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Graphs of trigonometric functions Function y \u003d sin x, its properties Transforming graphs of trigonometric functions by parallel transfer Transforming graphs of trigonometric functions by compressing and expanding For the curious ...

trigonometric functions The graph of the function y \u003d sin x is a sinusoid Function properties: D (y) \u003d R Periodic (T \u003d 2 ) Odd (sin (-x) \u003d -sin x) Zeros of the function: y \u003d 0, sin x \u003d 0 at x =  n, n  Z y=sin x

trigonometric functions Properties of the function y = sin x

trigonometric functions Function properties y= sin x 6. Intervals of monotonicity: the function increases on intervals of the form:  -  /2 +2  n ;  / 2+2  n   n  Z y = sin x

trigonometric functions Properties of the function y= sin x Intervals of monotonicity: the function decreases on intervals of the form:  /2 +2  n ; 3  / 2+2  n   n  Z y=sin x

trigonometric functions Function properties y \u003d sin x 7. Extreme points: X max \u003d  / 2 +2  n, n  Z X m in = -  / 2 +2  n, n  Z y \u003d sin x

trigonometric functions Properties of the function y \u003d sin x 8. Range of values: E(y) =  -1;1  y = sin x

trigonometric functions Transforming graphs of trigonometric functions The graph of the function y = f (x + b) is obtained from the graph of the function y \u003d f (x) by parallel translation by (-v) units along the abscissa The graph of the function y \u003d f (x) + a is obtained from the graph functions y \u003d f (x) by parallel translation by (a) units along the y-axis

Trigonometric functions

trigonometric functions Convert graphs of trigonometric functions y =sin (x+  /4) Plot the function: y=sin (x -  /6)

trigonometric functions Transforming graphs of trigonometric functions y = sin x +  Plot the function: y =sin (x -  /6)

trigonometric functions Transforming graphs of trigonometric functions y= sin x +  Graph the function: y=sin (x +  /2) remember the rules

trigonometric functions The graph of the function y \u003d cos x is a cosine List the properties of the function y \u003d cos x sin (x +  / 2) \u003d cos x

trigonometric functions Transforming the graphs of trigonometric functions by squeezing and stretching The graph of the function y = k f (x) is obtained from the graph of the function y = f(x) by stretching it k times (for k>1) along the y-axis The graph of the function y = k f (x ) is obtained from the graph of the function y = f(x) by compressing it k times (at 0

trigonometric functions Transform graphs of trigonometric functions by squeezing and stretching y=sin2x y=sin4x Y=sin0.5x remember the rules

trigonometric functions Transforming graphs of trigonometric functions by squeezing and stretching The graph of the function y \u003d f (kx) is obtained from the graph of the function y \u003d f (x) by squeezing it k times (for k> 1) along the abscissa The graph of the function y \u003d f (kx ) is obtained from the graph of the function y \u003d f (x) by stretching it k times (at 0

trigonometric functions Transform graphs of trigonometric functions by squeezing and stretching y = cos2x y = cos 0.5x remember the rules

trigonometric functions Transforming the graphs of trigonometric functions by squeezing and stretching The graphs of the functions y = -f (kx) and y=- k f(x) are obtained from the graphs of the functions y = f(kx) and y= k f(x), respectively, by mirroring them with respect to abscissa axis sine is an odd function, so sin(-kx) = - sin (kx) cosine is an even function, so cos(-kx) = cos(kx)

trigonometric functions Transform graphs of trigonometric functions by squeezing and stretching y=-sin3x y=sin3x remember the rules

trigonometric functions Transform graphs of trigonometric functions by squeezing and stretching y=2cosx y=-2cosx remember the rules

trigonometric functions Converting graphs of trigonometric functions by squeezing and stretching The graph of the function y = f (kx+b) is obtained from the graph of the function y = f(x) by translating it in parallel by (-to /k) units along the abscissa and by squeezing into k times (for k>1) or stretching k times (for 0

trigonometric functions Convert graphs of trigonometric functions by squeezing and stretching Y= cos(2x+  /3) y=cos(x+  /6) y= cos(2x+  /3) y= cos(2(x+  /6)) y = cos(2x+  /3) y= cos(2(x+  /6)) Y= cos(2x+  /3) y=cos2x remember the rules

trigonometric functions For the curious... See what the graphs of some other trigs look like. functions: y = 1 / cos x or y=sec x (read secons) y = cosec x or y= 1/ sin x read cosecons

On the topic: methodological developments, presentations and notes

DER "Conversion of graphs of trigonometric functions" Grades 10-11

Section of the curriculum: “Trigonometric functions”. Lesson type: digital educational resource of a combined algebra lesson. According to the form of presentation of the material: Combined (universal) DER with ...

Methodical development of a lesson in mathematics: "Conversion of graphs of trigonometric functions"

Methodical development of a lesson in mathematics: "Transformation of graphs of trigonometric functions" for tenth grade students. The lesson is accompanied by a presentation....

Trigonometric charts functions

• Function y = sinx, its properties

• Transforming graphs of trigonometric functions by parallel translation
• Convert graphs of trigonometric functions by compressing and expanding

• For the curious…
• Author

Function Graph y= sin x is sinusoid

y = sin x

Function Properties :

• D(y)=R2. Periodic (T=2  )

3. odd ( sin(-x)=-sin x) 4. Function nulls:

y=0, sinx=0 at x =  n, n  Z

Function properties y = sin x

y = sin x

5. Constancy intervals :

at 0 at X   (0+2  n ;  +2  n ) ,n  Z

at at x   ( -  +2  n ; 0+2  n),n  Z

Function properties y= sin x

6. Intervals of monotonicity :

the function increases over the intervals

type:  -  /2 +2  n ;  / 2+2  n   n  Z

Function properties y= sin x

Monotonic intervals:

the function decreases over the intervals

type:  /2 +2  n ; 3  / 2+2  n   n  Z

Function properties y = sin x

x min

x min

x max

x max

7 . extremum points :

x max =  / 2 +2  n , n  Z

x m in = -  / 2 +2  n , n  Z

Function properties y = sin x

8 . Range of values :

E(y) =  -1;1 

Graph Conversion trigonometric functions

• Graph of the function y = f(x +c) is obtained from the graph of the function y = f(x) parallel translation by (-in) units along the x-axis
• Graph of the function y = f(x )+a is obtained from the graph of the function y = f(x) parallel translation by (a) units along the y-axis

Plot

Functions y = sin(x+  /4 )

y = sin x

recall

rules

Plot

features: y=sin(x -  /6)

y=sin(x+  /4 )

Plot

features:

y = sin x + 

y=sin(x-  /6 )

y= sin x + 

Plot

features: y=sin (x +  /2)

recall

rules

Function Graph y= cos x is cosine wave

sin(x+  /2)=cos x

List Properties

functions y = cos x

by compression and stretching

• Graph of the function y = k f(x y= f(x) by stretching it into k times (when k1) along the y-axis
• Graph of the function y = k f (x ) is obtained from the graph of the function y= f(x) by compressing it into 1/k times (when 0 along the y-axis

by compression and stretching

y=0.5sinx

recall

rules

by compression and stretching

• Graph of the function y = f(kx ) is obtained from the graph of the function y= f(x) by compressing it into k times (when k1) along the abscissa
• Graph of the function y = f(kx ) is obtained from the graph of the function y= f(x) by stretching it into 1/k times (when 0 along the abscissa

by compression and stretching

y=cos2x

y = cos 0.5x

recall

rules

by compression and stretching

• Graphs of functions y = -f (kx ) and y=- f(x) obtained from function graphs y= f(kx) And y=kf(x) respectively, by mirroring them with respect to the x-axis
• sine is an odd function, so sin(-kx) = - sin(kx)

cosine is an even function, so cos(-kx) = cos(kx)

by compression and stretching

y= - 3sinx

y=3sinx

recall

rules

by compression and stretching

y=-2cosx

recall

rules

by compression and stretching

• Function Graph y= f (kx+b ) obtained from the graph of the function y= f(x) by transferring it in parallel to (-V /k) units along the x-axis and by shrinking into k times (when k1) or stretching in 1/k times (when 0 along the abscissa
• f(x+b) = f(k(x+b/k))

by compression and stretching

y= cos(2x+  /3)

y= cos(2(x+  /6))

y= cos(2x+  /3)

y= cos(2(x+  /6))

y=cos(x+  /6)

Y= cos(2x+  /3)

Y= cos(2x+  /3)

recall

rules

For the curious…

See what the graphs of some other trigs look like. functions :

y = cosec x or y= 1/sin x

read cosecons

y = 1 / cos x or y=sec x

( secons is read)

You can read about trigonometric functions in the works :

• Definition of trigonometric functions

• On periods of trigonometric functions

• Sine and cosine plots
• Tangent and cotangent plots
• Formulas casts
• The simplest trigonometric equations

Mathematic teacher

Derzhavin Lyceum

Petrozavodsk

Prisakar

Olga Borisovna

(mail : [email protected])

• Write me your

Graphing algorithm The graph of the function y = sin (x-a) can be obtained by parallel transfer of the graph of the function y = sinx along the Ox axis by a units to the right. The graph of the function y \u003d sin (x + a) can be obtained by parallel transfer of the graph of the function y \u003d sinx along the Ox axis by a units to the left.

0) can be obtained from the graph of the function y = sin x by its expansion (at 00) can be obtained from the graph of the function y = sin x by its expansion (at 0 7 Graphing Algorithm The graph of the function y = sin (Kx) (K>0) can be obtained from the graph of the function y = sin x by stretching it (when 01 is compressed by K times) along the Ox axis. 0) can be obtained from the graph of the function y \u003d sin x by stretching it (at 0 0) can be obtained from the graph of the function y \u003d sin x by its stretching (at 01 by shrinking by K times) along the Ox axis. "\u003e 0) can be obtained from the graph of the function y = sin x by its expansion (at 00) can be obtained from the plot of the function y = sin x by its expansion (at 0 title="Graphing Algorithm The graph of the function y = sin (Kx) (K>0) can be obtained from graph of the function y = sin x by its expansion (at 0

8 Squeeze and stretch to the ordinate Plot the function y = sin2 x Plot the function y = sin K > 1 squeeze 0 1 squeeze 0 1 squeeze 0 1 squeeze 0 1 squeeze 0 title="8 Сжатие и растяжение к оси ординат Построить график функции у = sin2 х Построить график функции у = sin K > 1 сжатие 0 !}

0) can be obtained from the graph of the function y \u003d sin x by stretching it (for K> 1 by stretching by K times) along the Oy axis. The plot of the function y = Ksin (x) (K>0) can be obtained from the plot of the function y = sinx with "title=" Graphing algorithm: The plot of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y \u003d sin x by stretching it (for K> 1 by stretching it by K times) along the Oy axis. The graph of the function y \u003d Кsin (x) (K> 0) can be obtained from the graph of the function y \u003d sinx with" class="link_thumb"> 9 !} Graphing algorithm: The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 stretching by K times) along the Oy axis. The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx by its compression (at 01 stretching by K times) along the Oy axis. The graph of the function y \u003d Ksin (x) (K> 0) can be obtained from the graph of the function y \u003d sinx its c "\u003e 0) can be obtained from the graph of the function y \u003d sin x by stretching it (for K> 1 by stretching K times) along the axis Oy. The graph of the function y \u003d Ksin (x) (K> 0) can be obtained from the graph of the function y \u003d sinx by its compression (with 01 stretching by K times) along the Oy axis. Graph of the function y \u003d Ksin (x) (K> 0) can be obtained from the graph of the function y = sinx with "title=" Graphing algorithm: The graph of the function y = Ksin (x) (K> 0) can be obtained from the graph of the function y = sin x by stretching it (at K> 1 by stretching K times) along the Oy axis.The graph of the function y \u003d Ksin (x) (K> 0) can be obtained from the graph of the function y \u003d sinx with"> title="Graphing algorithm: The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 stretching by K times) along the Oy axis. The graph of the function y \u003d Ksin (x) (K> 0) can be obtained from the graph of the function y \u003d sinx with">!}

1 stretch 0 1 stretch 0 10 10 Shrink and stretch to x-axis K > 1 stretch 0 1 stretch 0 1 stretch 0 1 stretch 0 1 stretch 0 title="10 Shrink and stretch to x-axis K > 1 stretch 0

13 Shift along the ordinate Plot the function y=sins+3 Plot the function y=sins-3 + up - down y = sinx y = sinx + 3 y = sinx y = sinx Graph transformation

X y 1 -2 Check: y 1 = sinx; y 2 = sinx + 2; y 3 = sinx

Summary of the lesson of algebra and the beginning of analysis in grade 10

on the topic: "Conversion of graphs of trigonometric functions"

The purpose of the lesson: to systematize knowledge on the topic "Properties and graphs of trigonometric functions y \u003d sin (x), y \u003d cos (x)".

Lesson objectives:

• repeat the properties of trigonometric functions y \u003d sin (x), y \u003d cos (x);
• repeat the reduction formulas;
• conversion of graphs of trigonometric functions;
• develop attention, memory, logical thinking; to activate mental activity, the ability to analyze, generalize and reason;
• education of industriousness, diligence in achieving the goal, interest in the subject.

Lesson equipment:ict

Lesson type: learning new

During the classes

Before the lesson, 2 students on the board build graphs from their homework.

Organizing time:

Hello guys!

Today in the lesson we will convert the graphs of trigonometric functions y \u003d sin (x), y \u003d cos (x).

Oral work:

Checking homework.

solving puzzles.

Learning new material

All transformations of function graphs are universal - they are suitable for all functions, including trigonometric ones. Here we confine ourselves to a brief reminder of the main transformations of graphs.

Transformation of graphs of functions.

The function y \u003d f (x) is given. We start building all graphs from the graph of this function, then we perform actions with it.

Function

What to do with the schedule

y = f(x) + a

We raise all points of the first graph by a units up.

y = f(x) – a

All points of the first graph are lowered by a units down.

y = f(x + a)

We shift all points of the first graph by a units to the left.

y = f (x - a)

We shift all points of the first graph by a units to the right.

y = a*f(x),a>1

We fix the zeros in place, we shift the upper points higher by a times, the lower ones we lower lower by a times.

The graph will "stretch" up and down, the zeros remain in place.

y = a*f(x), a<1

We fix the zeros, the upper points will go down a times, the lower ones will rise a times. The graph will “shrink” to the x-axis.

y=-f(x)

Mirror the first graph about the x-axis.

y = f(ax), a<1

Fix a point on the y-axis. Each segment on the x-axis is increased by a times. The graph will stretch from the y-axis in different directions.

y = f(ax), a>1

Fix a point on the ordinate axis, each segment on the abscissa axis is reduced by a times. The graph will “shrink” to the y-axis on both sides.

y= | f(x)|

The parts of the graph located under the x-axis are mirrored. The entire graph will be located in the upper half-plane.

Solution schemes.

1)y = sin x + 2.

We build a graph y \u003d sin x. We raise each point of the graph up by 2 units (zeros too).

2)y \u003d cos x - 3.

We build a graph y \u003d cos x. We lower each point of the graph down by 3 units.

3)y = cos (x - /2)

We build a graph y \u003d cos x. We shift all points n/2 to the right.

4) y = 2 sin x .

We build a graph y \u003d sin x. We leave the zeros in place, raise the upper points 2 times, lower the lower ones by the same amount.

PRACTICAL WORK Plotting trigonometric functions using the Advanced Grapher program.

Let's plot the function y = -cos 3x + 2.

1. Let's plot the function y \u003d cos x.
2. Reflect it about the x-axis.
3. This graph must be compressed three times along the x-axis.
4. Finally, such a graph must be lifted up by three units along the y-axis.

y = 0.5 sin x.

y=0.2 cos x-2

y = 5 cos 0 .5 x

y=-3sin(x+π).

2) Find the mistake and fix it.

V. Historical material. Euler's message.

Leonhard Euler is the greatest mathematician of the 18th century. Born in Switzerland. For many years he lived and worked in Russia, a member of the St. Petersburg Academy.

Why should we know and remember the name of this scientist?

By the beginning of the 18th century, trigonometry was still insufficiently developed: there were no symbols, formulas were written in words, it was difficult to assimilate them, the question of the signs of trigonometric functions in different quarters of the circle was also unclear, only angles or arcs were understood as an argument of a trigonometric function. Only in the works of Euler trigonometry received a modern look. It was he who began to consider the trigonometric function of a number, i.e. the argument came to be understood not only as arcs or degrees, but also as numbers. Euler deduced all trigonometric formulas from several basic ones, streamlined the question of the signs of the trigonometric function in different quarters of the circle. To designate trigonometric functions, he introduced symbols: sin x, cos x, tg x, ctg x.

On the threshold of the 18th century, a new direction appeared in the development of trigonometry - analytical. If before that the main goal of trigonometry was considered to be the solution of triangles, then Euler considered trigonometry as the science of trigonometric functions. The first part: the doctrine of function is part of the general doctrine of functions, which is studied in mathematical analysis. The second part: the solution of triangles - the chapter of geometry. Such innovations were made by Euler.

VI. Repetition

Independent work "Add the formula."

VII. Lesson summary:

1) What new did you learn at the lesson today?

2) What else do you want to know?

3) Grading.

Lesson 24

Target: consider the most common transformations of graphs of trigonometric functions.

I. Communication of the topic and purpose of the lesson

II. Repetition and consolidation of the material covered

1. Answers to questions on homework (analysis of unsolved problems).

2. Monitoring the assimilation of the material (written survey).

Option 1

sin x.

2. Find the main period of the function:

3. Plot the function

Option 2

1. Basic properties and graph of the function y \u003d cos x.

2. Find the main period of the function:

3. Plot the function

III. Learning new material

All transformations of function graphs, detailed in Chapter 1, are universal - they are suitable for all functions, including trigonometric ones. Therefore, we recommend repeating this topic. Here we confine ourselves to a brief reminder of the main transformations of graphs.

1. To plot the function y = f(x) + b it is necessary to move the graph of the function to | b | units along the y-axis - up at b > 0 and down at b< 0.

2. To plot a function graph y = mf(x) (where m > 0) it is necessary to stretch the graph of the function y = f(x) to m times along the y-axis. And for m > 1 there is really stretching in m times, for 0< m < 1 - сжатие в 1/ m раз.

3. To plot the function y = f (x + a ) it is necessary to transfer the graph of the function to | a | units along the x-axis - to the right at a< 0 и влево при а > 0.

4. To plot the function y = f(kx ) (where k > 0) it is necessary to compress the graph of the function y = f(x) to k times along the x-axis. And for k > 1 there is really compression in k times, for 0< k < 1 – растяжение в 1/ k times.

5. To plot the function y = - f(x ) you need a graph of the function y=f(x ) reflect about the x-axis (this transformation is a special case of transformation 2 for m = -1).

6. To plot the function y = f (-x) you need a graph of the function y=f(x ) to reflect about the y-axis (this transformation is a special case of transformation 4 for k = -1).

Example 1

Let's build a graph of the function y \u003d - cos 3 x + 2.

In accordance with rule 5, we need the graph of the function y \u003d cos x reflect about the x-axis. According to rule 3, this graph must be compressed three times along the x-axis. Finally, according to rule 1, such a graph must be raised up by three units along the y-axis.

It is also useful to recall the rules for converting graphs with modules.

1. To plot a function graph y=| f (x)| it is necessary to save a part of the graph of the function y \u003d f(x ), for which y ≥ 0. That part of the graph y = f(x ), for which< 0, надо симметрично отразить вверх относительно оси абсцисс.

2. To plot the function y = f (|x|) it is necessary to save a part of the graph of the function y \u003d f(x ), for which x ≥ 0. In addition, this part must be reflected symmetrically to the left with respect to the y-axis.

3. To plot the equation |y| = f (x) it is necessary to save a part of the graph of the function y \u003d f(x ), for which y ≥ 0. In addition, this part must be reflected symmetrically down relative to the x-axis.

Example 2

Let's plot the equation |y| = sin | x |.

Let's build a graph of the function y \u003d sin x for x ≥ 0. According to rule 2, this graph will be reflected to the left relative to the y-axis. Let us keep the parts of such a graph for which y ≥ 0. According to Rule 3, these parts will be reflected symmetrically down relative to the abscissa axis.

In more complex cases, the signs of the module must be disclosed.

Example 3

Let's build a graph of a complex function y \u003d cos(2x + |x|).

Recall that the argument of the cosine function is a function of the variable x, and therefore this function is complex. Let's expand the sign of the modulus and get:For two such intervals, we construct a graph of the function y(x ). We take into account that for x ≥ 0, the graph of the function y \u003d cos 3 x obtained from the graph of the function y = cos x by a factor of 3 along the x-axis.

Example 4

Let's plot the function

Using the formula of the square of the difference, we write the function in the formThe graph of the function consists of two parts. For x > 0, it is necessary to plot the function y \u003d 1 - cos X. It is obtained from the graph of the function y = cos x reflection about the abscissa axis and a shift of 1 unit up along the ordinate axis.

For x ≥ 0 we plot the function y = ( x -1)2 - 1. It is obtained from the graph of the function y \u003d x2 shifted 1 unit to the right along the x-axis and 1 unit up along the y-axis.

IV. Control questions (frontal survey)

1. Rules for transforming graphs of functions.

2. Transformation of graphs with modules.

V. Task in the lesson

§ 13, no. 2 (a, b); 3; 5; 7 (c, d); 8 (a, b); 9(a); 10 (b); 11 (a, b); 13 (c, d); 14; 17 (a, b); 19(b); 20 (a, c).

VI. Homework

§ 13, no. 2 (c, d); 4; 6; 7 (a, b); 8 (c, d); 9 (b); 10(a); 11 (c, d); 13 (a, b); 15; 17 (c, d); 19(a); 20 (b, d).

VII. Creative task

Plot the function graph, equations, inequalities:

VIII. Summing up the lesson