Stretching the y = sinx plot along the y-axis. Graph of the function y = sin x Graph of the function y sinx 3

Lesson and presentation on the topic: "Function y = sin (x). Definitions and properties"

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Manuals and simulators in the Integral online store for grade 10 from 1C
We solve problems in geometry. Interactive building tasks for grades 7-10
Software environment "1C: Mathematical Constructor 6.1"

What we will study:

Properties of the function Y = sin (X).
Function graph.
How to build a graph and its scale.
Examples.

Sine properties. Y = sin (X)

Guys, we already got acquainted with trigonometric functions of a numeric argument. Do you remember them?

Let's take a closer look at the function Y = sin (X)

Let's write down some properties of this function:
1) Domain of definition - a set of real numbers.
2) The function is odd. Let's remember the definition of an odd function. A function is called odd if the equality holds: y (-x) = - y (x). As we remember from the ghost formulas: sin (-x) = - sin (x). The definition has been fulfilled, so Y = sin (X) is an odd function.
3) The function Y = sin (X) increases on the segment and decreases on the segment [π / 2; π]. When we move along the first quarter (counterclockwise), the ordinate increases, and when we move along the second quarter, it decreases.

4) The function Y = sin (X) is bounded above and below. This property follows from the fact that
-1 ≤ sin (X) ≤ 1
5) The smallest value of the function is -1 (at x = - π / 2 + πk). The largest value of the function is 1 (at x = π / 2 + πk).

Let's use properties 1-5 to graph the function Y = sin (X). We will build our graph sequentially using our properties. Let's start building a graph on a segment.

Particular attention should be paid to the scale. On the ordinate axis it is more convenient to take a unit segment equal to 2 cells, and on the abscissa axis - to take a unit segment (two cells) equal to π / 3 (see figure).

Plot the sine x function, y = sin (x)

Let's calculate the values of the function on our segment:

Let's build a graph based on our points, taking into account the third property.

Conversion table for ghost formulas

Let's use the second property, which says that our function is odd, which means that it can be reflected symmetrically about the origin:

We know sin (x + 2π) = sin (x). This means that on the segment [- π; π] the graph looks the same as on the segment [π; 3π] or or [-3π; - π] and so on. It remains for us to carefully redraw the graph in the previous figure on the entire abscissa axis.

The graph of the function Y = sin (X) is called a sinusoid.

Let's write a few more properties according to the constructed graph:
6) The function Y = sin (X) increases on any segment of the form: [- π / 2 + 2πk; π / 2 + 2πk], k is an integer and decreases on any interval of the form: [π / 2 + 2πk; 3π / 2 + 2πk], k is an integer.
7) Function Y = sin (X) is a continuous function. Let's look at the graph of the function and make sure that our function has no discontinuities, which means continuity.
8) Range of values: segment [- 1; 1]. This is also clearly seen from the graph of the function.
9) Function Y = sin (X) is a periodic function. Let's look at the graph again and see that the function takes the same values at some intervals.

Examples of sine problems

1. Solve the equation sin (x) = x-π

Solution: Let's build 2 graphs of the function: y = sin (x) and y = x-π (see figure).
Our graphs intersect at one point A (π; 0), this is the answer: x = π

2. Plot the function y = sin (π / 6 + x) -1

Solution: The desired graph is obtained by moving the graph of the function y = sin (x) by π / 6 units to the left and 1 unit down.

Solution: Let's build a graph of the function and consider our segment [π / 2; 5π / 4].
The graph of the function shows that the largest and smallest values are reached at the ends of the segment, at points π / 2 and 5π / 4, respectively.
Answer: sin (π / 2) = 1 is the largest value, sin (5π / 4) = the smallest value.

Sine problems for independent solution

Solve the equation: sin (x) = x + 3π, sin (x) = x-5π
Plot function y = sin (π / 3 + x) -2
Plot function y = sin (-2π / 3 + x) +1
Find the largest and smallest value of the function y = sin (x) on an interval
Find the largest and smallest value of the function y = sin (x) on the segment [- π / 3; 5π / 6]

We found out that the behavior of trigonometric functions, and functions y = sin x in particular, on the whole number line (or for all values of the argument NS) is completely determined by its behavior in the interval 0 < NS < π / 2 .

Therefore, first of all, we will plot the function y = sin x precisely in this interval.

Let's compose the following table of the values of our function;

Marking the corresponding points on the coordinate plane and connecting them with a smooth line, we get the curve shown in the figure

The resulting curve could be constructed geometrically, without compiling a table of function values y = sin x .

1. Divide the first quarter of a circle of radius 1 into 8 equal parts. The ordinates of the points of division of the circle are the sines of the corresponding angles.

2.The first quarter of a circle corresponds to angles from 0 to π / 2 ... Therefore, on the axis NS take a segment and divide it into 8 equal parts.

3. Let's draw straight lines parallel to the axes NS, and from the division points, we will restore the perpendiculars to the intersection with the horizontal lines.

4. Connect the intersection points with a smooth line.

Now let's turn to the interval π / 2 < NS < π .
Each argument value NS from this interval can be represented as

x = π / 2 + φ

where 0 < φ < π / 2 ... By reduction formulas

sin ( π / 2 + φ ) = cos φ = sin ( π / 2 - φ ).

Axis points NS with abscissas π / 2 + φ and π / 2 - φ symmetrical to each other about the axis point NS with abscissa π / 2 , and the sinuses at these points are the same. This allows you to get a graph of the function y = sin x in the interval [ π / 2 , π ] by simple symmetric display of the graph of this function in the interval relative to the straight line NS = π / 2 .

Now using the property odd function y = sin x,

sin (- NS) = - sin NS,

it is easy to plot this function in the interval [- π , 0].

The function y = sin x is periodic with a period of 2π ;. Therefore, to plot the entire graph of this function, the curve shown in the figure is sufficient, continue to the left and right periodically with a period 2π .

The resulting curve is called sinusoid ... It is the graph of the function y = sin x.

The figure illustrates well all those properties of the function. y = sin x , which were previously proven by us. Let us recall these properties.

1) Function y = sin x defined for all values NS , so that the domain of its definition is the collection of all real numbers.

2) Function y = sin x limited. All values that it takes are in the range from -1 to 1, including these two numbers. Therefore, the range of variation of this function is determined by the inequality -1 < at < 1. When NS = π / 2 + 2k π function takes highest values equal to 1, and for x = - π / 2 + 2k π - the smallest values equal to - 1.

3) Function y = sin x is odd (the sinusoid is symmetrical about the origin).

4) Function y = sin x periodic with period 2 π .

5) In intervals 2n π < x < π + 2n π (n is any integer) it is positive, and in the intervals π + 2k π < NS < 2π + 2k π (k is any integer) it is negative. For x = k π the function vanishes. Therefore, these values of the argument x (0; ± π ; ± 2 π ; ...) are called zeros of the function y = sin x

6) In intervals - π / 2 + 2n π < NS < π / 2 + 2n π function y = sin x increases monotonically, and in intervals π / 2 + 2k π < NS < 3π / 2 + 2k π it decreases monotonically.

Pay special attention to the behavior of the function. y = sin x near point NS = 0 .

For example, sin 0.012 ≈ 0.012; sin (-0.05) ≈ -0,05;

sin 2 ° = sin π 2 / 180 = sin π / 90 ≈ 0,03 ≈ 0,03.

At the same time, it should be noted that for any values of x

| sin x| < | x | . (1)

Indeed, let the radius of the circle shown in the figure be 1,
a / AОВ = NS.

Then sin x= AC. But AC< АВ, а АВ, в свою очередь, меньше длины дуги АВ, на которую опирается угол NS... The length of this arc is obviously equal to NS, since the radius of the circle is 1. So, at 0< NS < π / 2

sin x< х.

Hence, due to the oddness of the function y = sin x it is easy to show that for - π / 2 < NS < 0

| sin x| < | x | .

Finally, at x = 0

| sin x | = | x |.

Thus, for | NS | < π / 2 inequality (1) is proved. In fact, this inequality is also true for | x | > π / 2 due to the fact that | sin NS | < 1, a π / 2 > 1

Exercises

1.On schedule function y = sin x determine: a) sin 2; b) sin 4; c) sin (-3).

2.On schedule function y = sin x determine which number is from the interval
[ - π / 2 , π / 2 ] has a sine equal to: a) 0.6; b) -0.8.

3. By function schedule y = sin x determine which numbers have a sine,
equal to 1/2.

4. Find approximately (without using tables): a) sin 1 °; b) sin 0.03;
c) sin (-0.015); d) sin (-2 ° 30 ").

How to plot the function y = sin x? First, let's look at the sine graph in the interval.

We take a single segment with a length of 2 cells of a notebook. Mark one on the Oy axis.

For convenience, we round the number π / 2 to 1.5 (and not to 1.6, as required by the rounding rules). In this case, a segment of length π / 2 corresponds to 3 cells.

On the Ox axis, we mark not unit segments, but segments of length π / 2 (every 3 cells). Accordingly, a segment of length π corresponds to 6 cells, a segment of length π / 6 - 1 cell.

With this choice of a unit segment, the graph depicted on a sheet of a notebook in a box corresponds as much as possible to the graph of the function y = sin x.

Let's compose a table of sine values in the interval:

We mark the obtained points on the coordinate plane:

Since y = sin x is an odd function, the sine graph is symmetrical about the origin - point O (0; 0). Taking this fact into account, we will continue to plot the graph to the left, then the points -π:

The function y = sin x is periodic with a period T = 2π. Therefore, the graph of the function, taken on the interval [-π; π], is repeated an infinite number of times to the right and to the left.

"Yoshkar-Ola technical school of service technologies"

Construction and study of the graph of the trigonometric function y = sinx in a table processorMS Excel

/ methodical development /

Yoshkar - Ola

Theme. Plotting and researching a trigonometric function graphy = sinx in MS Excel spreadsheet processor

Lesson type- integrated (gaining new knowledge)

Goals:

Didactic goal - explore the behavior of trigonometric function graphsy= sinxdepending on the odds using a computer

Educational:

1. Find out the change in the graph of the trigonometric function y= sin x depending on the coefficients

2. Show the introduction of computer technologies in teaching mathematics, the integration of two subjects: algebra and computer science.

3. To form the skills of using computer technologies in mathematics lessons

4. Strengthen the skills of researching functions and building their graphs

Developing:

1. To develop students' cognitive interest in academic disciplines and the ability to apply their knowledge in practical situations

2. Develop the ability to analyze, compare, highlight the main thing

3. Promote improvement general level student development

Upbringing :

1. To bring up independence, accuracy, diligence

2. Foster a culture of dialogue

Forms of work in the lesson - combined

Didactic equipment and equipment:

1. Computers

2. Multimedia projector

4. Handout material

5. Presentation slides

During the classes

I. Organization of the beginning of the lesson

Greetings from students and guests

Inspiration for the lesson

II... Goal setting and actualization of the topic

It takes a lot of time to study a function and build its graph, you have to perform a lot of cumbersome calculations, it is not convenient, computer technologies come to the rescue.

Today we will learn how to build graphs of trigonometric functions in the MS Excel 2007 spreadsheet environment.

The topic of our lesson is "Construction and study of the graph of a trigonometric function y= sinx in a table processor "

From the course of algebra, we know the scheme for studying a function and plotting its graph. Let's remember how to do this.

Slide 2

Function study diagram

1. Domain of the function (D (f))

2. Range of values of the function E (f)

3. Determination of parity

4. Frequency

5. Zeros of the function (y = 0)

6. Intervals of constancy (y> 0, y<0)

7. Intervals of monotony

8. Extrema of function

III. Primary assimilation of new educational material

Open MS Excel 2007.

Plot the function y = sin x

Plotting in a spreadsheet processorMS Excel 2007

The graph of this function will be plotted on the segment xЄ [-2π; 2π]

The argument values will be taken with a step , to make the graph more accurate.

Since the editor works with numbers, let's convert radians to numbers, knowing that P ≈ 3.14 ... (translation table in the handout).

1. Find the value of the function at the point x = -2P. For the rest of the argument, the editor calculates the corresponding values of the function automatically.

2. Now we have a table with the values of the argument and function. With this data, we have to plot this function using the Chart Wizard.

3. To build a graph, you need to select the required data range, lines with the values of the argument and function

4..jpg "width =" 667 "height =" 236 src = ">

We write conclusions in a notebook (Slide 5)

Output. The graph of a function of the form y = sinx + k is obtained from the graph of the function y = sinx using parallel translation along the OY axis by k units

If k> 0, then the graph is shifted up by k units

If k<0, то график смещается вниз на k единиц

Building and examining a function of the formy =k* sinx,k- const

Task 2. At work Liste2 plot functions in one coordinate system y= sinx y=2* sinx, y= * sinx, on the interval (-2π; 2π) and see how the graph view changes.

(In order not to re-set the value of the argument, let's copy the existing values. Now you need to set the formula, and build a graph from the resulting table.)

We compare the resulting graphs. Let's analyze together with the students the behavior of the graph of the trigonometric function depending on the coefficients. (Slide 6)

https://pandia.ru/text/78/510/images/image005_66.gif "width =" 16 "height =" 41 src = "> x , on the interval (-2π; 2π) and see how the graph view changes.

We compare the resulting graphs. Let's analyze together with the students the behavior of the graph of the trigonometric function depending on the coefficients. (Slide 8)

https://pandia.ru/text/78/510/images/image008_35.jpg "width =" 649 "height =" 281 src = ">

We write conclusions in a notebook (Slide 11)

Output. The graph of a function of the form y = sin (x + k) is obtained from the graph of the function y = sinx using parallel translation along the OX axis by k units

If k> 1, then the graph is shifted to the right along the OX axis

If 0

IV... Primary consolidation of the acquired knowledge

Differentiated cards with a task for building and researching a function using a graph


	Y = 6* sin (x)	Y =1-2 sinNS	Y =- sin(3x +)
1. Domain
2. Value range
3. Parity
4. Periodicity
5. Intervals of constancy


6. Gapsmonotony
The function is increasing
Function decreases
7. Function extrema
Minimum
Maximum

V... Organization of homework

Build a graph of the function y = -2 * sinx + 1, investigate and check the correctness of construction in the environment of a Microsoft Excel spreadsheet. (Slide 12)

VI... Reflection

Stretching the y = sinx plot along the y-axis. The function y = 3sinx is given. To plot its graph, you need to Stretch the graph y = sinx so that E (y): (-3; 3).

Picture 7 from the presentation "Build a graph of a function" to algebra lessons on the topic "Graph of a function"

Dimensions: 960 x 720 pixels, format: jpg. To download a picture for an algebra lesson for free, right-click on the image and click "Save Image As ...". To show pictures in the lesson, you can also download the presentation "Plot function graph.ppt" for free with all pictures in a zip-archive. The archive size is 327 KB.

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