Functiony = sinx
The graph of the function is a sinusoid.
The complete non-repeating portion of a sine wave is called a sine wave.
Half a sine wave is called a half sine wave (or arc).
Function propertiesy =
sinx:
3) This is an odd function. 4) This continuous function.
6) On the segment [-π/2; π/2] function increases on the interval [π/2; 3π/2] – decreases. 7) On intervals the function takes positive values. 8) Intervals of increasing function: [-π/2 + 2πn; π/2 + 2πn]. 9) Minimum points of the function: -π/2 + 2πn. |
To graph a function y= sin x It is convenient to use the following scales:
On a sheet of paper with a square, we take the length of two squares as a unit of segment.
On axis x Let's measure the length π. At the same time, for convenience, we present 3.14 in the form of 3 - that is, without a fraction. Then on a sheet of paper in a cell π will be 6 cells (three times 2 cells). And each cell will receive its own natural name (from the first to the sixth): π/6, π/3, π/2, 2π/3, 5π/6, π. These are the meanings x.
On the y-axis we mark 1, which includes two cells.
Let's create a table of function values using our values x:
√3 | √3 |
Next, let's create a schedule. It will turn out to be a half wave, highest point which (π/2; 1). This is the graph of the function y= sin x on the segment. Let's add a symmetrical half-wave to the constructed graph (symmetrical relative to the origin, that is, on the segment -π). The crest of this half-wave is under the x-axis with coordinates (-1; -1). The result will be a wave. This is the graph of the function y= sin x on the segment [-π; π].
You can continue the wave by constructing it on the segment [π; 3π], [π; 5π], [π; 7π], etc. On all these segments, the graph of the function will look the same as on the segment [-π; π]. You will get a continuous wavy line with identical waves.
Functiony = cosx.
The graph of a function is a sine wave (sometimes called a cosine wave).
Function propertiesy = cosx:
1) The domain of definition of a function is the set of real numbers. 2) The range of function values is the segment [–1; 1] 3) This is an even function. 4) This is a continuous function. 5) Coordinates of the intersection points of the graph: 6) On the segment the function decreases, on the segment [π; 2π] – increases. 7) On intervals [-π/2 + 2πn; π/2 + 2πn] function takes positive values. 8) Increasing intervals: [-π + 2πn; 2πn]. 9) Minimum points of the function: π + 2πn. 10) The function is limited from above and below. Lowest value functions –1, 11) This periodic function with period 2π (T = 2π) |
Functiony = mf(x).
Let's take the previous function y=cos x. As you already know, its graph is a sine wave. If we multiply the cosine of this function by a certain number m, then the wave will expand from the axis x(or will shrink, depending on the value of m).
This new wave will be the graph of the function y = mf(x), where m is any real number.
Thus, the function y = mf(x) is the familiar function y = f(x) multiplied by m.
Ifm< 1, то синусоида сжимается к оси x by the coefficientm. Ifm > 1, then the sinusoid is stretched from the axisx by the coefficientm.
When performing stretching or compression, you can first plot only one half-wave of a sine wave, and then complete the entire graph.
Functiony = f(kx).
If the function y =mf(x) leads to stretching of the sinusoid from the axis x or compression towards the axis x, then the function y = f(kx) leads to stretching from the axis y or compression towards the axis y.
Moreover, k is any real number.
At 0< k< 1 синусоида растягивается от оси y by the coefficientk. Ifk > 1, then the sinusoid is compressed towards the axisy by the coefficientk.
When graphing this function, you can first build one half-wave of a sine wave, and then use it to complete the entire graph.
Functiony = tgx.
Function graph y= tg x is a tangent.
It is enough to construct part of the graph in the interval from 0 to π/2, and then you can symmetrically continue it in the interval from 0 to 3π/2.
Function propertiesy = tgx:
Functiony = ctgx
Function graph y=ctg x is also a tangentoid (it is sometimes called a cotangentoid).
Function propertiesy = ctgx:
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Iron rusts without finding any use,
Still water rots or freezes in the cold,
and a person’s mind, not finding any use for itself, languishes.
Leonardo da Vinci
Technologies used: problem-based learning, critical thinking, communicative communication.
Goals:
- Development of cognitive interest in learning.
- Studying the properties of the function y = sin x.
- Formation of practical skills in constructing a graph of the function y = sin x based on the studied theoretical material.
Tasks:
1. Use the existing potential of knowledge about the properties of the function y = sin x in specific situations.
2. Apply conscious establishment of connections between analytical and geometric models of the function y = sin x.
Develop initiative, a certain willingness and interest in finding a solution; the ability to make decisions, not stop there, and defend your point of view.
To foster in students cognitive activity, a sense of responsibility, respect for each other, mutual understanding, mutual support, and self-confidence; culture of communication.
Lesson progress
Stage 1. Updating basic knowledge, motivating learning new material
"Entering the lesson."
There are 3 statements written on the board:
- Trigonometric sin equation t = a always has solutions.
- Schedule odd function can be constructed using a symmetry transformation about the Oy axis.
- Schedule trigonometric function can be constructed using one main half-wave.
Students discuss in pairs: are the statements true? (1 minute). The results of the initial discussion (yes, no) are then entered into the table in the "Before" column.
The teacher sets the goals and objectives of the lesson.
2. Updating knowledge (frontally on a model of a trigonometric circle).
We have already become acquainted with the function s = sin t.
1) What values can the variable t take. What is the scope of this function?
2) In what interval are the values of the expression sin t contained? Find the largest and smallest values of the function s = sin t.
3) Solve the equation sin t = 0.
4) What happens to the ordinate of a point as it moves along the first quarter? (the ordinate increases). What happens to the ordinate of a point as it moves along the second quarter? (the ordinate gradually decreases). How does this relate to the monotonicity of the function? (the function s = sin t increases on the segment and decreases on the segment ).
5) Let’s write the function s = sin t in the form y = sin x that is familiar to us (we will construct it in the usual xOy coordinate system) and compile a table of the values of this function.
| X | 0 | ||||||
| at | 0 | 1 | 0 |
Stage 2. Perception, comprehension, primary consolidation, involuntary memorization
Stage 4. Primary systematization of knowledge and methods of activity, their transfer and application in new situations
6. No. 10.18 (b,c)
Stage 5. Final control, correction, assessment and self-assessment
7. Return to the statements (beginning of the lesson), discuss using the properties of the trigonometric function y = sin x, and fill in the “After” column in the table.
8. D/z: clause 10, No. 10.7(a), 10.8(b), 10.11(b), 10.16(a)
In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t on coordinate circle and consider the graph of a function on a circle and a line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve several simple problems using the graph of a function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its basic properties and graph
When considering a function, it is important to associate each argument value with a single function value. This law of correspondence and is called a function.
Let us define the correspondence law for .
Any real number corresponds to a single point on the unit circle. A point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is associated with a single function value.
Obvious properties follow from the definition of sine.
The figure shows that 
because is the ordinate of a point on the unit circle.
Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding function values.
For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)
We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over the entire domain of definition (Fig. 3).
The main period of the function is This means that the graph can be obtained on a segment and then continued throughout the entire domain of definition.
Consider the properties of the function:
1) Scope of definition:
2) Range of values: 
3) Odd function:
4) Smallest positive period:
5) Coordinates of the points of intersection of the graph with the abscissa axis: 
6) Coordinates of the point of intersection of the graph with the ordinate axis:
7) Intervals at which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Decreasing intervals:
11) Minimum points: 
12) Minimum functions:
13) Maximum points: 
14) Maximum functions:
We looked at the properties of the function and its graph. The properties will be used repeatedly when solving problems.
References
1. Algebra and beginning of analysis, grade 10 (in two parts). Tutorial for educational institutions (profile level) ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for 10th grade ( training manual for students of schools and classes with in-depth study of mathematics).-M.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-Depth Study algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed.
A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.4, 16.5, 16.8.
Additional web resources
3. Educational portal to prepare for exams ().
In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t on the coordinate circle and consider the graph of the function on the circle and line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve several simple problems using the graph of a function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its basic properties and graph
When considering a function, it is important to associate each argument value with a single function value. This law of correspondence and is called a function.
Let us define the correspondence law for .
Any real number corresponds to a single point on the unit circle. A point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is associated with a single function value.
Obvious properties follow from the definition of sine.
The figure shows that because is the ordinate of a point on the unit circle.
Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is the central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding values of the function.
For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)
We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over the entire domain of definition (Fig. 3).
The main period of the function is This means that the graph can be obtained on a segment and then continued throughout the entire domain of definition.
Consider the properties of the function:
1) Scope of definition:
2) Range of values:
3) Odd function:
4) Smallest positive period:
5) Coordinates of the points of intersection of the graph with the abscissa axis:
6) Coordinates of the point of intersection of the graph with the ordinate axis:
7) Intervals at which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Decreasing intervals:
11) Minimum points:
12) Minimum functions:
13) Maximum points:
14) Maximum functions:
We looked at the properties of the function and its graph. The properties will be used repeatedly when solving problems.
References
1. Algebra and beginning of analysis, grade 10 (in two parts). Textbook for general education institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 (textbook for students of schools and classes with in-depth study of mathematics). - M.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-depth study of algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed.
A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.4, 16.5, 16.8.
Additional web resources
3. Educational portal for exam preparation ().
In this lesson we will take a detailed look at the function y = sin x, its basic properties and graph. At the beginning of the lesson, we will give the definition of the trigonometric function y = sin t on the coordinate circle and consider the graph of the function on the circle and line. Let's show the periodicity of this function on the graph and consider the main properties of the function. At the end of the lesson, we will solve several simple problems using the graph of a function and its properties.
Topic: Trigonometric functions
Lesson: Function y=sinx, its basic properties and graph
When considering a function, it is important to associate each argument value with a single function value. This law of correspondence and is called a function.
Let us define the correspondence law for .
Any real number corresponds to a single point on the unit circle. A point has a single ordinate, which is called the sine of the number (Fig. 1).
Each argument value is associated with a single function value.
Obvious properties follow from the definition of sine.
The figure shows that because is the ordinate of a point on the unit circle.
Consider the graph of the function. Let us recall the geometric interpretation of the argument. The argument is the central angle, measured in radians. Along the axis we will plot real numbers or angles in radians, along the axis the corresponding values of the function.
For example, an angle on the unit circle corresponds to a point on the graph (Fig. 2)
We have obtained a graph of the function in the area. But knowing the period of the sine, we can depict the graph of the function over the entire domain of definition (Fig. 3).
The main period of the function is This means that the graph can be obtained on a segment and then continued throughout the entire domain of definition.
Consider the properties of the function:
1) Scope of definition:
2) Range of values:
3) Odd function:
4) Smallest positive period:
5) Coordinates of the points of intersection of the graph with the abscissa axis:
6) Coordinates of the point of intersection of the graph with the ordinate axis:
7) Intervals at which the function takes positive values:
8) Intervals at which the function takes negative values:
9) Increasing intervals:
10) Decreasing intervals:
11) Minimum points:
12) Minimum functions:
13) Maximum points:
14) Maximum functions:
We looked at the properties of the function and its graph. The properties will be used repeatedly when solving problems.
References
1. Algebra and beginning of analysis, grade 10 (in two parts). Textbook for general education institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2009.
2. Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed. A. G. Mordkovich. -M.: Mnemosyne, 2007.
3. Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I. Algebra and mathematical analysis for grade 10 (textbook for students of schools and classes with in-depth study of mathematics). - M.: Prosveshchenie, 1996.
4. Galitsky M.L., Moshkovich M.M., Shvartsburd S.I. In-depth study of algebra and mathematical analysis.-M.: Education, 1997.
5. Collection of problems in mathematics for applicants to higher educational institutions (edited by M.I. Skanavi). - M.: Higher School, 1992.
6. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebraic simulator.-K.: A.S.K., 1997.
7. Sahakyan S.M., Goldman A.M., Denisov D.V. Problems on algebra and principles of analysis (a manual for students in grades 10-11 of general education institutions). - M.: Prosveshchenie, 2003.
8. Karp A.P. Collection of problems on algebra and principles of analysis: textbook. allowance for 10-11 grades. with depth studied Mathematics.-M.: Education, 2006.
Homework
Algebra and beginning of analysis, grade 10 (in two parts). Problem book for educational institutions (profile level), ed.
A. G. Mordkovich. -M.: Mnemosyne, 2007.
№№ 16.4, 16.5, 16.8.
Additional web resources
3. Educational portal for exam preparation ().
