microsoft Office Excel 2007. I used formulas to compile tables of argument and function values. I got the following table of values:
Using such a mathematical model, you can easily monitor changes in the graph of a linear function by changing the values of the coefficients in the table.
Also, using spreadsheets, I decided to monitor how the relative position of the graphs of two linear functions changes. Having built a new mathematical model in a spreadsheet, I got the following result:
By changing the coefficients of two linear functions, I was clearly convinced of the validity of the information I had learned about the properties of linear functions.
Conclusion
The linear function in algebra is considered the simplest. But at the same time it has many properties that are not immediately clear. Having built a mathematical model of a linear function in spreadsheets and examined it, the properties of a linear function became more clear to me. I was clearly able to see how the graph changes when the coefficients of the function change.
I think that the mathematical model I built will help seventh grade students independently explore the linear function and understand it better.
Bibliography
- Algebra textbook for 7th grade.
- Computer science textbook for 7th grade
- Wikipedia.org
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Object of study: linear function. Subject of research: mathematical model of a linear function.
Purpose of the work: to investigate a linear function in spreadsheets. Research objectives: to find and study information about the linear function; build a mathematical model of a linear function in a spreadsheet; explore a linear function using the constructed model.
A linear function is a function of the form y= k x+ b, where x is an argument, and k and b are some numbers (coefficients). The graph of a linear function is a straight line.
Consider a function y=kx+b such that k 0 , b=0 . View: y=kx In one coordinate system we will construct graphs of these functions: y=3x y=x y=-7x We will construct each graph with the corresponding color x 0 1 y 0 3 x 0 1 y 0 1 x 0 1 y 0 7
The graph of a linear function of the form y = k x passes through the origin. y=x y=3x y=-7x y x
Conclusion: The graph of a linear function of the form y = kx + b intersects the O Y axis at point (0; b).
Consider the function y=kx+b, where k=0. View: y=b In one coordinate system, construct graphs of functions: y=4 y=-3 y=0 We construct each graph with the appropriate color
The graph of a linear function of the form y = b runs parallel to the OX axis and intersects the O Y axis at point (0; b). y=4 y=-3 y=0 y x
In one coordinate system, construct graphs of functions: Y=2x Y=2x+ 3 Y=2x-4 We construct each graph with the appropriate color x 0 1 y 0 2 x 0 1 y 3 5 x 0 1 y -4 -2
Graphs of linear functions of the form y=kx+b are parallel if the coefficients of x are the same. y =2x+ 3 y =2x y =2x-4 y x
In one coordinate system we will construct graphs of functions: y=3x+4 Y= - 2x+4 We will construct graphs with the appropriate color x 0 1 y 4 7 x 0 1 y 4 2
The graphs of two linear functions of the form y=kx+b intersect if the coefficients of x are different. y x
In one coordinate system we will construct graphs of functions: y=0, 5x-2 y=-2x-4 y= 4 x-1 y=- 0, 2 5 x- 3 x 0 4 y x 0 -2 y -4 0 x 0 4 y -2 0 x 0 1 y -1 3 x 0 - 4 y -3 -2
y=0, 5x-2 y=-2x-4 y= 4 x-1 y=- 0, 2 5 x- 3 Graphs of two linear functions of the form y=kx+b are mutually perpendicular if the product of the coefficients of x is “ - 1" .
Therefore, the coefficient k is called the slope of the straight line - the graph of the function y=kx+ b. If k 0, then the angle of inclination of the graph to the O X axis is acute. The function increases. y x y x
Spreadsheet
Spreadsheet
Linear equations Algebraic condition Geometric derivation y = k 1 x+ b 1 k 1 = k 2, b 1 ≠ b 2 y = k 2 x+ b 2 k 1 = k 2, b 1 = b 2 k 1 ≠ k 2 k 1 * to 2 = -1 Lines are parallel Lines are coincident Lines are perpendicular Lines intersect
The mathematical model I built will help seventh grade students independently explore the linear function and understand it better.
Instructions
To find the coordinates of a point on a line, select it on the line and draw perpendicular lines on the coordinate axis. Determine what number the intersection point corresponds to, the intersection with the x axis is the abscissa value, that is, x1, the intersection with the y axis is the ordinate, y1.
Try to choose a point whose coordinates can be determined without fractional values, for convenience and accuracy of calculations. To construct an equation you need at least two points. Find the coordinates of another point belonging to this line (x2, y2).
Substitute the coordinate values into the equation of a straight line having the general form y=kx+b. You will get a system of two equations y1=kx1+b and y2=kx2+b. Solve this system, for example, in the following way.
Express b from the first equation and substitute into the second, find k, substitute into any equation and find b. For example, the solution to the system 1=2k+b and 3=5k+b will look like this: b=1-2k, 3=5k+(1-2k); 3k=2, k=1.5, b=1-2*1.5=-2. Thus, the equation of the straight line is y=1.5x-2.
Knowing two points belonging to a line, try to use the canonical equation of a line, it looks like this: (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1). Substitute the values (x1;y1) and (x2;y2), simplify. For example, points (2;3) and (-1;5) belong to the straight line (x-2)/(-1-2)=(y-3)/(5-3); -3(x-2)=2(y-3); -3x+6=2y-6; 2y=12-3x or y=6-1.5x.
To find the equation of a function that has a nonlinear graph, proceed as follows. View all standard charts y=x^2, y=x^3, y=√x, y=sinx, y=cosx, y=tgx, etc. If one of them reminds you of your schedule, use it as a basis.
Draw a standard graph of the base function on the same coordinate axis and find it from your graph. If the graph is moved several units up or down, this means that this number has been added to the function (for example, y=sinx+4). If the graph is moved to the right or left, it means that a number has been added to the argument (for example, y=sin (x+P/2).
An elongated graph in height indicates that the argument function is multiplied by some number (for example, y=2sinx). If the graph, on the contrary, is reduced in height, it means that the number in front of the function is less than 1.
Compare the graph of the base function and your function by width. If it is narrower, then x is preceded by a number greater than 1, wide - a number less than 1 (for example, y=sin0.5x).
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Perhaps the graph corresponds to the found equation only on a certain segment. In this case, indicate for which values of x the resulting equality holds.
A straight line is an algebraic line of the first order. In a Cartesian coordinate system on a plane, the equation of a straight line is given by an equation of the first degree.
You will need
- Knowledge of analytical geometry. Basic knowledge of algebra.
Instructions
The equation is given by two on which this straight line must pass. Let's make a ratio of the coordinates of these points. Let the first point have coordinates (x1,y1), and the second (x2,y2), then the equation of the straight line will be written as follows: (x-x1)/(x2-x1) = (y-y1)(y2-y1).
Let us transform the resulting straight line equation and express y explicitly in terms of x. After this operation, the equation of the straight line will take its final form: y=(x-x1)/((x2-x1)*(y2-y1))+y1.
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If one of the numbers in the denominator is zero, it means that the line is parallel to one of the coordinate axes.
Helpful advice
After you have written the equation of the line, check its correctness. To do this, substitute the coordinates of the points instead of the corresponding coordinates and make sure that the equality is satisfied.
It is often known that y depends linearly on x, and a graph of this dependence is given. In this case, it is possible to find out the equation of the line. First you need to select two points on a straight line.
Instructions
Find the selected points. To do this, lower the perpendiculars from the points on the coordinate axis and write down the numbers from the scale. So for point B from our example, the x coordinate is -2, and the y coordinate is 0. Similarly, for point A the coordinates will be (2;3).
It is known that the straight line has the form y = kx + b. We substitute the coordinates of the selected points into the equation in general form, then for point A we obtain the following equation: 3 = 2k + b. For point B we get another equation: 0 = -2k + b. Obviously, we have a system of two equations with two unknowns: k and b.
Then we solve the system in any convenient way. In our case, it is possible to add the equations of the system, since the unknown k is included in both equations with coefficients that are identical in magnitude but opposite in sign. Then we get 3 + 0 = 2k - 2k + b + b, or, what is the same: 3 = 2b. Thus b = 3/2. Substitute the found value of b into any of the equations to find k. Then 0 = -2k + 3/2, k = 3/4.
Let us substitute the found k and b into the general equation and obtain the desired equation of the straight line: y = 3x/4 + 3/2.
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The coefficient k is called the slope of the line and is equal to the tangent of the angle between the line and the x-axis.
A straight line can be drawn from two points. The coordinates of these points are “hidden” in the equation of the straight line. The equation will tell you all the secrets about the line: how it is rotated, on which side of the coordinate plane it is located, etc.
Instructions
More often it is required to build in a plane. Each point will have two coordinates: x, y. Pay attention to the equation, it obeys the general form: y=k*x ±b, where k, b are free numbers, and y, x are the same coordinates of all points on the line. From the general equation, that to find the y coordinate you need to know x coordinate The most interesting thing is that you can choose any value for the x coordinate: from the entire infinity of known numbers. Next, substitute x into the equation and solve it to find y. Example. Let the equation be given: y=4x-3. Come up with any two values for the coordinates of two points. For example, x1 = 1, x2 = 5. Substitute these values into the equations to find the y coordinates. y1 = 4*1 – 3 = 1. y2 = 4*5 – 3 = 17. We get two points A and B, A (1; 1) and B (5; 17).
You should plot the found points in the coordinate axis, connect them and see the very straight line that was described by the equation. To construct a straight line, you need to work in a Cartesian coordinate system. Draw the X and Y axes. Set the value to “zero” at the intersection point. Plot the numbers on the axes.
In the constructed system, mark the two points found in step 1. The principle of setting the indicated points: point A has coordinates x1 = 1, y1 = 1; on the X-axis select the number 1, on the Y-axis – the number 1. Point A is located at this point. Point B is given by the values x2 = 5, y2 = 17. By analogy, find point B on the graph. Connect A and B to make a straight line.
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The term solving a function as such is not used in mathematics. This formulation should be understood as performing certain actions on a given function in order to find a specific characteristic, as well as finding out the necessary data for constructing a graph of the function.
Instructions
You can consider an approximate diagram according to which the behavior of the function is appropriate and build its graph.
Find the domain of the function. Determine whether the function is even or odd. If you find the desired answer, continue only on the required half-axis. Determine whether the function is periodic. If the answer is positive, continue the study for only one period. Find the points and determine its behavior in the vicinity of these points.
Find the points of intersection of the graph of the function with the coordinate axes. Find them if they exist. Use the first derivative to examine a function for extrema and monotonicity intervals. Also conduct a study using the second derivative for convexity, concavity and inflection points. Select points to refine the function and calculate the function values at them. Construct a graph of the function, taking into account the results obtained from all studies conducted.
On the 0X axis, characteristic points should be identified: discontinuity points, x = 0, function zeros, extremum points, inflection points. These asymptotes will give a sketch of the graph of the function.
So, using a specific example of the function y=((x^2)+1)/(x-1), conduct a study using the first derivative. Rewrite the function as y=x+1+2/(x-1). The first derivative will be equal to y’=1-2/((x-1)^2).
Find critical points of the first kind: y’=0, (x-1)^2=2, the result will be two points: x1=1-sqrt2, x2=1+sqrt2. Mark the obtained values on the domain of definition of the function (Fig. 1).
Determine the sign of the derivative on each of the intervals. Based on the rule of alternating signs from “+” to “-” and from “-” to “+”, you get that the maximum point of the function is x1=1-sqrt2, and the minimum point is x2=1+sqrt2. The same conclusion can be drawn from the sign of the second derivative.
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